Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 67.7%

Time: 21.8s

Alternatives: 15

Speedup: N/A×

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(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != 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)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(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: 67.7% 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(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != 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)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(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: 67.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := \left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4\\ \mathbf{if}\;\mathsf{max}\left(t\_6, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_6, \frac{1}{\frac{1}{{t\_1}^{2} + \left({t\_0}^{2} + {t\_3}^{2}\right)}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_4}^{2}, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

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))
        (t_6 (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))))
   (if (<= (fmax t_6 (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3))) INFINITY)
     (log2
      (sqrt
       (fmax
        t_6
        (/ 1.0 (/ 1.0 (+ (pow t_1 2.0) (+ (pow t_0 2.0) (pow t_3 2.0))))))))
     (log2
      (sqrt (fmax (pow t_4 2.0) (* dY.u (* dY.u (pow (floor w) 2.0)))))))))

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;
	float t_6 = ((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4);
	float tmp;
	if (fmaxf(t_6, (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3))) <= ((float) INFINITY)) {
		tmp = log2f(sqrtf(fmaxf(t_6, (1.0f / (1.0f / (powf(t_1, 2.0f) + (powf(t_0, 2.0f) + powf(t_3, 2.0f))))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_4, 2.0f), (dY_46_u * (dY_46_u * powf(floorf(w), 2.0f))))));
	}
	return tmp;
}

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)
	t_6 = Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))
	tmp = Float32(0.0)
	if (((t_6 != t_6) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? t_6 : max(t_6, Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))))) <= Float32(Inf))
		tmp = log2(sqrt(((t_6 != t_6) ? Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))))) : ((Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))))) != Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))))))) ? t_6 : max(t_6, Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))))))))));
	else
		tmp = log2(sqrt((((t_4 ^ Float32(2.0)) != (t_4 ^ Float32(2.0))) ? Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))) : ((Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))) != Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) ? (t_4 ^ Float32(2.0)) : max((t_4 ^ Float32(2.0)), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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;
	t_6 = ((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4);
	tmp = single(0.0);
	if (max(t_6, (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3))) <= single(Inf))
		tmp = log2(sqrt(max(t_6, (single(1.0) / (single(1.0) / ((t_1 ^ single(2.0)) + ((t_0 ^ single(2.0)) + (t_3 ^ single(2.0)))))))));
	else
		tmp = log2(sqrt(max((t_4 ^ single(2.0)), (dY_46_u * (dY_46_u * (floor(w) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (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)))) < +inf.0

   1. Initial program 71.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing
   3. 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. flip3-+ N/A

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

      3. clear-num N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{1}{\frac{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}}}\right)}\right) \]

   4. Applied rewrites 71.5%

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

   if +inf.0 < (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 71.5%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 38.4

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

   8. Applied rewrites 38.4%

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

   10. Applied rewrites 38.4%

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

   11. Taylor expanded in dY.u around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-pow.f32 N/A

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

       7. lower-floor.f32 39.8

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

   13. Applied rewrites 39.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \frac{1}{\frac{1}{{\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_6 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.w \leq 0.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_1 \cdot t\_1 + t\_4 \cdot t\_4\right) + t\_6 \cdot t\_6, \mathsf{fma}\left(t\_0, dY.u \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_0, dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right), \left(t\_2 \cdot t\_2 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (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))
        (t_6 (* (floor d) dX.w)))
   (if (<= dY.w 0.5)
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_1 t_1) (* t_4 t_4)) (* t_6 t_6))
        (fma t_0 (* dY.u dY.u) (* (pow (floor h) 2.0) (* dY.v dY.v))))))
     (log2
      (sqrt
       (fmax
        (fma dX.u (* dX.u t_0) (* dX.w (* dX.w (pow (floor d) 2.0))))
        (+ (+ (* t_2 t_2) (* t_3 t_3)) (* t_5 t_5))))))))

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 = powf(floorf(w), 2.0f);
	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;
	float t_6 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_w <= 0.5f) {
		tmp = log2f(sqrtf(fmaxf((((t_1 * t_1) + (t_4 * t_4)) + (t_6 * t_6)), fmaf(t_0, (dY_46_u * dY_46_u), (powf(floorf(h), 2.0f) * (dY_46_v * dY_46_v))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_0), (dX_46_w * (dX_46_w * powf(floorf(d), 2.0f)))), (((t_2 * t_2) + (t_3 * t_3)) + (t_5 * t_5)))));
	}
	return tmp;
}

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 = floor(w) ^ Float32(2.0)
	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)
	t_6 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.5))
		tmp = log2(sqrt(((Float32(Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4)) + Float32(t_6 * t_6)) != Float32(Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4)) + Float32(t_6 * t_6))) ? fma(t_0, Float32(dY_46_u * dY_46_u), Float32((floor(h) ^ Float32(2.0)) * Float32(dY_46_v * dY_46_v))) : ((fma(t_0, Float32(dY_46_u * dY_46_u), Float32((floor(h) ^ Float32(2.0)) * Float32(dY_46_v * dY_46_v))) != fma(t_0, Float32(dY_46_u * dY_46_u), Float32((floor(h) ^ Float32(2.0)) * Float32(dY_46_v * dY_46_v)))) ? Float32(Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4)) + Float32(t_6 * t_6)) : max(Float32(Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4)) + Float32(t_6 * t_6)), fma(t_0, Float32(dY_46_u * dY_46_u), Float32((floor(h) ^ Float32(2.0)) * Float32(dY_46_v * dY_46_v))))))));
	else
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_0), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0))))) != fma(dX_46_u, Float32(dX_46_u * t_0), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0)))))) ? Float32(Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)) : ((Float32(Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)) != Float32(Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) + Float32(t_5 * t_5))) ? fma(dX_46_u, Float32(dX_46_u * t_0), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0))))) : max(fma(dX_46_u, Float32(dX_46_u * t_0), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0))))), Float32(Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 0.5

   1. Initial program 74.1%

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

   3. Taylor expanded in dY.w around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-floor.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 66.1

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

   5. Applied rewrites 66.1%

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

   if 0.5 < dY.w

   1. Initial program 61.9%

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

   3. Taylor expanded in dX.v around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower-floor.f32 59.3

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

   5. Applied rewrites 59.3%

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

Alternative 3: 62.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.v \leq 10000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right), \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_2 \cdot t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {t\_2}^{2}\right)}\right)\\ \end{array} \end{array} \]

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 d) dY.w)))
   (if (<= dX.v 10000000000.0)
     (log2
      (sqrt
       (fmax
        (fma
         dX.u
         (* dX.u (pow (floor w) 2.0))
         (* dX.w (* dX.w (pow (floor d) 2.0))))
        (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_2 t_2)))))
     (log2
      (sqrt
       (fmax
        (+ (pow (* (floor d) dX.w) 2.0) (pow (* (floor h) dX.v) 2.0))
        (pow t_2 2.0)))))))

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(d) * dY_46_w;
	float tmp;
	if (dX_46_v <= 10000000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * powf(floorf(w), 2.0f)), (dX_46_w * (dX_46_w * powf(floorf(d), 2.0f)))), (((t_0 * t_0) + (t_1 * t_1)) + (t_2 * t_2)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf((floorf(d) * dX_46_w), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f)), powf(t_2, 2.0f))));
	}
	return tmp;
}

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(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(10000000000.0))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0))))) != fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0)))))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2))) ? fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0))))) : max(fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(dX_46_w * Float32(dX_46_w * (floor(d) ^ Float32(2.0))))), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)))))));
	else
		tmp = log2(sqrt(((Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) != Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))) ? (t_2 ^ Float32(2.0)) : (((t_2 ^ Float32(2.0)) != (t_2 ^ Float32(2.0))) ? Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) : max(Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), (t_2 ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1e10

   1. Initial program 74.1%

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

   3. Taylor expanded in dX.v around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower-floor.f32 69.8

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

   5. Applied rewrites 69.8%

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

   if 1e10 < dX.v

   1. Initial program 49.9%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in dX.u around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-floor.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 48.9

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

   8. Applied rewrites 48.9%

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

   10. Applied rewrites 48.9%

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

Alternative 4: 56.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\\ \mathbf{if}\;dX.w \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \frac{1}{\frac{1}{t\_0}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0
         (+
          (pow (* (floor h) dY.v) 2.0)
          (+ (pow (* (floor w) dY.u) 2.0) (pow (* (floor d) dY.w) 2.0)))))
   (if (<= dX.w 20.0)
     (log2
      (sqrt (fmax (* (pow (floor w) 2.0) (* dX.u dX.u)) (/ 1.0 (/ 1.0 t_0)))))
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) t_0))))))

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 = powf((floorf(h) * dY_46_v), 2.0f) + (powf((floorf(w) * dY_46_u), 2.0f) + powf((floorf(d) * dY_46_w), 2.0f));
	float tmp;
	if (dX_46_w <= 20.0f) {
		tmp = log2f(sqrtf(fmaxf((powf(floorf(w), 2.0f) * (dX_46_u * dX_46_u)), (1.0f / (1.0f / t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), t_0)));
	}
	return tmp;
}

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((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0))))
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(20.0))
		tmp = log2(sqrt(((Float32((floor(w) ^ Float32(2.0)) * Float32(dX_46_u * dX_46_u)) != Float32((floor(w) ^ Float32(2.0)) * Float32(dX_46_u * dX_46_u))) ? Float32(Float32(1.0) / Float32(Float32(1.0) / t_0)) : ((Float32(Float32(1.0) / Float32(Float32(1.0) / t_0)) != Float32(Float32(1.0) / Float32(Float32(1.0) / t_0))) ? Float32((floor(w) ^ Float32(2.0)) * Float32(dX_46_u * dX_46_u)) : max(Float32((floor(w) ^ Float32(2.0)) * Float32(dX_46_u * dX_46_u)), Float32(Float32(1.0) / Float32(Float32(1.0) / t_0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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(h) * dY_46_v) ^ single(2.0)) + (((floor(w) * dY_46_u) ^ single(2.0)) + ((floor(d) * dY_46_w) ^ single(2.0)));
	tmp = single(0.0);
	if (dX_46_w <= single(20.0))
		tmp = log2(sqrt(max(((floor(w) ^ single(2.0)) * (dX_46_u * dX_46_u)), (single(1.0) / (single(1.0) / t_0)))));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 20

   1. Initial program 72.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing
   3. 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. flip3-+ N/A

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

      3. clear-num N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{1}{\frac{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}}}\right)}\right) \]

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 61.1

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

   7. Applied rewrites 61.1%

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

   if 20 < dX.w

   1. Initial program 69.4%

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

   3. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.4%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \frac{1}{\frac{1}{{\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\\ \mathbf{if}\;dX.w \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0
         (+
          (pow (* (floor h) dY.v) 2.0)
          (+ (pow (* (floor w) dY.u) 2.0) (pow (* (floor d) dY.w) 2.0)))))
   (if (<= dX.w 20.0)
     (log2 (sqrt (fmax (pow (* (floor w) dX.u) 2.0) t_0)))
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) t_0))))))

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 = powf((floorf(h) * dY_46_v), 2.0f) + (powf((floorf(w) * dY_46_u), 2.0f) + powf((floorf(d) * dY_46_w), 2.0f));
	float tmp;
	if (dX_46_w <= 20.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), t_0)));
	}
	return tmp;
}

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((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0))))
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(20.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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(h) * dY_46_v) ^ single(2.0)) + (((floor(w) * dY_46_u) ^ single(2.0)) + ((floor(d) * dY_46_w) ^ single(2.0)));
	tmp = single(0.0);
	if (dX_46_w <= single(20.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 20

   1. Initial program 72.1%

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

   3. Taylor expanded in dX.u around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 61.1

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 61.1%

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

   if 20 < dX.w

   1. Initial program 69.4%

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

   3. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.4%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.u \leq 448000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.w, dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right), t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.u 448000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (+
         (pow (* (floor h) dY.v) 2.0)
         (+ (pow (* (floor w) dY.u) 2.0) t_0)))))
     (log2
      (sqrt
       (fmax
        (fma
         dX.w
         (* dX.w (pow (floor d) 2.0))
         (* dX.u (* dX.u (pow (floor w) 2.0))))
        t_0))))))

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 = powf((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_u <= 448000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), (powf((floorf(h) * dY_46_v), 2.0f) + (powf((floorf(w) * dY_46_u), 2.0f) + t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_w, (dX_46_w * powf(floorf(d), 2.0f)), (dX_46_u * (dX_46_u * powf(floorf(w), 2.0f)))), t_0)));
	}
	return tmp;
}

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(d) * dY_46_w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(448000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_0)) : ((Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_0)) != Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_0)))))));
	else
		tmp = log2(sqrt(((fma(dX_46_w, Float32(dX_46_w * (floor(d) ^ Float32(2.0))), Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0))))) != fma(dX_46_w, Float32(dX_46_w * (floor(d) ^ Float32(2.0))), Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0)))))) ? t_0 : ((t_0 != t_0) ? fma(dX_46_w, Float32(dX_46_w * (floor(d) ^ Float32(2.0))), Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0))))) : max(fma(dX_46_w, Float32(dX_46_w * (floor(d) ^ Float32(2.0))), Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0))))), t_0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 448000

   1. Initial program 73.4%

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

   3. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 61.8%

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

   if 448000 < dX.u

   1. Initial program 61.2%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 31.5

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

   8. Applied rewrites 31.5%

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

   10. Applied rewrites 31.5%

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

   11. Taylor expanded in dX.v around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f32 N/A

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

       8. lower-pow.f32 N/A

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

       9. lower-floor.f32 N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

       13. lower-*.f32 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f32 N/A

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

       16. lower-pow.f32 N/A

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

       17. lower-floor.f32 55.9

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

   13. Applied rewrites 55.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 448000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.w, dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right), {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_2 := dY.w \cdot \left(dY.w \cdot t\_1\right)\\ \mathbf{if}\;dY.u \leq 5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, dX.u \cdot dX.u, t\_1 \cdot \left(dX.w \cdot dX.w\right)\right), t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot t\_0, t\_2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (pow (floor d) 2.0))
        (t_2 (* dY.w (* dY.w t_1))))
   (if (<= dY.u 5.0)
     (log2 (sqrt (fmax (fma t_0 (* dX.u dX.u) (* t_1 (* dX.w dX.w))) t_2)))
     (log2
      (sqrt
       (fmax (pow (* (floor d) dX.w) 2.0) (fma dY.u (* dY.u t_0) t_2)))))))

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 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(d), 2.0f);
	float t_2 = dY_46_w * (dY_46_w * t_1);
	float tmp;
	if (dY_46_u <= 5.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_0, (dX_46_u * dX_46_u), (t_1 * (dX_46_w * dX_46_w))), t_2)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), fmaf(dY_46_u, (dY_46_u * t_0), t_2))));
	}
	return tmp;
}

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 = floor(w) ^ Float32(2.0)
	t_1 = floor(d) ^ Float32(2.0)
	t_2 = Float32(dY_46_w * Float32(dY_46_w * t_1))
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(5.0))
		tmp = log2(sqrt(((fma(t_0, Float32(dX_46_u * dX_46_u), Float32(t_1 * Float32(dX_46_w * dX_46_w))) != fma(t_0, Float32(dX_46_u * dX_46_u), Float32(t_1 * Float32(dX_46_w * dX_46_w)))) ? t_2 : ((t_2 != t_2) ? fma(t_0, Float32(dX_46_u * dX_46_u), Float32(t_1 * Float32(dX_46_w * dX_46_w))) : max(fma(t_0, Float32(dX_46_u * dX_46_u), Float32(t_1 * Float32(dX_46_w * dX_46_w))), t_2)))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? fma(dY_46_u, Float32(dY_46_u * t_0), t_2) : ((fma(dY_46_u, Float32(dY_46_u * t_0), t_2) != fma(dY_46_u, Float32(dY_46_u * t_0), t_2)) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), fma(dY_46_u, Float32(dY_46_u * t_0), t_2))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 5

   1. Initial program 74.5%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in dX.v around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-floor.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 52.8

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

   8. Applied rewrites 52.8%

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

   if 5 < dY.u

   1. Initial program 64.3%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 47.4

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 35.4

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

   8. Applied rewrites 35.4%

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

   10. Applied rewrites 35.4%

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

   11. Taylor expanded in dY.v around 0

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

       1. *-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. *-inverses N/A

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

       4. associate-/l* N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f32 N/A

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

       11. lower-*.f32 N/A

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

       12. lower-pow.f32 N/A

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

       13. lower-floor.f32 N/A

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

       14. *-commutative N/A

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

       15. associate-*l/ N/A

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

       16. associate-/l* N/A

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

   13. Applied rewrites 54.6%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.u \cdot dX.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dX.w \cdot dX.w\right)\right), dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.u \leq 5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.w, dX.w \cdot t\_0, dX.u \cdot \left(dX.u \cdot t\_1\right)\right), {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot t\_1, dY.w \cdot \left(dY.w \cdot t\_0\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)) (t_1 (pow (floor w) 2.0)))
   (if (<= dY.u 5.0)
     (log2
      (sqrt
       (fmax
        (fma dX.w (* dX.w t_0) (* dX.u (* dX.u t_1)))
        (pow (* (floor d) dY.w) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (fma dY.u (* dY.u t_1) (* dY.w (* dY.w t_0)))))))))

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 = powf(floorf(d), 2.0f);
	float t_1 = powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_u <= 5.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_w, (dX_46_w * t_0), (dX_46_u * (dX_46_u * t_1))), powf((floorf(d) * dY_46_w), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), fmaf(dY_46_u, (dY_46_u * t_1), (dY_46_w * (dY_46_w * t_0))))));
	}
	return tmp;
}

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 = floor(d) ^ Float32(2.0)
	t_1 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(5.0))
		tmp = log2(sqrt(((fma(dX_46_w, Float32(dX_46_w * t_0), Float32(dX_46_u * Float32(dX_46_u * t_1))) != fma(dX_46_w, Float32(dX_46_w * t_0), Float32(dX_46_u * Float32(dX_46_u * t_1)))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? fma(dX_46_w, Float32(dX_46_w * t_0), Float32(dX_46_u * Float32(dX_46_u * t_1))) : max(fma(dX_46_w, Float32(dX_46_w * t_0), Float32(dX_46_u * Float32(dX_46_u * t_1))), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? fma(dY_46_u, Float32(dY_46_u * t_1), Float32(dY_46_w * Float32(dY_46_w * t_0))) : ((fma(dY_46_u, Float32(dY_46_u * t_1), Float32(dY_46_w * Float32(dY_46_w * t_0))) != fma(dY_46_u, Float32(dY_46_u * t_1), Float32(dY_46_w * Float32(dY_46_w * t_0)))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), fma(dY_46_u, Float32(dY_46_u * t_1), Float32(dY_46_w * Float32(dY_46_w * t_0))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 5

   1. Initial program 74.5%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 39.6

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

   8. Applied rewrites 39.6%

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

   10. Applied rewrites 39.6%

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

   11. Taylor expanded in dX.v around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f32 N/A

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

       8. lower-pow.f32 N/A

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

       9. lower-floor.f32 N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

       13. lower-*.f32 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f32 N/A

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

       16. lower-pow.f32 N/A

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

       17. lower-floor.f32 52.8

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

   13. Applied rewrites 52.8%

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

   if 5 < dY.u

   1. Initial program 64.3%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 47.4

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 35.4

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

   8. Applied rewrites 35.4%

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

   10. Applied rewrites 35.4%

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

   11. Taylor expanded in dY.v around 0

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

       1. *-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. *-inverses N/A

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

       4. associate-/l* N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f32 N/A

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

       11. lower-*.f32 N/A

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

       12. lower-pow.f32 N/A

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

       13. lower-floor.f32 N/A

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

       14. *-commutative N/A

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

       15. associate-*l/ N/A

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

       16. associate-/l* N/A

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

   13. Applied rewrites 54.6%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.w, dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right), {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\\ t_1 := dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 1:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \mathsf{fma}\left(dY.v \cdot dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot t\_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \mathsf{fma}\left(dY.u, t\_1, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dX.w) 2.0)) (t_1 (* dY.u (pow (floor w) 2.0))))
   (if (<= dY.w 1.0)
     (log2
      (sqrt (fmax t_0 (fma (* dY.v dY.v) (pow (floor h) 2.0) (* dY.u t_1)))))
     (log2
      (sqrt
       (fmax t_0 (fma dY.u t_1 (* dY.w (* dY.w (pow (floor d) 2.0))))))))))

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 = powf((floorf(d) * dX_46_w), 2.0f);
	float t_1 = dY_46_u * powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_w <= 1.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, fmaf((dY_46_v * dY_46_v), powf(floorf(h), 2.0f), (dY_46_u * t_1)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, fmaf(dY_46_u, t_1, (dY_46_w * (dY_46_w * powf(floorf(d), 2.0f)))))));
	}
	return tmp;
}

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(d) * dX_46_w) ^ Float32(2.0)
	t_1 = Float32(dY_46_u * (floor(w) ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(1.0))
		tmp = log2(sqrt(((t_0 != t_0) ? fma(Float32(dY_46_v * dY_46_v), (floor(h) ^ Float32(2.0)), Float32(dY_46_u * t_1)) : ((fma(Float32(dY_46_v * dY_46_v), (floor(h) ^ Float32(2.0)), Float32(dY_46_u * t_1)) != fma(Float32(dY_46_v * dY_46_v), (floor(h) ^ Float32(2.0)), Float32(dY_46_u * t_1))) ? t_0 : max(t_0, fma(Float32(dY_46_v * dY_46_v), (floor(h) ^ Float32(2.0)), Float32(dY_46_u * t_1)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? fma(dY_46_u, t_1, Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) : ((fma(dY_46_u, t_1, Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) != fma(dY_46_u, t_1, Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))))) ? t_0 : max(t_0, fma(dY_46_u, t_1, Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 1

   1. Initial program 74.1%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 37.9

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

   8. Applied rewrites 37.9%

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

   10. Applied rewrites 37.9%

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

   11. Taylor expanded in dY.w around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f32 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-pow.f32 N/A

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

       6. lower-floor.f32 N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f32 N/A

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

       11. lower-*.f32 N/A

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

       12. lower-pow.f32 N/A

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

       13. lower-floor.f32 50.8

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

   13. Applied rewrites 50.8%

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

   if 1 < dY.w

   1. Initial program 61.9%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 40.2

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

   8. Applied rewrites 40.2%

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

   10. Applied rewrites 40.2%

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

   11. Taylor expanded in dY.v around 0

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

       1. *-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. *-inverses N/A

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

       4. associate-/l* N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f32 N/A

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

       11. lower-*.f32 N/A

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

       12. lower-pow.f32 N/A

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

       13. lower-floor.f32 N/A

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

       14. *-commutative N/A

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

       15. associate-*l/ N/A

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

       16. associate-/l* N/A

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

   13. Applied rewrites 48.4%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 1:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.v \cdot dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\\ \mathbf{if}\;dX.u \leq 815000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot t\_0, t\_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 \cdot \left(dX.u \cdot dX.u\right), t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)) (t_1 (* dY.w (* dY.w (pow (floor d) 2.0)))))
   (if (<= dX.u 815000.0)
     (log2
      (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (fma dY.u (* dY.u t_0) t_1))))
     (log2 (sqrt (fmax (* t_0 (* dX.u dX.u)) t_1))))))

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 = powf(floorf(w), 2.0f);
	float t_1 = dY_46_w * (dY_46_w * powf(floorf(d), 2.0f));
	float tmp;
	if (dX_46_u <= 815000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), fmaf(dY_46_u, (dY_46_u * t_0), t_1))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 * (dX_46_u * dX_46_u)), t_1)));
	}
	return tmp;
}

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 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(815000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? fma(dY_46_u, Float32(dY_46_u * t_0), t_1) : ((fma(dY_46_u, Float32(dY_46_u * t_0), t_1) != fma(dY_46_u, Float32(dY_46_u * t_0), t_1)) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), fma(dY_46_u, Float32(dY_46_u * t_0), t_1))))));
	else
		tmp = log2(sqrt(((Float32(t_0 * Float32(dX_46_u * dX_46_u)) != Float32(t_0 * Float32(dX_46_u * dX_46_u))) ? t_1 : ((t_1 != t_1) ? Float32(t_0 * Float32(dX_46_u * dX_46_u)) : max(Float32(t_0 * Float32(dX_46_u * dX_46_u)), t_1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 815000

   1. Initial program 73.6%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 39.9

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

   8. Applied rewrites 39.9%

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

   10. Applied rewrites 39.9%

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

   11. Taylor expanded in dY.v around 0

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

       1. *-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. *-inverses N/A

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

       4. associate-/l* N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f32 N/A

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

       11. lower-*.f32 N/A

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

       12. lower-pow.f32 N/A

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

       13. lower-floor.f32 N/A

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

       14. *-commutative N/A

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

       15. associate-*l/ N/A

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

       16. associate-/l* N/A

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

   13. Applied rewrites 52.9%

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

   if 815000 < dX.u

   1. Initial program 60.3%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 49.8

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

   8. Applied rewrites 49.8%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 815000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dX.u \cdot dX.u\right), dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.u \leq 300000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.u 300000000.0)
     (log2
      (sqrt
       (fmax
        (+ (pow (* (floor d) dX.w) 2.0) (pow (* (floor h) dX.v) 2.0))
        t_0)))
     (log2 (sqrt (fmax (* dX.u (* dX.u (pow (floor w) 2.0))) t_0))))))

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 = powf((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_u <= 300000000.0f) {
		tmp = log2f(sqrtf(fmaxf((powf((floorf(d) * dX_46_w), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f)), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf((dX_46_u * (dX_46_u * powf(floorf(w), 2.0f))), t_0)));
	}
	return tmp;
}

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(d) * dY_46_w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(300000000.0))
		tmp = log2(sqrt(((Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) != Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))) ? t_0 : ((t_0 != t_0) ? Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) : max(Float32((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), t_0)))));
	else
		tmp = log2(sqrt(((Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0)))) != Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0))))) ? t_0 : ((t_0 != t_0) ? Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0)))) : max(Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0)))), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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(d) * dY_46_w) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(300000000.0))
		tmp = log2(sqrt(max((((floor(d) * dX_46_w) ^ single(2.0)) + ((floor(h) * dX_46_v) ^ single(2.0))), t_0)));
	else
		tmp = log2(sqrt(max((dX_46_u * (dX_46_u * (floor(w) ^ single(2.0)))), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 3e8

   1. Initial program 73.0%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in dX.u around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-floor.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 49.0

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

   8. Applied rewrites 49.0%

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

   10. Applied rewrites 49.1%

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

   if 3e8 < dX.u

   1. Initial program 61.5%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 30.6

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

   8. Applied rewrites 30.6%

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

   10. Applied rewrites 30.6%

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

   11. Taylor expanded in dX.u around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. lower-pow.f32 N/A

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

       8. lower-floor.f32 52.1

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

   13. Applied rewrites 52.1%

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

Alternative 12: 39.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 400000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, dY.u \cdot \left(dY.u \cdot t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 \cdot \left(dX.u \cdot dX.u\right), dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)))
   (if (<= dY.w 400000.0)
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (* dY.u (* dY.u t_0)))))
     (log2
      (sqrt
       (fmax (* t_0 (* dX.u dX.u)) (* dY.w (* dY.w (pow (floor d) 2.0)))))))))

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 = powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_w <= 400000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), (dY_46_u * (dY_46_u * t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 * (dX_46_u * dX_46_u)), (dY_46_w * (dY_46_w * powf(floorf(d), 2.0f))))));
	}
	return tmp;
}

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 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(400000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? Float32(dY_46_u * Float32(dY_46_u * t_0)) : ((Float32(dY_46_u * Float32(dY_46_u * t_0)) != Float32(dY_46_u * Float32(dY_46_u * t_0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), Float32(dY_46_u * Float32(dY_46_u * t_0)))))));
	else
		tmp = log2(sqrt(((Float32(t_0 * Float32(dX_46_u * dX_46_u)) != Float32(t_0 * Float32(dX_46_u * dX_46_u))) ? Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) : ((Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) != Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) ? Float32(t_0 * Float32(dX_46_u * dX_46_u)) : max(Float32(t_0 * Float32(dX_46_u * dX_46_u)), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_w <= single(400000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (dY_46_u * (dY_46_u * t_0)))));
	else
		tmp = log2(sqrt(max((t_0 * (dX_46_u * dX_46_u)), (dY_46_w * (dY_46_w * (floor(d) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 4e5

   1. Initial program 73.3%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 38.3

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

   8. Applied rewrites 38.3%

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

   10. Applied rewrites 38.3%

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

   11. Taylor expanded in dY.u around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-pow.f32 N/A

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

       7. lower-floor.f32 41.6

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

   13. Applied rewrites 41.6%

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

   if 4e5 < dY.w

   1. Initial program 61.6%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 48.5

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

   8. Applied rewrites 48.5%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 400000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dX.u \cdot dX.u\right), dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 400000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, dY.u \cdot \left(dY.u \cdot t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_0\right), {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)))
   (if (<= dY.w 400000.0)
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (* dY.u (* dY.u t_0)))))
     (log2 (sqrt (fmax (* dX.u (* dX.u t_0)) (pow (* (floor d) dY.w) 2.0)))))))

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 = powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_w <= 400000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), (dY_46_u * (dY_46_u * t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((dX_46_u * (dX_46_u * t_0)), powf((floorf(d) * dY_46_w), 2.0f))));
	}
	return tmp;
}

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 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(400000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? Float32(dY_46_u * Float32(dY_46_u * t_0)) : ((Float32(dY_46_u * Float32(dY_46_u * t_0)) != Float32(dY_46_u * Float32(dY_46_u * t_0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), Float32(dY_46_u * Float32(dY_46_u * t_0)))))));
	else
		tmp = log2(sqrt(((Float32(dX_46_u * Float32(dX_46_u * t_0)) != Float32(dX_46_u * Float32(dX_46_u * t_0))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? Float32(dX_46_u * Float32(dX_46_u * t_0)) : max(Float32(dX_46_u * Float32(dX_46_u * t_0)), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_w <= single(400000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (dY_46_u * (dY_46_u * t_0)))));
	else
		tmp = log2(sqrt(max((dX_46_u * (dX_46_u * t_0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 4e5

   1. Initial program 73.3%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 38.3

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

   8. Applied rewrites 38.3%

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

   10. Applied rewrites 38.3%

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

   11. Taylor expanded in dY.u around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-pow.f32 N/A

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

       7. lower-floor.f32 41.6

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

   13. Applied rewrites 41.6%

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

   if 4e5 < dY.w

   1. Initial program 61.6%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 38.8

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

   8. Applied rewrites 38.8%

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

   10. Applied rewrites 38.8%

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

   11. Taylor expanded in dX.u around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. lower-pow.f32 N/A

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

       8. lower-floor.f32 48.5

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

   13. Applied rewrites 48.5%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 400000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.w \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.w 20.0)
     (log2 (sqrt (fmax (* dX.u (* dX.u (pow (floor w) 2.0))) t_0)))
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) t_0))))))

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 = powf((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_w <= 20.0f) {
		tmp = log2f(sqrtf(fmaxf((dX_46_u * (dX_46_u * powf(floorf(w), 2.0f))), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), t_0)));
	}
	return tmp;
}

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(d) * dY_46_w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(20.0))
		tmp = log2(sqrt(((Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0)))) != Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0))))) ? t_0 : ((t_0 != t_0) ? Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0)))) : max(Float32(dX_46_u * Float32(dX_46_u * (floor(w) ^ Float32(2.0)))), t_0)))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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(d) * dY_46_w) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_w <= single(20.0))
		tmp = log2(sqrt(max((dX_46_u * (dX_46_u * (floor(w) ^ single(2.0)))), t_0)));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 20

   1. Initial program 72.1%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 33.3

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 33.3%

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

   11. Taylor expanded in dX.u around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. lower-pow.f32 N/A

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

       8. lower-floor.f32 39.7

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

   13. Applied rewrites 39.7%

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

   if 20 < dX.w

   1. Initial program 69.4%

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

   3. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 55.8

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

   8. Applied rewrites 55.8%

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

   10. Applied rewrites 55.8%

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

Alternative 15: 35.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (* (floor d) dY.w) 2.0)))))

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) {
	return log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf((floorf(d) * dY_46_w), 2.0f))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))))
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)
	tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
end

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in dY.w around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-floor.f32 56.2

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

5. Applied rewrites 56.2%

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

6. Taylor expanded in dX.w around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-floor.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 38.4

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

8. Applied rewrites 38.4%

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

10. Applied rewrites 38.4%

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

Reproduce

herbie shell --seed 2024223 
(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)))))))