Isotropic LOD (LOD)

Percentage Accurate: 67.5% → 71.1%

Time: 10.7s

Alternatives: 8

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: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 71.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

dY.w_m = fabs(dY_46_w);
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_m) {
	float t_0 = floorf(w) * floorf(w);
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = floorf(h) * dY_46_v;
	float t_4 = floorf(d) * dY_46_w_m;
	float t_5 = floorf(h) * dX_46_v;
	float t_6 = dX_46_w * floorf(d);
	float t_7 = floorf(d) * dX_46_w;
	float t_8 = dY_46_v * floorf(h);
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_1 * t_1) + (t_5 * t_5)) + (t_7 * t_7)), (((t_2 * t_2) + (t_3 * t_3)) + (t_4 * t_4))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_u), dX_46_u, fmaf(((dX_46_v * floorf(h)) * floorf(h)), dX_46_v, ((t_6 * dX_46_w) * floorf(d)))), fmaf((t_0 * dY_46_u), dY_46_u, fmaf((t_8 * floorf(h)), dY_46_v, (((dY_46_w_m * floorf(d)) * dY_46_w_m) * floorf(d)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_6 * floorf(d)), dX_46_w, (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w))), fmaf(expf((2.0f * logf(-dY_46_w_m))), (floorf(d) * floorf(d)), fmaf((dY_46_u * dY_46_u), t_0, (t_8 * t_8))))));
	}
	return tmp;
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	t_0 = Float32(floor(w) * floor(w))
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(floor(h) * dY_46_v)
	t_4 = Float32(floor(d) * dY_46_w_m)
	t_5 = Float32(floor(h) * dX_46_v)
	t_6 = Float32(dX_46_w * floor(d))
	t_7 = Float32(floor(d) * dX_46_w)
	t_8 = Float32(dY_46_v * floor(h))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_1 * t_1) + Float32(t_5 * t_5)) + Float32(t_7 * t_7)), Float32(Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) + Float32(t_4 * t_4))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_u), dX_46_u, fma(Float32(Float32(dX_46_v * floor(h)) * floor(h)), dX_46_v, Float32(Float32(t_6 * dX_46_w) * floor(d)))), fma(Float32(t_0 * dY_46_u), dY_46_u, fma(Float32(t_8 * floor(h)), dY_46_v, Float32(Float32(Float32(dY_46_w_m * floor(d)) * dY_46_w_m) * floor(d)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_6 * floor(d)), dX_46_w, Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w))), fma(exp(Float32(Float32(2.0) * log(Float32(-dY_46_w_m)))), Float32(floor(d) * floor(d)), fma(Float32(dY_46_u * dY_46_u), t_0, Float32(t_8 * t_8))))));
	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 67.5%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f32 N/A

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

   3. Applied rewrites 67.5%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. swap-sqr N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f32 N/A

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

   5. Applied rewrites 67.5%

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

   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 67.5%

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

   2. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.0

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

   4. Applied rewrites 60.0%

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-log.f32 59.9

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

   8. Applied rewrites 59.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-log.f32 N/A

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

      4. log-pow-rev N/A

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

      5. pow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. pow2 N/A

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

      8. log-pow N/A

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

      9. lower-*.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. lower-neg.f32 35.5

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

   10. Applied rewrites 35.5%

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

Alternative 2: 70.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

dY.w_m = fabs(dY_46_w);
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_m) {
	float t_0 = floorf(w) * floorf(w);
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = floorf(h) * dY_46_v;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(d) * dY_46_w_m;
	float t_6 = floorf(d) * dX_46_w;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_1 * t_1) + (t_4 * t_4)) + (t_6 * t_6)), (((t_2 * t_2) + (t_3 * t_3)) + (t_5 * t_5))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_u), dX_46_u, fmaf(((dX_46_v * floorf(h)) * floorf(h)), dX_46_v, (((dX_46_w * floorf(d)) * dX_46_w) * floorf(d)))), fmaf((t_0 * dY_46_u), dY_46_u, fmaf(((dY_46_v * floorf(h)) * floorf(h)), dY_46_v, (((dY_46_w_m * floorf(d)) * dY_46_w_m) * floorf(d)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(((floorf(h) * floorf(h)) * dX_46_v), dX_46_v, fmaf((t_1 * floorf(w)), dX_46_u, ((dX_46_w * dX_46_w) * (floorf(d) * floorf(d))))), -(((dY_46_u * floorf(w)) * dY_46_u) * floorf(w)))));
	}
	return tmp;
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	t_0 = Float32(floor(w) * floor(w))
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(floor(h) * dY_46_v)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(d) * dY_46_w_m)
	t_6 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4)) + Float32(t_6 * t_6)), Float32(Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) + Float32(t_5 * t_5))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_u), dX_46_u, fma(Float32(Float32(dX_46_v * floor(h)) * floor(h)), dX_46_v, Float32(Float32(Float32(dX_46_w * floor(d)) * dX_46_w) * floor(d)))), fma(Float32(t_0 * dY_46_u), dY_46_u, fma(Float32(Float32(dY_46_v * floor(h)) * floor(h)), dY_46_v, Float32(Float32(Float32(dY_46_w_m * floor(d)) * dY_46_w_m) * floor(d)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(Float32(floor(h) * floor(h)) * dX_46_v), dX_46_v, fma(Float32(t_1 * floor(w)), dX_46_u, Float32(Float32(dX_46_w * dX_46_w) * Float32(floor(d) * floor(d))))), Float32(-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 (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 67.5%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f32 N/A

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

   3. Applied rewrites 67.5%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. swap-sqr N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f32 N/A

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

   5. Applied rewrites 67.5%

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

   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 67.5%

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

   3. Applied rewrites 67.5%

      \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}\right) \]
   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 67.5

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

      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. rem-sqrt-square-rev N/A

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

      16. lift-*.f32 N/A

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

      17. lower-sqrt.f32 64.2

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift-*.f32 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f32 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f32 N/A

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

      25. lower-*.f32 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in dY.u around -inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right)\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      5. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      8. lower-floor.f32 46.2

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

   10. Applied rewrites 46.1%

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

Alternative 3: 70.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

dY.w_m = fabs(dY_46_w);
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_m) {
	float t_0 = floorf(w) * dX_46_u;
	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_m;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = (dY_46_u * floorf(w)) * dY_46_u;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_3 * t_3)) + (t_5 * t_5)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(((dY_46_w_m * floorf(d)) * dY_46_w_m), floorf(d), fmaf(t_6, floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(((floorf(h) * floorf(h)) * dX_46_v), dX_46_v, fmaf((t_0 * floorf(w)), dX_46_u, ((dX_46_w * dX_46_w) * (floorf(d) * floorf(d))))), -(t_6 * floorf(w)))));
	}
	return tmp;
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	t_0 = Float32(floor(w) * dX_46_u)
	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_m)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(Float32(dY_46_u * floor(w)) * dY_46_u)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))))) <= 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 * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(Float32(Float32(dY_46_w_m * floor(d)) * dY_46_w_m), floor(d), fma(t_6, floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(Float32(floor(h) * floor(h)) * dX_46_v), dX_46_v, fma(Float32(t_0 * floor(w)), dX_46_u, Float32(Float32(dX_46_w * dX_46_w) * Float32(floor(d) * floor(d))))), Float32(-Float32(t_6 * 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 67.5%

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

   3. Applied rewrites 67.5%

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

   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 67.5%

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

   3. Applied rewrites 67.5%

      \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}\right) \]
   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 67.5

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

      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. rem-sqrt-square-rev N/A

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

      16. lift-*.f32 N/A

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

      17. lower-sqrt.f32 64.2

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift-*.f32 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f32 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f32 N/A

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

      25. lower-*.f32 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in dY.u around -inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right)\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      5. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      8. lower-floor.f32 46.2

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

   10. Applied rewrites 46.1%

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

Alternative 4: 61.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} dY.w_m = \left|dY.w\right| \\ \begin{array}{l} t_0 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ \mathbf{if}\;dX.v \leq 15000000512:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left|\left\lfloor d\right\rfloor \cdot dX.w\right| \cdot \left\lfloor d\right\rfloor , dX.w, t\_0 \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \mathsf{fma}\left(\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\_m, dY.w\_m, \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(t\_0, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left(-\left\lfloor w\right\rfloor \right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.w_m = (fabs.f32 dY.w)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w_m)
 :precision binary32
 (let* ((t_0 (* (* dX.u (floor w)) dX.u)))
   (if (<= dX.v 15000000512.0)
     (log2
      (sqrt
       (fmax
        (fma (* (fabs (* (floor d) dX.w)) (floor d)) dX.w (* t_0 (floor w)))
        (fma
         (* (* (floor w) (floor w)) dY.u)
         dY.u
         (fma
          (* (* (floor d) (floor d)) dY.w_m)
          dY.w_m
          (* (* (* dY.v (floor h)) dY.v) (floor h)))))))
     (log2
      (sqrt
       (fmax
        (fma
         (* (* dX.w (floor d)) dX.w)
         (floor d)
         (fma t_0 (floor w) (* (* (* dX.v (floor h)) dX.v) (floor h))))
        (* (* (* dY.u (floor w)) dY.u) (- (floor w)))))))))

C

dY.w_m = fabs(dY_46_w);
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_m) {
	float t_0 = (dX_46_u * floorf(w)) * dX_46_u;
	float tmp;
	if (dX_46_v <= 15000000512.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((fabsf((floorf(d) * dX_46_w)) * floorf(d)), dX_46_w, (t_0 * floorf(w))), fmaf(((floorf(w) * floorf(w)) * dY_46_u), dY_46_u, fmaf(((floorf(d) * floorf(d)) * dY_46_w_m), dY_46_w_m, (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(t_0, floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), (((dY_46_u * floorf(w)) * dY_46_u) * -floorf(w)))));
	}
	return tmp;
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	t_0 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(15000000512.0))
		tmp = log2(sqrt(fmax(fma(Float32(abs(Float32(floor(d) * dX_46_w)) * floor(d)), dX_46_w, Float32(t_0 * floor(w))), fma(Float32(Float32(floor(w) * floor(w)) * dY_46_u), dY_46_u, fma(Float32(Float32(floor(d) * floor(d)) * dY_46_w_m), dY_46_w_m, Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(t_0, floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * Float32(-floor(w))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 15000000500

   1. Initial program 67.5%

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

   2. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.0

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

   4. Applied rewrites 60.0%

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 60.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 56.4

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

   8. Applied rewrites 56.4%

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

   10. Applied rewrites 56.4%

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

   if 15000000500 < dX.v

   1. Initial program 67.5%

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

   3. Applied rewrites 67.5%

      \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}\right) \]
   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 67.5

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

      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. rem-sqrt-square-rev N/A

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

      16. lift-*.f32 N/A

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

      17. lower-sqrt.f32 64.2

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift-*.f32 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f32 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f32 N/A

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

      25. lower-*.f32 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in dY.u around -inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right)\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      5. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      8. lower-floor.f32 46.2

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. mul-1-neg N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f32 N/A

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

      10. lift-pow.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. pow1/2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

      14. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

      15. unpow-prod-down N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)}\right)\right)\right)}\right) \]

      19. metadata-eval N/A

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

      20. sqr-pow N/A

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

      21. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{1}\right)\right)}\right) \]

      22. unpow1 N/A

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

   10. Applied rewrites 46.2%

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

Alternative 5: 58.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} dY.w_m = \left|dY.w\right| \\ \begin{array}{l} t_0 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_1 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_2 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ \mathbf{if}\;dX.v \leq 15000000512:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot \left\lfloor d\right\rfloor , dX.w, t\_2 \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(dY.w\_m \cdot dY.w\_m, \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor , \mathsf{fma}\left(dY.u \cdot dY.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_1 \cdot t\_1\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(t\_2, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left(-\left\lfloor w\right\rfloor \right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.w_m = (fabs.f32 dY.w)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w_m)
 :precision binary32
 (let* ((t_0 (* dX.w (floor d)))
        (t_1 (* dY.v (floor h)))
        (t_2 (* (* dX.u (floor w)) dX.u)))
   (if (<= dX.v 15000000512.0)
     (log2
      (sqrt
       (fmax
        (fma (* t_0 (floor d)) dX.w (* t_2 (floor w)))
        (fma
         (* dY.w_m dY.w_m)
         (* (floor d) (floor d))
         (fma (* dY.u dY.u) (* (floor w) (floor w)) (* t_1 t_1))))))
     (log2
      (sqrt
       (fmax
        (fma
         (* t_0 dX.w)
         (floor d)
         (fma t_2 (floor w) (* (* (* dX.v (floor h)) dX.v) (floor h))))
        (* (* (* dY.u (floor w)) dY.u) (- (floor w)))))))))

C

dY.w_m = fabs(dY_46_w);
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_m) {
	float t_0 = dX_46_w * floorf(d);
	float t_1 = dY_46_v * floorf(h);
	float t_2 = (dX_46_u * floorf(w)) * dX_46_u;
	float tmp;
	if (dX_46_v <= 15000000512.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * floorf(d)), dX_46_w, (t_2 * floorf(w))), fmaf((dY_46_w_m * dY_46_w_m), (floorf(d) * floorf(d)), fmaf((dY_46_u * dY_46_u), (floorf(w) * floorf(w)), (t_1 * t_1))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_w), floorf(d), fmaf(t_2, floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), (((dY_46_u * floorf(w)) * dY_46_u) * -floorf(w)))));
	}
	return tmp;
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	t_0 = Float32(dX_46_w * floor(d))
	t_1 = Float32(dY_46_v * floor(h))
	t_2 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(15000000512.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * floor(d)), dX_46_w, Float32(t_2 * floor(w))), fma(Float32(dY_46_w_m * dY_46_w_m), Float32(floor(d) * floor(d)), fma(Float32(dY_46_u * dY_46_u), Float32(floor(w) * floor(w)), Float32(t_1 * t_1))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_w), floor(d), fma(t_2, floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * Float32(-floor(w))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 15000000500

   1. Initial program 67.5%

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

   2. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.0

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

   4. Applied rewrites 60.0%

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

   6. Applied rewrites 60.0%

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

   if 15000000500 < dX.v

   1. Initial program 67.5%

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

   3. Applied rewrites 67.5%

      \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}\right) \]
   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 67.5

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

      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. rem-sqrt-square-rev N/A

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

      16. lift-*.f32 N/A

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

      17. lower-sqrt.f32 64.2

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift-*.f32 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f32 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f32 N/A

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

      25. lower-*.f32 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in dY.u around -inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right)\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      5. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      8. lower-floor.f32 46.2

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. mul-1-neg N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f32 N/A

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

      10. lift-pow.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. pow1/2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

      14. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

      15. unpow-prod-down N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)}\right)\right)\right)}\right) \]

      19. metadata-eval N/A

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

      20. sqr-pow N/A

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

      21. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{1}\right)\right)}\right) \]

      22. unpow1 N/A

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

   10. Applied rewrites 46.2%

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

Alternative 6: 55.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.w_m = \left|dY.w\right| \\ \begin{array}{l} t_0 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_1 := dY.v \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;dY.w\_m \leq 60:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), dY.u \cdot \left(\left|dY.u \cdot \left\lfloor w\right\rfloor \right| \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(\left|t\_0\right| \cdot \left\lfloor d\right\rfloor \right), \mathsf{fma}\left(dY.w\_m \cdot dY.w\_m, \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor , \mathsf{fma}\left(dY.u \cdot dY.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_1 \cdot t\_1\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.w_m = (fabs.f32 dY.w)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w_m)
 :precision binary32
 (let* ((t_0 (* dX.w (floor d))) (t_1 (* dY.v (floor h))))
   (if (<= dY.w_m 60.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* t_0 dX.w)
         (floor d)
         (fma
          (* (* dX.u (floor w)) dX.u)
          (floor w)
          (* (* (* dX.v (floor h)) dX.v) (floor h))))
        (* dY.u (* (fabs (* dY.u (floor w))) (floor w))))))
     (log2
      (sqrt
       (fmax
        (* dX.w (* (fabs t_0) (floor d)))
        (fma
         (* dY.w_m dY.w_m)
         (* (floor d) (floor d))
         (fma (* dY.u dY.u) (* (floor w) (floor w)) (* t_1 t_1)))))))))

C

dY.w_m = fabs(dY_46_w);
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_m) {
	float t_0 = dX_46_w * floorf(d);
	float t_1 = dY_46_v * floorf(h);
	float tmp;
	if (dY_46_w_m <= 60.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_w), floorf(d), fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), (dY_46_u * (fabsf((dY_46_u * floorf(w))) * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((dX_46_w * (fabsf(t_0) * floorf(d))), fmaf((dY_46_w_m * dY_46_w_m), (floorf(d) * floorf(d)), fmaf((dY_46_u * dY_46_u), (floorf(w) * floorf(w)), (t_1 * t_1))))));
	}
	return tmp;
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	t_0 = Float32(dX_46_w * floor(d))
	t_1 = Float32(dY_46_v * floor(h))
	tmp = Float32(0.0)
	if (dY_46_w_m <= Float32(60.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), Float32(dY_46_u * Float32(abs(Float32(dY_46_u * floor(w))) * floor(w))))));
	else
		tmp = log2(sqrt(fmax(Float32(dX_46_w * Float32(abs(t_0) * floor(d))), fma(Float32(dY_46_w_m * dY_46_w_m), Float32(floor(d) * floor(d)), fma(Float32(dY_46_u * dY_46_u), Float32(floor(w) * floor(w)), Float32(t_1 * t_1))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 60

   1. Initial program 67.5%

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

   3. Applied rewrites 67.5%

      \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}\right) \]
   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 67.5

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

      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. rem-sqrt-square-rev N/A

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

      16. lift-*.f32 N/A

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

      17. lower-sqrt.f32 64.2

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift-*.f32 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f32 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f32 N/A

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

      25. lower-*.f32 64.2

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

   5. Applied rewrites 64.2%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{dY.u \cdot \left(\left|dY.u \cdot \left\lfloor w\right\rfloor \right| \cdot \left\lfloor w\right\rfloor \right)}\right)}\right) \]
   9. 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 49.8

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

   10. Applied rewrites 49.8%

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

   if 60 < dY.w

   1. Initial program 67.5%

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

   2. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.0

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

   4. Applied rewrites 60.0%

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 60.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 56.4

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

   8. Applied rewrites 56.4%

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

   9. Taylor expanded in dX.u around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-fabs.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-floor.f32 N/A

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

       6. lower-floor.f32 49.8

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

   11. Applied rewrites 49.8%

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

Alternative 7: 51.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.w_m = \left|dY.w\right| \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dX.w \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dX.u \leq 15000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(\left|t\_1\right| \cdot \left\lfloor d\right\rfloor \right), \mathsf{fma}\left(dY.w\_m \cdot dY.w\_m, \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor , \mathsf{fma}\left(dY.u \cdot dY.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_0 \cdot t\_0\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1 \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left(-\left\lfloor w\right\rfloor \right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.w_m = (fabs.f32 dY.w)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w_m)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h))) (t_1 (* dX.w (floor d))))
   (if (<= dX.u 15000000.0)
     (log2
      (sqrt
       (fmax
        (* dX.w (* (fabs t_1) (floor d)))
        (fma
         (* dY.w_m dY.w_m)
         (* (floor d) (floor d))
         (fma (* dY.u dY.u) (* (floor w) (floor w)) (* t_0 t_0))))))
     (log2
      (sqrt
       (fmax
        (fma
         (* t_1 dX.w)
         (floor d)
         (fma
          (* (* dX.u (floor w)) dX.u)
          (floor w)
          (* (* (* dX.v (floor h)) dX.v) (floor h))))
        (* (* (* dY.u (floor w)) dY.u) (- (floor w)))))))))

C

dY.w_m = fabs(dY_46_w);
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_m) {
	float t_0 = dY_46_v * floorf(h);
	float t_1 = dX_46_w * floorf(d);
	float tmp;
	if (dX_46_u <= 15000000.0f) {
		tmp = log2f(sqrtf(fmaxf((dX_46_w * (fabsf(t_1) * floorf(d))), fmaf((dY_46_w_m * dY_46_w_m), (floorf(d) * floorf(d)), fmaf((dY_46_u * dY_46_u), (floorf(w) * floorf(w)), (t_0 * t_0))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_1 * dX_46_w), floorf(d), fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), (((dY_46_u * floorf(w)) * dY_46_u) * -floorf(w)))));
	}
	return tmp;
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32(dX_46_w * floor(d))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(15000000.0))
		tmp = log2(sqrt(fmax(Float32(dX_46_w * Float32(abs(t_1) * floor(d))), fma(Float32(dY_46_w_m * dY_46_w_m), Float32(floor(d) * floor(d)), fma(Float32(dY_46_u * dY_46_u), Float32(floor(w) * floor(w)), Float32(t_0 * t_0))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_1 * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * Float32(-floor(w))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1.5e7

   1. Initial program 67.5%

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

   2. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.0

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

   4. Applied rewrites 60.0%

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 60.0

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. rem-exp-log N/A

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

      15. lower-fabs.f32 56.4

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

   8. Applied rewrites 56.4%

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

   9. Taylor expanded in dX.u around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-fabs.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-floor.f32 N/A

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

       6. lower-floor.f32 49.8

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

   11. Applied rewrites 49.8%

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

   if 1.5e7 < dX.u

   1. Initial program 67.5%

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

   3. Applied rewrites 67.5%

      \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}\right) \]
   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 67.5

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

      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. rem-sqrt-square-rev N/A

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

      16. lift-*.f32 N/A

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

      17. lower-sqrt.f32 64.2

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift-*.f32 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f32 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f32 N/A

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

      25. lower-*.f32 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in dY.u around -inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right)\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      5. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

      8. lower-floor.f32 46.2

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

      2. mul-1-neg N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f32 N/A

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

      10. lift-pow.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. pow1/2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

      14. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

      15. unpow-prod-down N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)}\right)\right)\right)}\right) \]

      19. metadata-eval N/A

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

      20. sqr-pow N/A

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

      21. metadata-eval N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{1}\right)\right)}\right) \]

      22. unpow1 N/A

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

   10. Applied rewrites 46.2%

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

Alternative 8: 46.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

dY.w_m = fabs(dY_46_w);
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_m) {
	return log2f(sqrtf(fmaxf(fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), (((dY_46_u * floorf(w)) * dY_46_u) * -floorf(w)))));
}

Julia

dY.w_m = abs(dY_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w_m)
	return log2(sqrt(fmax(fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * Float32(-floor(w))))))
end

Derivation

1. Initial program 67.5%

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

3. Applied rewrites 67.5%

   \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}\right) \]
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 67.5

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

   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. rem-sqrt-square-rev N/A

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

   16. lift-*.f32 N/A

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

   17. lower-sqrt.f32 64.2

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. lift-*.f32 N/A

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

   21. swap-sqr N/A

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

   22. lift-*.f32 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f32 N/A

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

   25. lower-*.f32 64.2

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

5. Applied rewrites 64.2%

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

6. Taylor expanded in dY.u around -inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right)\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

   5. lower-floor.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}}\right)\right)\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

   8. lower-floor.f32 46.2

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)\right)}\right) \]

8. Applied rewrites 46.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \color{blue}{-1 \cdot \left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), -1 \cdot \color{blue}{\left({dY.u}^{2} \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)\right)}\right)}\right) \]

   2. mul-1-neg N/A

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

   3. lift-*.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lift-*.f32 N/A

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

   13. pow1/2 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

   14. lift-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)}\right) \]

   15. unpow-prod-down N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

   17. metadata-eval N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\frac{1}{2}}\right)\right)\right)}\right) \]

   18. metadata-eval N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{\left(\frac{1}{2}\right)}\right)\right)\right)}\right) \]

   19. metadata-eval N/A

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

   20. sqr-pow N/A

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

   21. metadata-eval N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{neg}\left(\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{1}\right)\right)}\right) \]

   22. unpow1 N/A

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

10. Applied rewrites 46.2%

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

Reproduce

herbie shell --seed 2025140 
(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)))))))