Isotropic LOD (LOD)

Percentage Accurate: 68.0% → 71.4%

Time: 12.2s

Alternatives: 14

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: 68.0% 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: 71.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, 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) * dX_46_u_m;
	float t_1 = floorf(w) * dY_46_u;
	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 = t_4 * t_4;
	float t_6 = t_2 * t_2;
	float t_7 = floorf(h) * dY_46_v;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + t_6) + t_5), (((t_1 * t_1) + (t_7 * t_7)) + (t_3 * t_3))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u_m * floorf(w)) * dX_46_u_m), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((expf((logf((-dX_46_u_m * floorf(w))) * 2.0f)) + t_6) + t_5), fmaf(powf(dY_46_u, 2.0f), powf(floorf(w), 2.0f), (powf(dY_46_w, 2.0f) * powf(floorf(d), 2.0f))))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u_m)
	t_1 = Float32(floor(w) * dY_46_u)
	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(t_4 * t_4)
	t_6 = Float32(t_2 * t_2)
	t_7 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + t_6) + t_5), Float32(Float32(Float32(t_1 * t_1) + Float32(t_7 * t_7)) + Float32(t_3 * t_3))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(exp(Float32(log(Float32(Float32(-dX_46_u_m) * floor(w))) * Float32(2.0))) + t_6) + t_5), fma((dY_46_u ^ Float32(2.0)), (floor(w) ^ Float32(2.0)), Float32((dY_46_w ^ Float32(2.0)) * (floor(d) ^ 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 68.0%

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

   3. Applied rewrites 68.0%

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

   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 68.0%

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

   2. Taylor expanded in dY.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.8

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

   4. Applied rewrites 60.8%

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

      1. lift-*.f32 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 36.6

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

   6. Applied rewrites 36.6%

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

Alternative 2: 69.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, 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) * dX_46_u_m;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = t_2 * t_2;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(d) * dY_46_w;
	float t_6 = t_5 * t_5;
	float t_7 = floorf(d) * dX_46_w;
	float t_8 = dY_46_u * floorf(w);
	float t_9 = (t_0 * t_0) + (t_4 * t_4);
	float tmp;
	if (log2f(sqrtf(fmaxf((t_9 + (t_7 * t_7)), (((t_1 * t_1) + t_3) + t_6)))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u_m * floorf(w)) * dX_46_u_m), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf((t_8 * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_9 + expf((logf((-dX_46_w * floorf(d))) * 2.0f))), ((powf(expf(2.0f), logf(t_8)) + t_3) + t_6))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u_m)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(t_2 * t_2)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(d) * dY_46_w)
	t_6 = Float32(t_5 * t_5)
	t_7 = Float32(floor(d) * dX_46_w)
	t_8 = Float32(dY_46_u * floor(w))
	t_9 = Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(t_9 + Float32(t_7 * t_7)), Float32(Float32(Float32(t_1 * t_1) + t_3) + t_6)))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(t_8 * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(Float32(t_9 + exp(Float32(log(Float32(Float32(-dX_46_w) * floor(d))) * Float32(2.0)))), Float32(Float32((exp(Float32(2.0)) ^ log(t_8)) + t_3) + t_6))));
	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 68.0%

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

   3. Applied rewrites 68.0%

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

   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 68.0%

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

      1. lift-*.f32 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 57.4

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

   3. Applied rewrites 57.4%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. exp-to-pow N/A

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

      7. lift-log.f32 N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 40.0

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

   5. Applied rewrites 40.0%

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

Alternative 3: 69.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, 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) * dX_46_u_m;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = t_2 * t_2;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(d) * dY_46_w;
	float t_6 = t_5 * t_5;
	float t_7 = floorf(d) * dX_46_w;
	float t_8 = dY_46_u * floorf(w);
	float t_9 = (t_0 * t_0) + (t_4 * t_4);
	float tmp;
	if (log2f(sqrtf(fmaxf((t_9 + (t_7 * t_7)), (((t_1 * t_1) + t_3) + t_6)))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u_m * floorf(w)) * dX_46_u_m), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf((t_8 * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_9 + expf((logf((-dX_46_w * floorf(d))) * 2.0f))), ((expf((2.0f * logf(t_8))) + t_3) + t_6))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u_m)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(t_2 * t_2)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(d) * dY_46_w)
	t_6 = Float32(t_5 * t_5)
	t_7 = Float32(floor(d) * dX_46_w)
	t_8 = Float32(dY_46_u * floor(w))
	t_9 = Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(t_9 + Float32(t_7 * t_7)), Float32(Float32(Float32(t_1 * t_1) + t_3) + t_6)))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(t_8 * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(Float32(t_9 + exp(Float32(log(Float32(Float32(-dX_46_w) * floor(d))) * Float32(2.0)))), Float32(Float32(exp(Float32(Float32(2.0) * log(t_8))) + t_3) + t_6))));
	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 68.0%

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

   3. Applied rewrites 68.0%

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

   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 68.0%

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

      1. lift-*.f32 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 57.4

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

   3. Applied rewrites 57.4%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. exp-to-pow N/A

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

      7. lift-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-exp.f32 40.0

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 40.0

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

   5. Applied rewrites 40.0%

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

Alternative 4: 68.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, 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) * dX_46_u_m;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h))));
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = (dX_46_w * floorf(d)) * dX_46_w;
	float t_7 = floorf(w) * dY_46_u;
	float t_8 = (dX_46_u_m * floorf(w)) * dX_46_u_m;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_2 * t_2)) + (t_5 * t_5)), (((t_7 * t_7) + (t_1 * t_1)) + (t_4 * t_4))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_6, floorf(d), fmaf(t_8, floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), t_3)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_6, floorf(d), fmaf(t_8, floorf(w), expf((logf((-dX_46_v * floorf(h))) * 2.0f)))), t_3)));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u_m)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))))
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(Float32(dX_46_w * floor(d)) * dX_46_w)
	t_7 = Float32(floor(w) * dY_46_u)
	t_8 = Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_5 * t_5)), Float32(Float32(Float32(t_7 * t_7) + Float32(t_1 * t_1)) + Float32(t_4 * t_4))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(t_6, floor(d), fma(t_8, floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), t_3)));
	else
		tmp = log2(sqrt(fmax(fma(t_6, floor(d), fma(t_8, floor(w), exp(Float32(log(Float32(Float32(-dX_46_v) * floor(h))) * Float32(2.0))))), t_3)));
	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 68.0%

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

   3. Applied rewrites 68.0%

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

   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 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-log.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 57.4

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

   5. Applied rewrites 57.4%

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

Alternative 5: 68.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, 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) * dX_46_u_m;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = (t_0 * t_0) + (t_3 * t_3);
	float t_7 = ((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4);
	float tmp;
	if (log2f(sqrtf(fmaxf((t_6 + (t_5 * t_5)), t_7))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u_m * floorf(w)) * dX_46_u_m), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_6 + expf((logf((-dX_46_w * floorf(d))) * 2.0f))), t_7)));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u_m)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(Float32(t_0 * t_0) + Float32(t_3 * t_3))
	t_7 = Float32(Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(t_6 + Float32(t_5 * t_5)), t_7))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(Float32(t_6 + exp(Float32(log(Float32(Float32(-dX_46_w) * floor(d))) * Float32(2.0)))), t_7)));
	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 68.0%

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

   3. Applied rewrites 68.0%

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

   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 68.0%

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

      1. lift-*.f32 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 57.4

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

   3. Applied rewrites 57.4%

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

Alternative 6: 68.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, 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) * dX_46_u_m;
	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 = dY_46_u * floorf(w);
	float t_6 = (dX_46_w * floorf(d)) * dX_46_w;
	float t_7 = floorf(w) * dY_46_u;
	float t_8 = (dX_46_u_m * floorf(w)) * dX_46_u_m;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_2 * t_2)) + (t_4 * t_4)), (((t_7 * t_7) + (t_1 * t_1)) + (t_3 * t_3))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_6, floorf(d), fmaf(t_8, floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf((t_5 * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_6, floorf(d), fmaf(t_8, floorf(w), (((expf(logf(-dX_46_v)) * expf(logf(floorf(h)))) * dX_46_v) * floorf(h)))), (dY_46_u * (fabsf(t_5) * floorf(w))))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u_m)
	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(dY_46_u * floor(w))
	t_6 = Float32(Float32(dX_46_w * floor(d)) * dX_46_w)
	t_7 = Float32(floor(w) * dY_46_u)
	t_8 = Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_7 * t_7) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(t_6, floor(d), fma(t_8, floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(t_5 * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(fma(t_6, floor(d), fma(t_8, floor(w), Float32(Float32(Float32(exp(log(Float32(-dX_46_v))) * exp(log(floor(h)))) * dX_46_v) * floor(h)))), Float32(dY_46_u * Float32(abs(t_5) * floor(w))))));
	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 68.0%

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

   3. Applied rewrites 68.0%

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

   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 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 68.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 63.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fabs.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-floor.f32 N/A

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

      6. lower-floor.f32 49.5

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

   8. Applied rewrites 49.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 49.5

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

      4. rem-exp-log N/A

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

      5. lift-log.f32 N/A

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

      6. exp-fabs N/A

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

      7. lift-log.f32 N/A

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

      8. rem-exp-log N/A

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

      9. neg-fabs N/A

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

      10. rem-exp-log N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

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

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

      14. lift-neg.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. lift-log.f32 N/A

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

      17. exp-fabs N/A

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

      18. lift-log.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. log-prod N/A

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

      21. exp-sum N/A

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

      22. lower-*.f32 N/A

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

   10. Applied rewrites 23.7%

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

Alternative 7: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u\_m \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\_m, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}\right) \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (fma
     (* (* dX.w (floor d)) dX.w)
     (floor d)
     (fma
      (* (* dX.u_m (floor w)) dX.u_m)
      (floor w)
      (* (* (* dX.v (floor h)) dX.v) (floor h))))
    (fma
     (* (* dY.w (floor d)) dY.w)
     (floor d)
     (fma
      (* (* dY.u (floor w)) dY.u)
      (floor w)
      (* (* (* dY.v (floor h)) dY.v) (floor h))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u_m * floorf(w)) * dX_46_u_m), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))))
end

Derivation

1. Initial program 68.0%

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

3. Applied rewrites 68.0%

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

Alternative 8: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := dY.w \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;dY.u \leq 60000002048:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor d\right\rfloor \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dX.w, \mathsf{fma}\left(dX.v \cdot dX.v, t\_2, \left(dX.u\_m \cdot dX.u\_m\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right), \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, t\_1 \cdot t\_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left\lfloor d\right\rfloor , dX.w, \mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u\_m, t\_0\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\_m - t\_0\right)\right), \mathsf{fma}\left(t\_2 \cdot dY.v, dY.v, \mathsf{fma}\left(t\_1 \cdot \left\lfloor d\right\rfloor , dY.w, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* dY.w (floor d)))
        (t_2 (* (floor h) (floor h))))
   (if (<= dY.u 60000002048.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* (floor d) (* (floor d) dX.w))
         dX.w
         (fma (* dX.v dX.v) t_2 (* (* dX.u_m dX.u_m) (* (floor w) (floor w)))))
        (fma (* (* dY.v (floor h)) (floor h)) dY.v (* t_1 t_1)))))
     (log2
      (sqrt
       (fmax
        (fma
         (* (* dX.w (floor d)) (floor d))
         dX.w
         (* (fma (floor w) dX.u_m t_0) (- (* (floor w) dX.u_m) t_0)))
        (fma
         (* t_2 dY.v)
         dY.v
         (fma
          (* t_1 (floor d))
          dY.w
          (* (* (* dY.u (floor w)) dY.u) (floor w))))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = dY_46_w * floorf(d);
	float t_2 = floorf(h) * floorf(h);
	float tmp;
	if (dY_46_u <= 60000002048.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((floorf(d) * (floorf(d) * dX_46_w)), dX_46_w, fmaf((dX_46_v * dX_46_v), t_2, ((dX_46_u_m * dX_46_u_m) * (floorf(w) * floorf(w))))), fmaf(((dY_46_v * floorf(h)) * floorf(h)), dY_46_v, (t_1 * t_1)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * floorf(d)), dX_46_w, (fmaf(floorf(w), dX_46_u_m, t_0) * ((floorf(w) * dX_46_u_m) - t_0))), fmaf((t_2 * dY_46_v), dY_46_v, fmaf((t_1 * floorf(d)), dY_46_w, (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w)))))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(dY_46_w * floor(d))
	t_2 = Float32(floor(h) * floor(h))
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(60000002048.0))
		tmp = log2(sqrt(fmax(fma(Float32(floor(d) * Float32(floor(d) * dX_46_w)), dX_46_w, fma(Float32(dX_46_v * dX_46_v), t_2, Float32(Float32(dX_46_u_m * dX_46_u_m) * Float32(floor(w) * floor(w))))), fma(Float32(Float32(dY_46_v * floor(h)) * floor(h)), dY_46_v, Float32(t_1 * t_1)))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * floor(d)), dX_46_w, Float32(fma(floor(w), dX_46_u_m, t_0) * Float32(Float32(floor(w) * dX_46_u_m) - t_0))), fma(Float32(t_2 * dY_46_v), dY_46_v, fma(Float32(t_1 * floor(d)), dY_46_w, Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 60000002000

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in dY.u around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.9

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

   6. Applied rewrites 60.9%

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

   8. Applied rewrites 60.9%

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

   if 60000002000 < dY.u

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-log.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 60.6%

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

Alternative 9: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dY.w \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dY.u \leq 60000002048:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor d\right\rfloor \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dX.w, \mathsf{fma}\left(dX.v \cdot dX.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(dX.u\_m \cdot dX.u\_m\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right), \mathsf{fma}\left(t\_0 \cdot \left\lfloor h\right\rfloor , dY.v, t\_1 \cdot t\_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(\left|dX.w \cdot \left\lfloor d\right\rfloor \right| \cdot \left\lfloor d\right\rfloor \right), \mathsf{fma}\left(t\_1 \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left|dY.u \cdot \left\lfloor w\right\rfloor \right| \cdot dY.u, \left\lfloor w\right\rfloor , \left(t\_0 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h))) (t_1 (* dY.w (floor d))))
   (if (<= dY.u 60000002048.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* (floor d) (* (floor d) dX.w))
         dX.w
         (fma
          (* dX.v dX.v)
          (* (floor h) (floor h))
          (* (* dX.u_m dX.u_m) (* (floor w) (floor w)))))
        (fma (* t_0 (floor h)) dY.v (* t_1 t_1)))))
     (log2
      (sqrt
       (fmax
        (* dX.w (* (fabs (* dX.w (floor d))) (floor d)))
        (fma
         (* t_1 dY.w)
         (floor d)
         (fma
          (* (fabs (* dY.u (floor w))) dY.u)
          (floor w)
          (* (* t_0 dY.v) (floor h))))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = dY_46_v * floorf(h);
	float t_1 = dY_46_w * floorf(d);
	float tmp;
	if (dY_46_u <= 60000002048.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((floorf(d) * (floorf(d) * dX_46_w)), dX_46_w, fmaf((dX_46_v * dX_46_v), (floorf(h) * floorf(h)), ((dX_46_u_m * dX_46_u_m) * (floorf(w) * floorf(w))))), fmaf((t_0 * floorf(h)), dY_46_v, (t_1 * t_1)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((dX_46_w * (fabsf((dX_46_w * floorf(d))) * floorf(d))), fmaf((t_1 * dY_46_w), floorf(d), fmaf((fabsf((dY_46_u * floorf(w))) * dY_46_u), floorf(w), ((t_0 * dY_46_v) * floorf(h)))))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32(dY_46_w * floor(d))
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(60000002048.0))
		tmp = log2(sqrt(fmax(fma(Float32(floor(d) * Float32(floor(d) * dX_46_w)), dX_46_w, fma(Float32(dX_46_v * dX_46_v), Float32(floor(h) * floor(h)), Float32(Float32(dX_46_u_m * dX_46_u_m) * Float32(floor(w) * floor(w))))), fma(Float32(t_0 * floor(h)), dY_46_v, Float32(t_1 * t_1)))));
	else
		tmp = log2(sqrt(fmax(Float32(dX_46_w * Float32(abs(Float32(dX_46_w * floor(d))) * floor(d))), fma(Float32(t_1 * dY_46_w), floor(d), fma(Float32(abs(Float32(dY_46_u * floor(w))) * dY_46_u), floor(w), Float32(Float32(t_0 * dY_46_v) * floor(h)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 60000002000

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in dY.u around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.9

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

   6. Applied rewrites 60.9%

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

   8. Applied rewrites 60.9%

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

   if 60000002000 < dY.u

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 68.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 63.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 63.9

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

   5. Applied rewrites 63.9%

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

      1. rem-exp-log N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lift-log.f32 N/A

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

      6. exp-fabs N/A

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

      7. lift-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. rem-exp-log N/A

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

      12. lower-fabs.f32 59.1

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 59.1

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

   7. Applied rewrites 59.1%

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

   8. Taylor expanded in dX.w around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fabs.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-floor.f32 N/A

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

      6. lower-floor.f32 43.4

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

   10. Applied rewrites 43.4%

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

Alternative 10: 51.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_1 := \left|dY.u \cdot \left\lfloor w\right\rfloor \right|\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.v \leq 3.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u\_m \cdot dX.u\_m\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor w\right\rfloor , t\_2 \cdot t\_2\right)\right), dY.u \cdot \left(t\_1 \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(\left|t\_0\right| \cdot \left\lfloor d\right\rfloor \right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(t\_1 \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* dX.w (floor d)))
        (t_1 (fabs (* dY.u (floor w))))
        (t_2 (* (floor h) dX.v)))
   (if (<= dY.v 3.5)
     (log2
      (sqrt
       (fmax
        (fma
         (* t_0 dX.w)
         (floor d)
         (fma (* (* dX.u_m dX.u_m) (floor w)) (floor w) (* t_2 t_2)))
        (* dY.u (* t_1 (floor w))))))
     (log2
      (sqrt
       (fmax
        (* dX.w (* (fabs t_0) (floor d)))
        (fma
         (* (* dY.w (floor d)) dY.w)
         (floor d)
         (fma
          (* t_1 dY.u)
          (floor w)
          (* (* (* dY.v (floor h)) dY.v) (floor h))))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = dX_46_w * floorf(d);
	float t_1 = fabsf((dY_46_u * floorf(w)));
	float t_2 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_v <= 3.5f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_w), floorf(d), fmaf(((dX_46_u_m * dX_46_u_m) * floorf(w)), floorf(w), (t_2 * t_2))), (dY_46_u * (t_1 * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((dX_46_w * (fabsf(t_0) * floorf(d))), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), fmaf((t_1 * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dX_46_w * floor(d))
	t_1 = abs(Float32(dY_46_u * floor(w)))
	t_2 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(3.5))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * dX_46_u_m) * floor(w)), floor(w), Float32(t_2 * t_2))), Float32(dY_46_u * Float32(t_1 * floor(w))))));
	else
		tmp = log2(sqrt(fmax(Float32(dX_46_w * Float32(abs(t_0) * floor(d))), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(t_1 * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 3.5

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 68.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 63.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fabs.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-floor.f32 N/A

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

      6. lower-floor.f32 49.5

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

   8. Applied rewrites 49.5%

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

   10. Applied rewrites 49.5%

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

   if 3.5 < dY.v

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 68.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 63.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 63.9

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

   5. Applied rewrites 63.9%

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

      1. rem-exp-log N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lift-log.f32 N/A

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

      6. exp-fabs N/A

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

      7. lift-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. rem-exp-log N/A

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

      12. lower-fabs.f32 59.1

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 59.1

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

   7. Applied rewrites 59.1%

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

   8. Taylor expanded in dX.w around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fabs.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-floor.f32 N/A

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

      6. lower-floor.f32 43.4

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

   10. Applied rewrites 43.4%

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

Alternative 11: 49.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	return log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u_m * dX_46_u_m) * floorf(w)), floorf(w), (t_0 * t_0))), (dY_46_u * (fabsf((dY_46_u * floorf(w))) * floorf(w))))));
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	return log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * dX_46_u_m) * floor(w)), floor(w), Float32(t_0 * t_0))), Float32(dY_46_u * Float32(abs(Float32(dY_46_u * floor(w))) * floor(w))))))
end

Derivation

1. Initial program 68.0%

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

3. Applied rewrites 68.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 68.0

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

   4. rem-exp-log N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. exp-fabs N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. rem-exp-log N/A

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

   15. lower-fabs.f32 63.9

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 63.9

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

5. Applied rewrites 63.9%

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

6. Taylor expanded in dY.u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fabs.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-floor.f32 N/A

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

   6. lower-floor.f32 49.5

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

8. Applied rewrites 49.5%

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

10. Applied rewrites 49.5%

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

Alternative 12: 47.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.u \leq 60000002048:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u\_m \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\_m, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \left(\left(\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u\_m, t\_0\right), \left\lfloor w\right\rfloor \cdot dX.u\_m - t\_0, \left(dX.w \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right)\right), dY.u \cdot \left(\left|dY.u \cdot \left\lfloor w\right\rfloor \right| \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v)))
   (if (<= dY.u 60000002048.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* (* dX.w (floor d)) dX.w)
         (floor d)
         (fma
          (* (* dX.u_m (floor w)) dX.u_m)
          (floor w)
          (* (* (* dX.v (floor h)) dX.v) (floor h))))
        (* (* (* (- dY.u) (floor w)) (floor w)) dY.u))))
     (log2
      (sqrt
       (fmax
        (fma
         (fma (floor w) dX.u_m t_0)
         (- (* (floor w) dX.u_m) t_0)
         (* (* dX.w dX.w) (* (floor d) (floor d))))
        (* dY.u (* (fabs (* dY.u (floor w))) (floor w)))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_u <= 60000002048.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u_m * floorf(w)) * dX_46_u_m), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), (((-dY_46_u * floorf(w)) * floorf(w)) * dY_46_u))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(fmaf(floorf(w), dX_46_u_m, t_0), ((floorf(w) * dX_46_u_m) - t_0), ((dX_46_w * dX_46_w) * (floorf(d) * floorf(d)))), (dY_46_u * (fabsf((dY_46_u * floorf(w))) * floorf(w))))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(60000002048.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u_m * floor(w)) * dX_46_u_m), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), Float32(Float32(Float32(Float32(-dY_46_u) * floor(w)) * floor(w)) * dY_46_u))));
	else
		tmp = log2(sqrt(fmax(fma(fma(floor(w), dX_46_u_m, t_0), Float32(Float32(floor(w) * dX_46_u_m) - t_0), Float32(Float32(dX_46_w * dX_46_w) * Float32(floor(d) * floor(d)))), Float32(dY_46_u * Float32(abs(Float32(dY_46_u * floor(w))) * floor(w))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 60000002000

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 68.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 63.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fabs.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-floor.f32 N/A

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

      6. lower-floor.f32 49.5

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

   8. Applied rewrites 49.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 49.5

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

   10. Applied rewrites 45.9%

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

   if 60000002000 < dY.u

   1. Initial program 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 68.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 63.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fabs.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-floor.f32 N/A

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

      6. lower-floor.f32 49.5

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

   8. Applied rewrites 49.5%

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

   10. Applied rewrites 36.3%

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

Alternative 13: 36.3% accurate, 1.5× speedup

Math

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

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v)))
   (log2
    (sqrt
     (fmax
      (fma
       (fma (floor w) dX.u_m t_0)
       (- (* (floor w) dX.u_m) t_0)
       (* (* dX.w dX.w) (* (floor d) (floor d))))
      (* dY.u (* (fabs (* dY.u (floor w))) (floor w))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	return log2f(sqrtf(fmaxf(fmaf(fmaf(floorf(w), dX_46_u_m, t_0), ((floorf(w) * dX_46_u_m) - t_0), ((dX_46_w * dX_46_w) * (floorf(d) * floorf(d)))), (dY_46_u * (fabsf((dY_46_u * floorf(w))) * floorf(w))))));
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	return log2(sqrt(fmax(fma(fma(floor(w), dX_46_u_m, t_0), Float32(Float32(floor(w) * dX_46_u_m) - t_0), Float32(Float32(dX_46_w * dX_46_w) * Float32(floor(d) * floor(d)))), Float32(dY_46_u * Float32(abs(Float32(dY_46_u * floor(w))) * floor(w))))))
end

Derivation

1. Initial program 68.0%

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

3. Applied rewrites 68.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 68.0

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

   4. rem-exp-log N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. exp-fabs N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. rem-exp-log N/A

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

   15. lower-fabs.f32 63.9

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 63.9

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

5. Applied rewrites 63.9%

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

6. Taylor expanded in dY.u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fabs.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-floor.f32 N/A

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

   6. lower-floor.f32 49.5

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

8. Applied rewrites 49.5%

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

10. Applied rewrites 36.3%

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

Alternative 14: 30.1% accurate, 1.5× speedup

Math

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

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v)))
   (log2
    (sqrt
     (fmax
      (fma
       (* (* dX.w (floor d)) (floor d))
       dX.w
       (* (fma (floor w) dX.u_m t_0) (- (* (floor w) dX.u_m) t_0)))
      (* (* (* (- dY.u) (floor w)) (floor w)) dY.u))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	return log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * floorf(d)), dX_46_w, (fmaf(floorf(w), dX_46_u_m, t_0) * ((floorf(w) * dX_46_u_m) - t_0))), (((-dY_46_u * floorf(w)) * floorf(w)) * dY_46_u))));
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	return log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * floor(d)), dX_46_w, Float32(fma(floor(w), dX_46_u_m, t_0) * Float32(Float32(floor(w) * dX_46_u_m) - t_0))), Float32(Float32(Float32(Float32(-dY_46_u) * floor(w)) * floor(w)) * dY_46_u))))
end

Derivation

1. Initial program 68.0%

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

3. Applied rewrites 68.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 68.0

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

   4. rem-exp-log N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. exp-fabs N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. rem-exp-log N/A

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

   15. lower-fabs.f32 63.9

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 63.9

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

5. Applied rewrites 63.9%

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

6. Taylor expanded in dY.u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fabs.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-floor.f32 N/A

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

   6. lower-floor.f32 49.5

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

8. Applied rewrites 49.5%

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

10. Applied rewrites 30.1%

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

Reproduce

herbie shell --seed 2025156 
(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)))))))