Isotropic LOD (LOD)

Percentage Accurate: 68.5% → 72.0%

Time: 15.9s

Alternatives: 15

Speedup: 0.5×

Specification

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(1 \leq d \land d \leq 4096\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.w\right| \land \left|dX.w\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.w\right| \land \left|dY.w\right| \leq 10^{+20}\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(fmax(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Initial Program: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(fmax(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Alternative 1: 72.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\_m\\ t_3 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_6 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_7 := t\_0 \cdot dX.u\\ t_8 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_9 := \left(t\_5 \cdot dX.w\right) \cdot dX.w\\ t_10 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_11 := t\_5 \cdot dY.w\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_1 \cdot t\_1 + t\_4 \cdot t\_4\right) + t\_10 \cdot t\_10, \left(t\_2 \cdot t\_2 + t\_8 \cdot t\_8\right) + t\_6 \cdot t\_6\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_7, dX.u, \mathsf{fma}\left(t\_3 \cdot dX.v, dX.v, t\_9\right)\right), \mathsf{fma}\left(t\_0 \cdot dY.u\_m, dY.u\_m, \mathsf{fma}\left(t\_11, dY.w, \left(t\_3 \cdot dY.v\right) \cdot dY.v\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_7, dX.u, t\_9\right), \mathsf{fma}\left(t\_11, dY.w, e^{\log \left(\left(-dY.u\_m\right) \cdot \left\lfloor w\right\rfloor \right) \cdot 2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) dY.u_m))
        (t_3 (pow (floor h) 2.0))
        (t_4 (* (floor h) dX.v))
        (t_5 (pow (floor d) 2.0))
        (t_6 (* (floor d) dY.w))
        (t_7 (* t_0 dX.u))
        (t_8 (* (floor h) dY.v))
        (t_9 (* (* t_5 dX.w) dX.w))
        (t_10 (* (floor d) dX.w))
        (t_11 (* t_5 dY.w)))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_1 t_1) (* t_4 t_4)) (* t_10 t_10))
           (+ (+ (* t_2 t_2) (* t_8 t_8)) (* t_6 t_6)))))
        100.0)
     (log2
      (sqrt
       (fmax
        (fma t_7 dX.u (fma (* t_3 dX.v) dX.v t_9))
        (fma (* t_0 dY.u_m) dY.u_m (fma t_11 dY.w (* (* t_3 dY.v) dY.v))))))
     (log2
      (sqrt
       (fmax
        (fma t_7 dX.u t_9)
        (fma t_11 dY.w (exp (* (log (* (- dY.u_m) (floor w))) 2.0)))))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u_m;
	float t_3 = powf(floorf(h), 2.0f);
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = powf(floorf(d), 2.0f);
	float t_6 = floorf(d) * dY_46_w;
	float t_7 = t_0 * dX_46_u;
	float t_8 = floorf(h) * dY_46_v;
	float t_9 = (t_5 * dX_46_w) * dX_46_w;
	float t_10 = floorf(d) * dX_46_w;
	float t_11 = t_5 * dY_46_w;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_1 * t_1) + (t_4 * t_4)) + (t_10 * t_10)), (((t_2 * t_2) + (t_8 * t_8)) + (t_6 * t_6))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_7, dX_46_u, fmaf((t_3 * dX_46_v), dX_46_v, t_9)), fmaf((t_0 * dY_46_u_m), dY_46_u_m, fmaf(t_11, dY_46_w, ((t_3 * dY_46_v) * dY_46_v))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_7, dX_46_u, t_9), fmaf(t_11, dY_46_w, expf((logf((-dY_46_u_m * floorf(w))) * 2.0f))))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u_m)
	t_3 = floor(h) ^ Float32(2.0)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = floor(d) ^ Float32(2.0)
	t_6 = Float32(floor(d) * dY_46_w)
	t_7 = Float32(t_0 * dX_46_u)
	t_8 = Float32(floor(h) * dY_46_v)
	t_9 = Float32(Float32(t_5 * dX_46_w) * dX_46_w)
	t_10 = Float32(floor(d) * dX_46_w)
	t_11 = Float32(t_5 * dY_46_w)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4)) + Float32(t_10 * t_10)), Float32(Float32(Float32(t_2 * t_2) + Float32(t_8 * t_8)) + Float32(t_6 * t_6))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(t_7, dX_46_u, fma(Float32(t_3 * dX_46_v), dX_46_v, t_9)), fma(Float32(t_0 * dY_46_u_m), dY_46_u_m, fma(t_11, dY_46_w, Float32(Float32(t_3 * dY_46_v) * dY_46_v))))));
	else
		tmp = log2(sqrt(fmax(fma(t_7, dX_46_u, t_9), fma(t_11, dY_46_w, exp(Float32(log(Float32(Float32(-dY_46_u_m) * floor(w))) * Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 100

   1. Initial program 99.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left({dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)\right)}\right)} \]

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 6.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 10.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor }\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)}\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)\right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      10. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      11. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u}, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      16. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 13.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 16.2%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, e^{\log \left(\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot 2}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_2 := \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\_m\right) \cdot dY.u\_m\\ t_3 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_4 := t\_3 \cdot t\_3\\ \mathbf{if}\;dY.w \leq 1000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + t\_4, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, t\_2\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_4, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, t\_2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor h) dX.v))
        (t_2 (* (* (pow (floor w) 2.0) dY.u_m) dY.u_m))
        (t_3 (* (floor d) dX.w))
        (t_4 (* t_3 t_3)))
   (if (<= dY.w 1000000.0)
     (log2
      (sqrt
       (fmax
        (+ (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0)) t_4)
        (fma (* (pow (floor h) 2.0) dY.v) dY.v t_2))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_0 t_0) (* t_1 t_1)) t_4)
        (fma (* (pow (floor d) 2.0) dY.w) dY.w t_2)))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = (powf(floorf(w), 2.0f) * dY_46_u_m) * dY_46_u_m;
	float t_3 = floorf(d) * dX_46_w;
	float t_4 = t_3 * t_3;
	float tmp;
	if (dY_46_w <= 1000000.0f) {
		tmp = log2f(sqrtf(fmaxf(((powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f)) + t_4), fmaf((powf(floorf(h), 2.0f) * dY_46_v), dY_46_v, t_2))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_1 * t_1)) + t_4), fmaf((powf(floorf(d), 2.0f) * dY_46_w), dY_46_w, t_2))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u_m) * dY_46_u_m)
	t_3 = Float32(floor(d) * dX_46_w)
	t_4 = Float32(t_3 * t_3)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(1000000.0))
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) + t_4), fma(Float32((floor(h) ^ Float32(2.0)) * dY_46_v), dY_46_v, t_2))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + t_4), fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, t_2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 1e6

   1. Initial program 69.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} \cdot dY.v\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-floor.f32 63.3

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   5. Applied rewrites 63.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      3. lower-+.f32 63.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      6. lift-pow.f32 63.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      9. lower-*.f32 63.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      12. lower-pow.f32 63.3

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      13. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-*.f32 63.3

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   7. Applied rewrites 63.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   8. Taylor expanded in dY.w around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   10. Applied rewrites 64.3%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   if 1e6 < dY.w

   1. Initial program 54.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 56.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.w \leq 1000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + t\_0 \cdot t\_0, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\_m\right) \cdot dY.u\_m\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {t\_0}^{2}\right) + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)))
   (if (<= dY.w 1000000.0)
     (log2
      (sqrt
       (fmax
        (+
         (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0))
         (* t_0 t_0))
        (fma
         (* (pow (floor h) 2.0) dY.v)
         dY.v
         (* (* (pow (floor w) 2.0) dY.u_m) dY.u_m)))))
     (log2
      (sqrt
       (fmax
        (+
         (+ (pow (* (floor w) dX.u) 2.0) (pow t_0 2.0))
         (pow (* (floor h) dX.v) 2.0))
        (+ (pow (* (floor d) dY.w) 2.0) (pow (* (floor w) dY.u_m) 2.0))))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_w <= 1000000.0f) {
		tmp = log2f(sqrtf(fmaxf(((powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f)) + (t_0 * t_0)), fmaf((powf(floorf(h), 2.0f) * dY_46_v), dY_46_v, ((powf(floorf(w), 2.0f) * dY_46_u_m) * dY_46_u_m)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf((floorf(w) * dX_46_u), 2.0f) + powf(t_0, 2.0f)) + powf((floorf(h) * dX_46_v), 2.0f)), (powf((floorf(d) * dY_46_w), 2.0f) + powf((floorf(w) * dY_46_u_m), 2.0f)))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(1000000.0))
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) + Float32(t_0 * t_0)), fma(Float32((floor(h) ^ Float32(2.0)) * dY_46_v), dY_46_v, Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u_m) * dY_46_u_m)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u_m) ^ Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 1e6

   1. Initial program 69.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} \cdot dY.v\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-floor.f32 63.3

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   5. Applied rewrites 63.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      3. lower-+.f32 63.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      6. lift-pow.f32 63.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      9. lower-*.f32 63.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      12. lower-pow.f32 63.3

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      13. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-*.f32 63.3

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   7. Applied rewrites 63.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   8. Taylor expanded in dY.w around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   10. Applied rewrites 64.3%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   if 1e6 < dY.w

   1. Initial program 54.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 56.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   7. Applied rewrites 56.2%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\right) + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 64.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.v \leq 210000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {t\_0}^{2}\right) + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + t\_0 \cdot t\_0, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)))
   (if (<= dY.v 210000.0)
     (log2
      (sqrt
       (fmax
        (+
         (+ (pow (* (floor w) dX.u) 2.0) (pow t_0 2.0))
         (pow (* (floor h) dX.v) 2.0))
        (+ (pow (* (floor d) dY.w) 2.0) (pow (* (floor w) dY.u_m) 2.0)))))
     (log2
      (sqrt
       (fmax
        (+
         (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0))
         (* t_0 t_0))
        (fma
         (* (pow (floor d) 2.0) dY.w)
         dY.w
         (* (* (pow (floor h) 2.0) dY.v) dY.v))))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_v <= 210000.0f) {
		tmp = log2f(sqrtf(fmaxf(((powf((floorf(w) * dX_46_u), 2.0f) + powf(t_0, 2.0f)) + powf((floorf(h) * dX_46_v), 2.0f)), (powf((floorf(d) * dY_46_w), 2.0f) + powf((floorf(w) * dY_46_u_m), 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f)) + (t_0 * t_0)), fmaf((powf(floorf(d), 2.0f) * dY_46_w), dY_46_w, ((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v)))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(210000.0))
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u_m) ^ Float32(2.0))))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) + Float32(t_0 * t_0)), fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 2.1e5

   1. Initial program 69.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 65.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   7. Applied rewrites 65.4%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\right) + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)} \]

   if 2.1e5 < dY.v

   1. Initial program 57.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} \cdot dY.v\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-floor.f32 55.4

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   5. Applied rewrites 55.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      3. lower-+.f32 55.4

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      6. lift-pow.f32 55.4

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      9. lower-*.f32 55.4

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      12. lower-pow.f32 55.4

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      13. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-*.f32 55.4

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   7. Applied rewrites 55.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\\ t_1 := {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\\ t_2 := {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\\ t_3 := {\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2}\\ \mathbf{if}\;dX.u \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_2 + t\_0, t\_1 + \left(t\_3 + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + t\_0\right) + t\_2, t\_1 + t\_3\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dX.w) 2.0))
        (t_1 (pow (* (floor d) dY.w) 2.0))
        (t_2 (pow (* (floor h) dX.v) 2.0))
        (t_3 (pow (* (floor w) dY.u_m) 2.0)))
   (if (<= dX.u 3.999999975690116e-8)
     (log2
      (sqrt (fmax (+ t_2 t_0) (+ t_1 (+ t_3 (pow (* (floor h) dY.v) 2.0))))))
     (log2
      (sqrt
       (fmax (+ (+ (pow (* (floor w) dX.u) 2.0) t_0) t_2) (+ t_1 t_3)))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf((floorf(d) * dX_46_w), 2.0f);
	float t_1 = powf((floorf(d) * dY_46_w), 2.0f);
	float t_2 = powf((floorf(h) * dX_46_v), 2.0f);
	float t_3 = powf((floorf(w) * dY_46_u_m), 2.0f);
	float tmp;
	if (dX_46_u <= 3.999999975690116e-8f) {
		tmp = log2f(sqrtf(fmaxf((t_2 + t_0), (t_1 + (t_3 + powf((floorf(h) * dY_46_v), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf((floorf(w) * dX_46_u), 2.0f) + t_0) + t_2), (t_1 + t_3))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dX_46_w) ^ Float32(2.0)
	t_1 = Float32(floor(d) * dY_46_w) ^ Float32(2.0)
	t_2 = Float32(floor(h) * dX_46_v) ^ Float32(2.0)
	t_3 = Float32(floor(w) * dY_46_u_m) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(3.999999975690116e-8))
		tmp = log2(sqrt(fmax(Float32(t_2 + t_0), Float32(t_1 + Float32(t_3 + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + t_0) + t_2), Float32(t_1 + t_3))));
	end
	return tmp
end

MATLAB

dY.u_m = abs(dY_46_u);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = (floor(d) * dX_46_w) ^ single(2.0);
	t_1 = (floor(d) * dY_46_w) ^ single(2.0);
	t_2 = (floor(h) * dX_46_v) ^ single(2.0);
	t_3 = (floor(w) * dY_46_u_m) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(3.999999975690116e-8))
		tmp = log2(sqrt(max((t_2 + t_0), (t_1 + (t_3 + ((floor(h) * dY_46_v) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max(((((floor(w) * dX_46_u) ^ single(2.0)) + t_0) + t_2), (t_1 + t_3))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 3.99999998e-8

   1. Initial program 69.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v}, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.w \cdot dX.w\right)}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)} \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dX.w\right) \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      14. lower-floor.f32 63.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dX.w\right) \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 63.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. Applied rewrites 63.8%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)} \]

   if 3.99999998e-8 < dX.u

   1. Initial program 63.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 60.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   7. Applied rewrites 60.0%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\right) + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.u \leq 0.006000000052154064:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1 \cdot dX.u, dX.u, \left(t\_0 \cdot dX.w\right) \cdot dX.w\right), \mathsf{fma}\left(t\_0 \cdot dY.w, dY.w, \left(t\_1 \cdot dY.u\_m\right) \cdot dY.u\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)) (t_1 (pow (floor w) 2.0)))
   (if (<= dX.u 0.006000000052154064)
     (log2
      (sqrt
       (fmax
        (+ (pow (* (floor h) dX.v) 2.0) (pow (* (floor d) dX.w) 2.0))
        (+
         (pow (* (floor d) dY.w) 2.0)
         (+ (pow (* (floor w) dY.u_m) 2.0) (pow (* (floor h) dY.v) 2.0))))))
     (log2
      (sqrt
       (fmax
        (fma (* t_1 dX.u) dX.u (* (* t_0 dX.w) dX.w))
        (fma (* t_0 dY.w) dY.w (* (* t_1 dY.u_m) dY.u_m))))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float t_1 = powf(floorf(w), 2.0f);
	float tmp;
	if (dX_46_u <= 0.006000000052154064f) {
		tmp = log2f(sqrtf(fmaxf((powf((floorf(h) * dX_46_v), 2.0f) + powf((floorf(d) * dX_46_w), 2.0f)), (powf((floorf(d) * dY_46_w), 2.0f) + (powf((floorf(w) * dY_46_u_m), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_1 * dX_46_u), dX_46_u, ((t_0 * dX_46_w) * dX_46_w)), fmaf((t_0 * dY_46_w), dY_46_w, ((t_1 * dY_46_u_m) * dY_46_u_m)))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	t_1 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(0.006000000052154064))
		tmp = log2(sqrt(fmax(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w) ^ Float32(2.0))), Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u_m) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_1 * dX_46_u), dX_46_u, Float32(Float32(t_0 * dX_46_w) * dX_46_w)), fma(Float32(t_0 * dY_46_w), dY_46_w, Float32(Float32(t_1 * dY_46_u_m) * dY_46_u_m)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.00600000005

   1. Initial program 69.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v}, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.w \cdot dX.w\right)}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)} \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dX.w\right) \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      14. lower-floor.f32 64.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dX.w\right) \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 64.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. Applied rewrites 64.6%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)} \]

   if 0.00600000005 < dX.u

   1. Initial program 61.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 57.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor }\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)}\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)\right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      10. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      11. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u}, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      16. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 54.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.w \leq 0.029999999329447746:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + t\_0 \cdot t\_0, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left({\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)\right)}^{0.25}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)))
   (if (<= dY.w 0.029999999329447746)
     (log2
      (sqrt
       (fmax
        (+
         (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0))
         (* t_0 t_0))
        (* (* (pow (floor h) 2.0) dY.v) dY.v))))
     (log2
      (pow
       (pow
        (fmax
         (pow (* (floor w) dX.u) 2.0)
         (+
          (pow (* (floor d) dY.w) 2.0)
          (+ (pow (* (floor w) dY.u_m) 2.0) (pow (* (floor h) dY.v) 2.0))))
        0.25)
       2.0)))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_w <= 0.029999999329447746f) {
		tmp = log2f(sqrtf(fmaxf(((powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f)) + (t_0 * t_0)), ((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v))));
	} else {
		tmp = log2f(powf(powf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (powf((floorf(d) * dY_46_w), 2.0f) + (powf((floorf(w) * dY_46_u_m), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f)))), 0.25f), 2.0f));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.029999999329447746))
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) + Float32(t_0 * t_0)), Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v))));
	else
		tmp = log2(((fmax((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u_m) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))))) ^ Float32(0.25)) ^ Float32(2.0)));
	end
	return tmp
end

MATLAB

dY.u_m = abs(dY_46_u);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = floor(d) * dX_46_w;
	tmp = single(0.0);
	if (dY_46_w <= single(0.029999999329447746))
		tmp = log2(sqrt(max(((((dX_46_v * floor(h)) ^ single(2.0)) + ((dX_46_u * floor(w)) ^ single(2.0))) + (t_0 * t_0)), (((floor(h) ^ single(2.0)) * dY_46_v) * dY_46_v))));
	else
		tmp = log2(((max(((floor(w) * dX_46_u) ^ single(2.0)), (((floor(d) * dY_46_w) ^ single(2.0)) + (((floor(w) * dY_46_u_m) ^ single(2.0)) + ((floor(h) * dY_46_v) ^ single(2.0))))) ^ single(0.25)) ^ single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 0.0299999993

   1. Initial program 69.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} \cdot dY.v\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-floor.f32 64.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   5. Applied rewrites 64.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      3. lower-+.f32 64.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      6. lift-pow.f32 64.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      9. lower-*.f32 64.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      12. lower-pow.f32 64.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      13. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-*.f32 64.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   7. Applied rewrites 64.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   8. Taylor expanded in dY.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.v}^{2} \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 57.6%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot \color{blue}{dY.v}\right)}\right) \]

   if 0.0299999993 < dY.w

   1. Initial program 60.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 57.6

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 57.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. Applied rewrites 57.6%

      \[\leadsto \log_{2} \color{blue}{\left({\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)\right)}^{0.25}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.w \leq 0.029999999329447746:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + t\_0 \cdot t\_0, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)))
   (if (<= dY.w 0.029999999329447746)
     (log2
      (sqrt
       (fmax
        (+
         (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0))
         (* t_0 t_0))
        (* (* (pow (floor h) 2.0) dY.v) dY.v))))
     (log2
      (sqrt
       (fmax
        (* (* (pow (floor w) 2.0) dX.u) dX.u)
        (fma
         (* (pow (floor d) 2.0) dY.w)
         dY.w
         (+ (pow (* (floor w) dY.u_m) 2.0) (pow (* (floor h) dY.v) 2.0)))))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_w <= 0.029999999329447746f) {
		tmp = log2f(sqrtf(fmaxf(((powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f)) + (t_0 * t_0)), ((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(w), 2.0f) * dX_46_u) * dX_46_u), fmaf((powf(floorf(d), 2.0f) * dY_46_w), dY_46_w, (powf((floorf(w) * dY_46_u_m), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.029999999329447746))
		tmp = log2(sqrt(fmax(Float32(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) + Float32(t_0 * t_0)), Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u) * dX_46_u), fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32((Float32(floor(w) * dY_46_u_m) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 0.0299999993

   1. Initial program 69.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} \cdot dY.v\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-floor.f32 64.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   5. Applied rewrites 64.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      3. lower-+.f32 64.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      6. lift-pow.f32 64.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      9. lower-*.f32 64.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} + \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      12. lower-pow.f32 64.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      13. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

      15. lower-*.f32 64.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   7. Applied rewrites 64.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right) \]

   8. Taylor expanded in dY.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.v}^{2} \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 57.6%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot \color{blue}{dY.v}\right)}\right) \]

   if 0.0299999993 < dY.w

   1. Initial program 60.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 57.6

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 57.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right)} \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      4. swap-sqr N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot dY.w\right)}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot \left(dY.w \cdot dY.w\right)\right)}\right) \]

      6. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot \left(dY.w \cdot dY.w\right)\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)} \cdot dY.w\right)}\right) \]

      9. lower-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}\right)}\right) \]

      11. lower-fma.f32 57.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}\right)}\right) \]

   7. Applied rewrites 57.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.w \leq 10000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(t\_0 \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(t\_0 \cdot dX.w\right) \cdot dX.w\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)))
   (if (<= dX.w 10000.0)
     (log2
      (sqrt
       (fmax
        (* (* (pow (floor w) 2.0) dX.u) dX.u)
        (fma
         (* t_0 dY.w)
         dY.w
         (+ (pow (* (floor w) dY.u_m) 2.0) (pow (* (floor h) dY.v) 2.0))))))
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor h) 2.0) dX.v) dX.v (* (* t_0 dX.w) dX.w))
        (+ (pow (* dY.w (floor d)) 2.0) (pow (* dY.u_m (floor w)) 2.0))))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float tmp;
	if (dX_46_w <= 10000.0f) {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(w), 2.0f) * dX_46_u) * dX_46_u), fmaf((t_0 * dY_46_w), dY_46_w, (powf((floorf(w) * dY_46_u_m), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(h), 2.0f) * dX_46_v), dX_46_v, ((t_0 * dX_46_w) * dX_46_w)), (powf((dY_46_w * floorf(d)), 2.0f) + powf((dY_46_u_m * floorf(w)), 2.0f)))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(10000.0))
		tmp = log2(sqrt(fmax(Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u) * dX_46_u), fma(Float32(t_0 * dY_46_w), dY_46_w, Float32((Float32(floor(w) * dY_46_u_m) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32((floor(h) ^ Float32(2.0)) * dX_46_v), dX_46_v, Float32(Float32(t_0 * dX_46_w) * dX_46_w)), Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_u_m * floor(w)) ^ Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 1e4

   1. Initial program 68.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 57.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 57.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right)} \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      4. swap-sqr N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot dY.w\right)}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot \left(dY.w \cdot dY.w\right)\right)}\right) \]

      6. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot \left(dY.w \cdot dY.w\right)\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)} \cdot dY.w\right)}\right) \]

      9. lower-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}\right)}\right) \]

      11. lower-fma.f32 57.7

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}\right)}\right) \]

   7. Applied rewrites 57.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)}\right) \]

   if 1e4 < dX.w

   1. Initial program 62.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 38.9

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 38.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} \cdot dY.u\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-floor.f32 35.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 35.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   10. Applied rewrites 35.5%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.u around 0

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v}, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       6. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       7. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dX.v, dX.v, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.w \cdot dX.w\right)}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       10. associate-*r* N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       11. lower-*.f32 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       12. lower-*.f32 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)} \cdot dX.w\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       13. lower-pow.f32 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dX.w\right) \cdot dX.w\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

       14. lower-floor.f32 57.1

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dX.w\right) \cdot dX.w\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

   13. Applied rewrites 57.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\\ t_1 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_2 := t\_1 \cdot dY.w\\ \mathbf{if}\;dX.w \leq 15000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 \cdot dX.u, \mathsf{fma}\left(t\_2, dY.w, {\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, dX.u, \left(t\_1 \cdot dX.w\right) \cdot dX.w\right), t\_2 \cdot dY.w\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (pow (floor w) 2.0) dX.u))
        (t_1 (pow (floor d) 2.0))
        (t_2 (* t_1 dY.w)))
   (if (<= dX.w 15000.0)
     (log2
      (sqrt
       (fmax
        (* t_0 dX.u)
        (fma
         t_2
         dY.w
         (+ (pow (* (floor w) dY.u_m) 2.0) (pow (* (floor h) dY.v) 2.0))))))
     (log2 (sqrt (fmax (fma t_0 dX.u (* (* t_1 dX.w) dX.w)) (* t_2 dY.w)))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f) * dX_46_u;
	float t_1 = powf(floorf(d), 2.0f);
	float t_2 = t_1 * dY_46_w;
	float tmp;
	if (dX_46_w <= 15000.0f) {
		tmp = log2f(sqrtf(fmaxf((t_0 * dX_46_u), fmaf(t_2, dY_46_w, (powf((floorf(w) * dY_46_u_m), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_0, dX_46_u, ((t_1 * dX_46_w) * dX_46_w)), (t_2 * dY_46_w))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = Float32((floor(w) ^ Float32(2.0)) * dX_46_u)
	t_1 = floor(d) ^ Float32(2.0)
	t_2 = Float32(t_1 * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(15000.0))
		tmp = log2(sqrt(fmax(Float32(t_0 * dX_46_u), fma(t_2, dY_46_w, Float32((Float32(floor(w) * dY_46_u_m) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(fmax(fma(t_0, dX_46_u, Float32(Float32(t_1 * dX_46_w) * dX_46_w)), Float32(t_2 * dY_46_w))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 15000

   1. Initial program 68.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 57.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 57.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right)} \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      4. swap-sqr N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot dY.w\right)}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot \left(dY.w \cdot dY.w\right)\right)}\right) \]

      6. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot \left(dY.w \cdot dY.w\right)\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)} \cdot dY.w\right)}\right) \]

      9. lower-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}\right)}\right) \]

      11. lower-fma.f32 57.7

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}\right)}\right) \]

   7. Applied rewrites 57.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)}\right) \]

   if 15000 < dX.w

   1. Initial program 62.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 61.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor }\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)}\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)\right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      10. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      11. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u}, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      16. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 54.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   9. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), {dY.w}^{2} \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 50.7%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot \color{blue}{dY.w}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.w \leq 15000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\_m\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left(t\_0 \cdot dX.w\right) \cdot dX.w\right), \left(t\_0 \cdot dY.w\right) \cdot dY.w\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)))
   (if (<= dX.w 15000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor w) dX.u) 2.0)
        (+
         (pow (* (floor d) dY.w) 2.0)
         (+ (pow (* (floor w) dY.u_m) 2.0) (pow (* (floor h) dY.v) 2.0))))))
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor w) 2.0) dX.u) dX.u (* (* t_0 dX.w) dX.w))
        (* (* t_0 dY.w) dY.w)))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float tmp;
	if (dX_46_w <= 15000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (powf((floorf(d) * dY_46_w), 2.0f) + (powf((floorf(w) * dY_46_u_m), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(w), 2.0f) * dX_46_u), dX_46_u, ((t_0 * dX_46_w) * dX_46_w)), ((t_0 * dY_46_w) * dY_46_w))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(15000.0))
		tmp = log2(sqrt(fmax((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u_m) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32((floor(w) ^ Float32(2.0)) * dX_46_u), dX_46_u, Float32(Float32(t_0 * dX_46_w) * dX_46_w)), Float32(Float32(t_0 * dY_46_w) * dY_46_w))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 15000

   1. Initial program 68.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 57.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 57.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 57.7%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)} \]

   if 15000 < dX.w

   1. Initial program 62.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 61.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor }\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)}\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)\right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      10. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      11. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u}, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      16. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 54.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   9. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), {dY.w}^{2} \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 50.7%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot \color{blue}{dY.w}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.u\_m \leq 30000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left(t\_0 \cdot dX.w\right) \cdot dX.w\right), \left(t\_0 \cdot dY.w\right) \cdot dY.w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.25}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)))
   (if (<= dY.u_m 30000.0)
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor w) 2.0) dX.u) dX.u (* (* t_0 dX.w) dX.w))
        (* (* t_0 dY.w) dY.w))))
     (log2
      (pow
       (pow
        (fmax
         (pow (* dX.u (floor w)) 2.0)
         (+ (pow (* dY.w (floor d)) 2.0) (pow (* dY.u_m (floor w)) 2.0)))
        0.25)
       2.0)))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float tmp;
	if (dY_46_u_m <= 30000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(w), 2.0f) * dX_46_u), dX_46_u, ((t_0 * dX_46_w) * dX_46_w)), ((t_0 * dY_46_w) * dY_46_w))));
	} else {
		tmp = log2f(powf(powf(fmaxf(powf((dX_46_u * floorf(w)), 2.0f), (powf((dY_46_w * floorf(d)), 2.0f) + powf((dY_46_u_m * floorf(w)), 2.0f))), 0.25f), 2.0f));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u_m <= Float32(30000.0))
		tmp = log2(sqrt(fmax(fma(Float32((floor(w) ^ Float32(2.0)) * dX_46_u), dX_46_u, Float32(Float32(t_0 * dX_46_w) * dX_46_w)), Float32(Float32(t_0 * dY_46_w) * dY_46_w))));
	else
		tmp = log2(((fmax((Float32(dX_46_u * floor(w)) ^ Float32(2.0)), Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_u_m * floor(w)) ^ Float32(2.0)))) ^ Float32(0.25)) ^ Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 3e4

   1. Initial program 68.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 61.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor }\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)}\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)\right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      10. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      11. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u}, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      16. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 53.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   9. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), {dY.w}^{2} \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot \color{blue}{dY.w}\right)}\right) \]

   if 3e4 < dY.u

   1. Initial program 59.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 55.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 55.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} \cdot dY.u\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-floor.f32 53.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 53.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   10. Applied rewrites 53.6%

       \[\leadsto \log_{2} \color{blue}{\left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.25}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := t\_0 \cdot dX.u\\ t_2 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.u\_m \leq 30000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1, dX.u, \left(t\_2 \cdot dX.w\right) \cdot dX.w\right), \left(t\_2 \cdot dY.w\right) \cdot dY.w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_1 \cdot dX.u, \mathsf{fma}\left(t\_0, dY.u\_m \cdot dY.u\_m, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)) (t_1 (* t_0 dX.u)) (t_2 (pow (floor d) 2.0)))
   (if (<= dY.u_m 30000.0)
     (log2
      (sqrt (fmax (fma t_1 dX.u (* (* t_2 dX.w) dX.w)) (* (* t_2 dY.w) dY.w))))
     (log2
      (sqrt
       (fmax
        (* t_1 dX.u)
        (fma t_0 (* dY.u_m dY.u_m) (pow (* dY.w (floor d)) 2.0))))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = t_0 * dX_46_u;
	float t_2 = powf(floorf(d), 2.0f);
	float tmp;
	if (dY_46_u_m <= 30000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_1, dX_46_u, ((t_2 * dX_46_w) * dX_46_w)), ((t_2 * dY_46_w) * dY_46_w))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_1 * dX_46_u), fmaf(t_0, (dY_46_u_m * dY_46_u_m), powf((dY_46_w * floorf(d)), 2.0f)))));
	}
	return tmp;
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(t_0 * dX_46_u)
	t_2 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u_m <= Float32(30000.0))
		tmp = log2(sqrt(fmax(fma(t_1, dX_46_u, Float32(Float32(t_2 * dX_46_w) * dX_46_w)), Float32(Float32(t_2 * dY_46_w) * dY_46_w))));
	else
		tmp = log2(sqrt(fmax(Float32(t_1 * dX_46_u), fma(t_0, Float32(dY_46_u_m * dY_46_u_m), (Float32(dY_46_w * floor(d)) ^ Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 3e4

   1. Initial program 68.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor }\right)\right)\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)}\right)\right) \cdot {dY.u}^{2}\right)}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor w\right\rfloor \right)\right)\right)} \cdot {dY.u}^{2}\right)}\right) \]

      11. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot {dY.u}^{2}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w + \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   5. Applied rewrites 61.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor }\right)\right)\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \left(\mathsf{neg}\left(\color{blue}{\left\lfloor d\right\rfloor \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)}\right)\right) \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right) \cdot \left(\mathsf{neg}\left(\left\lfloor d\right\rfloor \right)\right)\right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      10. sqr-neg-rev N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right)} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      11. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dX.w}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u + \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u}, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      16. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u, dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dX.w}^{2}}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 53.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right)}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   9. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), {dY.w}^{2} \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot \color{blue}{dY.w}\right)}\right) \]

   if 3e4 < dY.u

   1. Initial program 59.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 55.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 55.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} \cdot dY.u\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

      15. lower-floor.f32 53.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   8. Applied rewrites 53.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 53.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{dY.u \cdot dY.u}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\_m\right) \cdot dY.u\_m\right)\right)}\right) \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (pow (* dX.u (floor w)) 2.0)
    (fma
     (* (pow (floor d) 2.0) dY.w)
     dY.w
     (* (* (pow (floor w) 2.0) dY.u_m) dY.u_m))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(powf((dX_46_u * floorf(w)), 2.0f), fmaf((powf(floorf(d), 2.0f) * dY_46_w), dY_46_w, ((powf(floorf(w), 2.0f) * dY_46_u_m) * dY_46_u_m)))));
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	return log2(sqrt(fmax((Float32(dX_46_u * floor(w)) ^ Float32(2.0)), fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u_m) * dY_46_u_m)))))
end

Derivation

1. Initial program 67.1%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
2. Add Preprocessing

3. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. lower-floor.f32 53.9

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

5. Applied rewrites 53.9%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

6. Taylor expanded in dY.v around 0

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

   8. lower-floor.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)\right)}\right) \]

   10. unpow2 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}\right) \]

   11. associate-*r* N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} \cdot dY.u\right)\right)}\right) \]

   14. lower-pow.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   15. lower-floor.f32 45.6

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

8. Applied rewrites 45.6%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 45.6%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]
11. Add Preprocessing

Alternative 15: 46.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} dY.u_m = \left|dY.u\right| \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \end{array} \]

FPCore

dY.u_m = (fabs.f32 dY.u)
(FPCore (w h d dX.u dX.v dX.w dY.u_m dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (pow (* dX.u (floor w)) 2.0)
    (+ (pow (* dY.w (floor d)) 2.0) (pow (* dY.u_m (floor w)) 2.0))))))

C

dY.u_m = fabs(dY_46_u);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u_m, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(powf((dX_46_u * floorf(w)), 2.0f), (powf((dY_46_w * floorf(d)), 2.0f) + powf((dY_46_u_m * floorf(w)), 2.0f)))));
}

Julia

dY.u_m = abs(dY_46_u)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	return log2(sqrt(fmax((Float32(dX_46_u * floor(w)) ^ Float32(2.0)), Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_u_m * floor(w)) ^ Float32(2.0))))))
end

MATLAB

dY.u_m = abs(dY_46_u);
function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u_m, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((dX_46_u * floor(w)) ^ single(2.0)), (((dY_46_w * floor(d)) ^ single(2.0)) + ((dY_46_u_m * floor(w)) ^ single(2.0))))));
end

Derivation

1. Initial program 67.1%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
2. Add Preprocessing

3. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. lower-floor.f32 53.9

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

5. Applied rewrites 53.9%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

6. Taylor expanded in dY.v around 0

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w} + {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w}, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

   8. lower-floor.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w, dY.w, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)\right)}\right) \]

   10. unpow2 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}\right) \]

   11. associate-*r* N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)\right)}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} \cdot dY.u\right)\right)}\right) \]

   14. lower-pow.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

   15. lower-floor.f32 45.6

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right) \]

8. Applied rewrites 45.6%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)}\right) \]

10. Applied rewrites 45.6%

    \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024350 
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :name "Isotropic LOD (LOD)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1.0 d) (<= d 4096.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dX.w)) (<= (fabs dX.w) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (and (<= 1e-20 (fabs dY.w)) (<= (fabs dY.w) 1e+20)))
  (log2 (sqrt (fmax (+ (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (* (* (floor d) dX.w) (* (floor d) dX.w))) (+ (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v))) (* (* (floor d) dY.w) (* (floor d) dY.w)))))))