Isotropic LOD (LOD)

Percentage Accurate: 68.0% → 70.6%

Time: 50.2s

Alternatives: 18

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((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: 68.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((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: 70.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_7 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;\mathsf{max}\left(\left(t\_6 \cdot t\_6 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5, \left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4\right) \leq 1.5000000027488779 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_6, t\_6, t\_7 \cdot \left(dX.v \cdot dX.v\right)\right) + t\_1 \cdot \left(dX.w \cdot dX.w\right), \mathsf{fma}\left(t\_0, t\_0, t\_7 \cdot \left(dY.v \cdot dY.v\right)\right) + t\_1 \cdot \left(dY.w \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_3}^{2}, {t\_4}^{2}\right)}\right)\\ \end{array} \end{array} \]

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 d) (floor d)))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor h) dX.v))
        (t_4 (* (floor d) dY.w))
        (t_5 (* (floor d) dX.w))
        (t_6 (* (floor w) dX.u))
        (t_7 (* (floor h) (floor h))))
   (if (<=
        (fmax
         (+ (+ (* t_6 t_6) (* t_3 t_3)) (* t_5 t_5))
         (+ (+ (* t_0 t_0) (* t_2 t_2)) (* t_4 t_4)))
        1.5000000027488779e+38)
     (log2
      (sqrt
       (fmax
        (+ (fma t_6 t_6 (* t_7 (* dX.v dX.v))) (* t_1 (* dX.w dX.w)))
        (+ (fma t_0 t_0 (* t_7 (* dY.v dY.v))) (* t_1 (* dY.w dY.w))))))
     (log2 (sqrt (fmax (pow t_3 2.0) (pow t_4 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(d) * floorf(d);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = floorf(w) * dX_46_u;
	float t_7 = floorf(h) * floorf(h);
	float tmp;
	if (fmaxf((((t_6 * t_6) + (t_3 * t_3)) + (t_5 * t_5)), (((t_0 * t_0) + (t_2 * t_2)) + (t_4 * t_4))) <= 1.5000000027488779e+38f) {
		tmp = log2f(sqrtf(fmaxf((fmaf(t_6, t_6, (t_7 * (dX_46_v * dX_46_v))) + (t_1 * (dX_46_w * dX_46_w))), (fmaf(t_0, t_0, (t_7 * (dY_46_v * dY_46_v))) + (t_1 * (dY_46_w * dY_46_w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_3, 2.0f), powf(t_4, 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(d) * floor(d))
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(floor(w) * dX_46_u)
	t_7 = Float32(floor(h) * floor(h))
	tmp = Float32(0.0)
	if (((Float32(Float32(Float32(t_6 * t_6) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)) != Float32(Float32(Float32(t_6 * t_6) + Float32(t_3 * t_3)) + Float32(t_5 * t_5))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_6 * t_6) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)) : max(Float32(Float32(Float32(t_6 * t_6) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))))) <= Float32(1.5000000027488779e+38))
		tmp = log2(sqrt(((Float32(fma(t_6, t_6, Float32(t_7 * Float32(dX_46_v * dX_46_v))) + Float32(t_1 * Float32(dX_46_w * dX_46_w))) != Float32(fma(t_6, t_6, Float32(t_7 * Float32(dX_46_v * dX_46_v))) + Float32(t_1 * Float32(dX_46_w * dX_46_w)))) ? Float32(fma(t_0, t_0, Float32(t_7 * Float32(dY_46_v * dY_46_v))) + Float32(t_1 * Float32(dY_46_w * dY_46_w))) : ((Float32(fma(t_0, t_0, Float32(t_7 * Float32(dY_46_v * dY_46_v))) + Float32(t_1 * Float32(dY_46_w * dY_46_w))) != Float32(fma(t_0, t_0, Float32(t_7 * Float32(dY_46_v * dY_46_v))) + Float32(t_1 * Float32(dY_46_w * dY_46_w)))) ? Float32(fma(t_6, t_6, Float32(t_7 * Float32(dX_46_v * dX_46_v))) + Float32(t_1 * Float32(dX_46_w * dX_46_w))) : max(Float32(fma(t_6, t_6, Float32(t_7 * Float32(dX_46_v * dX_46_v))) + Float32(t_1 * Float32(dX_46_w * dX_46_w))), Float32(fma(t_0, t_0, Float32(t_7 * Float32(dY_46_v * dY_46_v))) + Float32(t_1 * Float32(dY_46_w * dY_46_w))))))));
	else
		tmp = log2(sqrt((((t_3 ^ Float32(2.0)) != (t_3 ^ Float32(2.0))) ? (t_4 ^ Float32(2.0)) : (((t_4 ^ Float32(2.0)) != (t_4 ^ Float32(2.0))) ? (t_3 ^ Float32(2.0)) : max((t_3 ^ Float32(2.0)), (t_4 ^ Float32(2.0)))))));
	end
	return 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)))) < 1.5e38

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   if 1.5e38 < (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 9.6%

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

   3. Taylor expanded in w around 0 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in dY.u around 0 12.3%

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

      1. *-commutative 12.3%

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

   8. Simplified 12.3%

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

   9. Taylor expanded in dX.v around inf 17.9%

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

       1. unpow2 17.9%

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

       2. unpow2 17.9%

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

       3. swap-sqr 17.9%

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

       4. unpow2 17.9%

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

   11. Simplified 17.9%

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

   12. Taylor expanded in dY.w around inf 20.6%

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

       1. *-commutative 20.6%

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

   14. Simplified 20.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \leq 1.5000000027488779 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor w\right\rfloor \cdot dX.u, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor w\right\rfloor \cdot dY.u, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \mathbf{if}\;\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right) \leq 1.5000000027488779 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(t\_5, t\_2\right), t\_4\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_3, \mathsf{hypot}\left(t\_0, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_2}^{2}, {t\_3}^{2}\right)}\right)\\ \end{array} \end{array} \]

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 h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (if (<=
        (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)))
        1.5000000027488779e+38)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot t_5 t_2) t_4) 2.0)
        (pow (hypot t_3 (hypot t_0 t_1)) 2.0))))
     (log2 (sqrt (fmax (pow t_2 2.0) (pow t_3 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 tmp;
	if (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))) <= 1.5000000027488779e+38f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(hypotf(t_5, t_2), t_4), 2.0f), powf(hypotf(t_3, hypotf(t_0, t_1)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_2, 2.0f), powf(t_3, 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)
	tmp = Float32(0.0)
	if (((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))))) <= Float32(1.5000000027488779e+38))
		tmp = log2(sqrt((((hypot(hypot(t_5, t_2), t_4) ^ Float32(2.0)) != (hypot(hypot(t_5, t_2), t_4) ^ Float32(2.0))) ? (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)) : (((hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)) != (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0))) ? (hypot(hypot(t_5, t_2), t_4) ^ Float32(2.0)) : max((hypot(hypot(t_5, t_2), t_4) ^ Float32(2.0)), (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((t_2 ^ Float32(2.0)) != (t_2 ^ Float32(2.0))) ? (t_3 ^ Float32(2.0)) : (((t_3 ^ Float32(2.0)) != (t_3 ^ Float32(2.0))) ? (t_2 ^ Float32(2.0)) : max((t_2 ^ Float32(2.0)), (t_3 ^ 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;
	tmp = single(0.0);
	if (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))) <= single(1.5000000027488779e+38))
		tmp = log2(sqrt(max((hypot(hypot(t_5, t_2), t_4) ^ single(2.0)), (hypot(t_3, hypot(t_0, t_1)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_2 ^ single(2.0)), (t_3 ^ 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)))) < 1.5e38

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0 100.0%

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

   5. Simplified 100.0%

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

   if 1.5e38 < (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 9.6%

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

   3. Taylor expanded in w around 0 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in dY.u around 0 12.3%

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

      1. *-commutative 12.3%

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

   8. Simplified 12.3%

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

   9. Taylor expanded in dX.v around inf 17.9%

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

       1. unpow2 17.9%

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

       2. unpow2 17.9%

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

       3. swap-sqr 17.9%

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

       4. unpow2 17.9%

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

   11. Simplified 17.9%

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

   12. Taylor expanded in dY.w around inf 20.6%

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

       1. *-commutative 20.6%

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

   14. Simplified 20.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \leq 1.5000000027488779 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right) \cdot 0.5}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (exp
   (*
    (log
     (fmax
      (pow
       (hypot (* (floor d) dX.w) (hypot (* (floor w) dX.u) (* (floor h) dX.v)))
       2.0)
      (pow (hypot (* (floor d) dY.w) (* (floor w) dY.u)) 2.0)))
    0.5))))

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) {
	return log2f(expf((logf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), powf(hypotf((floorf(d) * dY_46_w), (floorf(w) * dY_46_u)), 2.0f))) * 0.5f)));
}

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(exp(Float32(log((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)))))) * Float32(0.5))))
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(exp((log(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), (hypot((floor(d) * dY_46_w), (floor(w) * dY_46_u)) ^ single(2.0)))) * single(0.5))));
end

Derivation

1. Initial program 68.2%

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

4. Applied egg-rr 67.5%

   \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right) \cdot 0.5}\right)} \]

5. Taylor expanded in dY.u around inf 64.4%

   \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \color{blue}{dY.u \cdot \left\lfloor w\right\rfloor }\right)\right)}^{2}\right)\right) \cdot 0.5}\right) \]
6. Step-by-step derivation

   1. *-commutative 64.4%

      \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \color{blue}{\left\lfloor w\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right) \cdot 0.5}\right) \]

7. Simplified 64.4%

   \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \color{blue}{\left\lfloor w\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right) \cdot 0.5}\right) \]
8. Add Preprocessing

Alternative 4: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.w \leq 500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, t\_2\right), t\_3\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_2}^{2} + {t\_3}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(t\_1, t\_0\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dX.w)))
   (if (<= dY.w 500.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot (* (floor w) dX.u) t_2) t_3) 2.0)
        (pow (hypot t_0 t_1) 2.0))))
     (log2
      (sqrt
       (fmax
        (+ (pow t_2 2.0) (pow t_3 2.0))
        (pow (hypot (* (floor d) dY.w) (hypot t_1 t_0)) 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(h) * dY_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_w <= 500.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(hypotf((floorf(w) * dX_46_u), t_2), t_3), 2.0f), powf(hypotf(t_0, t_1), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(t_2, 2.0f) + powf(t_3, 2.0f)), powf(hypotf((floorf(d) * dY_46_w), hypotf(t_1, t_0)), 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(h) * dY_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(500.0))
		tmp = log2(sqrt((((hypot(hypot(Float32(floor(w) * dX_46_u), t_2), t_3) ^ Float32(2.0)) != (hypot(hypot(Float32(floor(w) * dX_46_u), t_2), t_3) ^ Float32(2.0))) ? (hypot(t_0, t_1) ^ Float32(2.0)) : (((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? (hypot(hypot(Float32(floor(w) * dX_46_u), t_2), t_3) ^ Float32(2.0)) : max((hypot(hypot(Float32(floor(w) * dX_46_u), t_2), t_3) ^ Float32(2.0)), (hypot(t_0, t_1) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32((t_2 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) != Float32((t_2 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(t_1, t_0)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(t_1, t_0)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(t_1, t_0)) ^ Float32(2.0))) ? Float32((t_2 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) : max(Float32((t_2 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))), (hypot(Float32(floor(d) * dY_46_w), hypot(t_1, t_0)) ^ 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(h) * dY_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dX_46_w;
	tmp = single(0.0);
	if (dY_46_w <= single(500.0))
		tmp = log2(sqrt(max((hypot(hypot((floor(w) * dX_46_u), t_2), t_3) ^ single(2.0)), (hypot(t_0, t_1) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((t_2 ^ single(2.0)) + (t_3 ^ single(2.0))), (hypot((floor(d) * dY_46_w), hypot(t_1, t_0)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 500

   1. Initial program 68.1%

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

   3. Taylor expanded in w around 0 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in dY.w around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. unpow2 63.2%

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

      4. swap-sqr 63.2%

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

      5. unpow2 63.2%

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

      6. *-commutative 63.2%

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

      7. unpow2 63.2%

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

      8. unpow2 63.2%

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

      9. swap-sqr 63.2%

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

      10. unpow2 63.2%

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

      11. rem-square-sqrt 63.2%

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

      12. unpow2 63.2%

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

      13. unpow2 63.2%

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

      14. hypot-undefine 63.2%

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

   8. Simplified 63.2%

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

   if 500 < dY.w

   1. Initial program 68.8%

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

   3. Taylor expanded in w around 0 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in dX.u around 0 69.0%

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

      1. +-commutative 69.0%

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

      2. unpow2 69.0%

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

      3. unpow2 69.0%

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

      4. swap-sqr 69.0%

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

      5. unpow2 69.0%

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

      6. unpow2 69.0%

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

      7. unpow2 69.0%

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

      8. swap-sqr 69.0%

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

      9. unpow2 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor h\right\rfloor \cdot dY.v, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ \mathbf{if}\;dY.u \leq 60000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, t\_1\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\mathsf{hypot}\left(t\_1, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0
         (pow
          (hypot
           (hypot (* (floor w) dX.u) (* (floor h) dX.v))
           (* (floor d) dX.w))
          2.0))
        (t_1 (* (floor h) dY.v)))
   (if (<= dY.u 60000.0)
     (log2 (sqrt (fmax t_0 (pow (hypot (* (floor d) dY.w) t_1) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (hypot t_1 (* (floor w) dY.u)) 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(hypotf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), (floorf(d) * dX_46_w)), 2.0f);
	float t_1 = floorf(h) * dY_46_v;
	float tmp;
	if (dY_46_u <= 60000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf((floorf(d) * dY_46_w), t_1), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf(t_1, (floorf(w) * dY_46_u)), 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 = hypot(hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)
	t_1 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(60000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (hypot(Float32(floor(d) * dY_46_w), t_1) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), t_1) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), t_1) ^ Float32(2.0))) ? t_0 : max(t_0, (hypot(Float32(floor(d) * dY_46_w), t_1) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? t_0 : max(t_0, (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ 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 = hypot(hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)), (floor(d) * dX_46_w)) ^ single(2.0);
	t_1 = floor(h) * dY_46_v;
	tmp = single(0.0);
	if (dY_46_u <= single(60000.0))
		tmp = log2(sqrt(max(t_0, (hypot((floor(d) * dY_46_w), t_1) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, (hypot(t_1, (floor(w) * dY_46_u)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 6e4

   1. Initial program 67.9%

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

   3. Taylor expanded in w around 0 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in dY.u around 0 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

   if 6e4 < dY.u

   1. Initial program 69.5%

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

   3. Taylor expanded in w around 0 69.5%

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

   5. Simplified 69.5%

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

   6. Taylor expanded in dY.w around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. unpow2 64.2%

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

      3. unpow2 64.2%

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

      4. swap-sqr 64.2%

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

      5. unpow2 64.2%

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

      6. *-commutative 64.2%

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

      7. unpow2 64.2%

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

      8. unpow2 64.2%

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

      9. swap-sqr 64.2%

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

      10. unpow2 64.2%

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

      11. rem-square-sqrt 64.2%

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

      12. unpow2 64.2%

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

      13. unpow2 64.2%

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

      14. hypot-undefine 64.2%

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

   8. Simplified 64.2%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 60000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor h\right\rfloor \cdot dY.v, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u)))
   (if (<= dY.u 1000000.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot t_0 (* (floor h) dX.v)) (* (floor d) dX.w)) 2.0)
        (pow (hypot (* (floor d) dY.w) (* (floor h) dY.v)) 2.0))))
     (log2
      (sqrt
       (fmax
        (+
         (fma t_0 t_0 (* (* (floor h) (floor h)) (* dX.v dX.v)))
         (* (* (floor d) (floor d)) (* dX.w dX.w)))
        (pow (* (floor w) dY.u) 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) * dX_46_u;
	float tmp;
	if (dY_46_u <= 1000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(hypotf(t_0, (floorf(h) * dX_46_v)), (floorf(d) * dX_46_w)), 2.0f), powf(hypotf((floorf(d) * dY_46_w), (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((fmaf(t_0, t_0, ((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v))) + ((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w))), powf((floorf(w) * dY_46_u), 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) * dX_46_u)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(1000000.0))
		tmp = log2(sqrt((((hypot(hypot(t_0, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) != (hypot(hypot(t_0, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (hypot(hypot(t_0, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) : max((hypot(hypot(t_0, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))) != Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w)))) ? (Float32(floor(w) * dY_46_u) ^ Float32(2.0)) : (((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) ? Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))) : max(Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))), (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 1e6

   1. Initial program 68.1%

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

   3. Taylor expanded in w around 0 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in dY.u around 0 61.9%

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

      1. *-commutative 61.9%

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

   8. Simplified 61.9%

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

   if 1e6 < dY.u

   1. Initial program 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in dY.v around inf 49.9%

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

   6. Taylor expanded in dY.u around inf 62.8%

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

      1. *-commutative 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. swap-sqr 62.8%

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

      5. unpow2 62.8%

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

   8. Simplified 62.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 1000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor w\right\rfloor \cdot dX.u, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ \mathbf{if}\;dY.w \leq 500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, t\_0, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), {t\_1}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(t\_1, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u)) (t_1 (* (floor w) dY.u)))
   (if (<= dY.w 500.0)
     (log2
      (sqrt
       (fmax
        (+
         (fma t_0 t_0 (* (* (floor h) (floor h)) (* dX.v dX.v)))
         (* (* (floor d) (floor d)) (* dX.w dX.w)))
        (pow t_1 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (pow
         (hypot (* (floor d) dY.w) (hypot t_1 (* (floor h) dY.v)))
         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) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float tmp;
	if (dY_46_w <= 500.0f) {
		tmp = log2f(sqrtf(fmaxf((fmaf(t_0, t_0, ((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v))) + ((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w))), powf(t_1, 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf(t_1, (floorf(h) * dY_46_v))), 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) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(500.0))
		tmp = log2(sqrt(((Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))) != Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w)))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))) : max(Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))), (t_1 ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 500

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in dY.v around inf 55.4%

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

   6. Taylor expanded in dY.u around inf 60.7%

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

      1. *-commutative 60.7%

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

      2. unpow2 60.7%

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

      3. unpow2 60.7%

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

      4. swap-sqr 60.7%

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

      5. unpow2 60.7%

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

   8. Simplified 60.7%

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

   if 500 < dY.w

   1. Initial program 68.8%

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

   3. Taylor expanded in w around 0 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in dX.v around inf 66.0%

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

      1. unpow2 64.8%

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

      2. unpow2 64.8%

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

      3. swap-sqr 64.9%

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

      4. unpow2 64.9%

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

   8. Simplified 66.1%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor w\right\rfloor \cdot dX.u, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dY.u \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, t\_0, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), {t\_1}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u)) (t_1 (* (floor d) dY.w)))
   (if (<= dY.u 500000.0)
     (log2
      (sqrt
       (fmax
        (+
         (fma t_0 t_0 (* (* (floor h) (floor h)) (* dX.v dX.v)))
         (* (* (floor d) (floor d)) (* dX.w dX.w)))
        (pow t_1 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (pow
         (hypot t_1 (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         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) * dX_46_u;
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dY_46_u <= 500000.0f) {
		tmp = log2f(sqrtf(fmaxf((fmaf(t_0, t_0, ((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v))) + ((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w))), powf(t_1, 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf(t_1, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 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) * dX_46_u)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(500000.0))
		tmp = log2(sqrt(((Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))) != Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w)))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))) : max(Float32(fma(t_0, t_0, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))) + Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))), (t_1 ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 5e5

   1. Initial program 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in dY.v around inf 57.0%

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

   6. Taylor expanded in dY.w around inf 58.3%

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

      1. *-commutative 58.3%

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

      2. unpow2 58.3%

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

      3. unpow2 58.3%

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

      4. swap-sqr 58.3%

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

      5. unpow2 58.3%

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

   8. Simplified 58.3%

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

   if 5e5 < dY.u

   1. Initial program 69.5%

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

   3. Taylor expanded in w around 0 69.5%

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

   5. Simplified 69.5%

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

   6. Taylor expanded in dX.v around inf 61.9%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. swap-sqr 39.0%

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

      4. unpow2 39.0%

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

   8. Simplified 61.9%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor w\right\rfloor \cdot dX.u, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.u \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, t\_1\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {t\_0}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_1}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)) (t_1 (* (floor h) dX.v)))
   (if (<= dY.u 500000.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot (* (floor w) dX.u) t_1) (* (floor d) dX.w)) 2.0)
        (pow t_0 2.0))))
     (log2
      (sqrt
       (fmax
        (pow t_1 2.0)
        (pow
         (hypot t_0 (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         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(d) * dY_46_w;
	float t_1 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_u <= 500000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(hypotf((floorf(w) * dX_46_u), t_1), (floorf(d) * dX_46_w)), 2.0f), powf(t_0, 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_1, 2.0f), powf(hypotf(t_0, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 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) * dY_46_w)
	t_1 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(500000.0))
		tmp = log2(sqrt((((hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) != (hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) : max((hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)), (t_0 ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : max((t_1 ^ Float32(2.0)), (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ 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(d) * dY_46_w;
	t_1 = floor(h) * dX_46_v;
	tmp = single(0.0);
	if (dY_46_u <= single(500000.0))
		tmp = log2(sqrt(max((hypot(hypot((floor(w) * dX_46_u), t_1), (floor(d) * dX_46_w)) ^ single(2.0)), (t_0 ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_1 ^ single(2.0)), (hypot(t_0, hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 5e5

   1. Initial program 67.9%

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

   3. Taylor expanded in w around 0 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in dY.u around 0 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

   9. Taylor expanded in dY.w around inf 58.3%

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

       1. *-commutative 39.0%

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

   11. Simplified 58.3%

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

   if 5e5 < dY.u

   1. Initial program 69.5%

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

   3. Taylor expanded in w around 0 69.5%

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

   5. Simplified 69.5%

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

   6. Taylor expanded in dX.v around inf 61.9%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. swap-sqr 39.0%

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

      4. unpow2 39.0%

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

   8. Simplified 61.9%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ \mathbf{if}\;dX.v \leq 184000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(t\_1, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)) (t_1 (* (floor w) dY.u)))
   (if (<= dX.v 184000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (pow (hypot t_0 (hypot t_1 (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt (fmax (pow (* (floor h) dX.v) 2.0) (pow (hypot t_0 t_1) 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(d) * dY_46_w;
	float t_1 = floorf(w) * dY_46_u;
	float tmp;
	if (dX_46_v <= 184000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf(t_0, hypotf(t_1, (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf(t_0, t_1), 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) * dY_46_w)
	t_1 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(184000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(t_0, hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(t_0, hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(t_0, hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(t_0, t_1) ^ Float32(2.0)) : (((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(t_0, t_1) ^ 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(d) * dY_46_w;
	t_1 = floor(w) * dY_46_u;
	tmp = single(0.0);
	if (dX_46_v <= single(184000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot(t_0, hypot(t_1, (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), (hypot(t_0, t_1) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 184000

   1. Initial program 73.2%

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

   3. Taylor expanded in w around 0 73.2%

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

   5. Simplified 73.2%

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

   6. Taylor expanded in dX.w around inf 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. swap-sqr 59.5%

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

      4. unpow2 59.5%

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

   8. Simplified 59.5%

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

   if 184000 < dX.v

   1. Initial program 50.8%

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

   3. Taylor expanded in w around 0 50.8%

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

   5. Simplified 50.8%

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

   6. Taylor expanded in dX.v around inf 48.4%

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

      1. unpow2 45.0%

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

      2. unpow2 45.0%

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

      3. swap-sqr 45.1%

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

      4. unpow2 45.1%

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

   8. Simplified 48.5%

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

   9. Taylor expanded in dY.u around inf 50.3%

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

       1. *-commutative 52.6%

          \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \color{blue}{\left\lfloor w\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right) \cdot 0.5}\right) \]

   11. Simplified 50.3%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 184000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.w \leq 125000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, t\_0\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, t\_0\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v)) (t_1 (* (floor d) dY.w)))
   (if (<= dX.w 125000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (pow (hypot t_1 (hypot (* (floor w) dY.u) t_0)) 2.0))))
     (log2
      (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (hypot t_1 t_0) 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(h) * dY_46_v;
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_w <= 125000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf(t_1, hypotf((floorf(w) * dY_46_u), t_0)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf(t_1, t_0), 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(h) * dY_46_v)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(125000000.0))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) : (((hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) != (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ 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))) ? (hypot(t_1, t_0) ^ Float32(2.0)) : (((hypot(t_1, t_0) ^ Float32(2.0)) != (hypot(t_1, t_0) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(t_1, t_0) ^ 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(h) * dY_46_v;
	t_1 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_w <= single(125000000.0))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), (hypot(t_1, hypot((floor(w) * dY_46_u), t_0)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot(t_1, t_0) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 1.25e8

   1. Initial program 71.4%

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

   3. Taylor expanded in w around 0 71.4%

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

   5. Simplified 71.4%

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

   6. Taylor expanded in dX.v around inf 56.2%

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

      1. unpow2 45.9%

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

      2. unpow2 45.9%

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

      3. swap-sqr 45.9%

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

      4. unpow2 45.9%

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

   8. Simplified 56.2%

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

   if 1.25e8 < dX.w

   1. Initial program 50.9%

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

   3. Taylor expanded in w around 0 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in dY.u around 0 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

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

   9. Taylor expanded in dX.w around inf 41.1%

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

       1. unpow2 45.9%

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

       2. unpow2 45.9%

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

       3. swap-sqr 45.9%

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

       4. unpow2 45.9%

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

   11. Simplified 41.1%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 125000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.w \leq 12000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)))
   (if (<= dX.w 12000000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (pow (hypot t_0 (* (floor w) dY.u)) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (pow (hypot t_0 (* (floor h) dY.v)) 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(d) * dY_46_w;
	float tmp;
	if (dX_46_w <= 12000000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf(t_0, (floorf(w) * dY_46_u)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf(t_0, (floorf(h) * dY_46_v)), 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) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(12000000000.0))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ 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))) ? (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ 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(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_w <= single(12000000000.0))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), (hypot(t_0, (floor(w) * dY_46_u)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot(t_0, (floor(h) * dY_46_v)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 1.2e10

   1. Initial program 70.5%

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

   3. Taylor expanded in w around 0 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in dX.v around inf 55.8%

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

      1. unpow2 45.7%

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

      2. unpow2 45.7%

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

      3. swap-sqr 45.7%

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

      4. unpow2 45.7%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in dY.u around inf 50.5%

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

       1. *-commutative 66.4%

          \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \color{blue}{\left\lfloor w\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right) \cdot 0.5}\right) \]

   11. Simplified 50.5%

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

   if 1.2e10 < dX.w

   1. Initial program 51.7%

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

   3. Taylor expanded in w around 0 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in dY.u around 0 47.4%

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

      1. *-commutative 47.4%

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

   8. Simplified 47.4%

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

   9. Taylor expanded in dX.w around inf 45.8%

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

       1. unpow2 50.2%

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

       2. unpow2 50.2%

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

       3. swap-sqr 50.2%

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

       4. unpow2 50.2%

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

   11. Simplified 45.8%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 12000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.2% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor h) dX.v) 2.0)) (t_1 (* (floor d) dY.w)))
   (if (<= dY.u 2.0)
     (log2 (sqrt (fmax t_0 (pow (hypot t_1 (* (floor h) dY.v)) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (hypot t_1 (* (floor w) dY.u)) 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(h) * dX_46_v), 2.0f);
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dY_46_u <= 2.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf(t_1, (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf(t_1, (floorf(w) * dY_46_u)), 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(h) * dX_46_v) ^ Float32(2.0)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(2.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? t_0 : max(t_0, (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? t_0 : max(t_0, (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ 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(h) * dX_46_v) ^ single(2.0);
	t_1 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dY_46_u <= single(2.0))
		tmp = log2(sqrt(max(t_0, (hypot(t_1, (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, (hypot(t_1, (floor(w) * dY_46_u)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 2

   1. Initial program 68.2%

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

   3. Taylor expanded in w around 0 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in dY.u around 0 61.7%

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

      1. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in dX.v around inf 46.2%

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

       1. unpow2 46.2%

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

       2. unpow2 46.2%

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

       3. swap-sqr 46.2%

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

       4. unpow2 46.2%

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

   11. Simplified 46.2%

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

   if 2 < dY.u

   1. Initial program 68.3%

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

   3. Taylor expanded in w around 0 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in dX.v around inf 56.2%

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

      1. unpow2 38.8%

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

      2. unpow2 38.8%

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

      3. swap-sqr 38.8%

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

      4. unpow2 38.8%

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

   8. Simplified 56.2%

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

   9. Taylor expanded in dY.u around inf 55.1%

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

       1. *-commutative 67.0%

          \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \color{blue}{\left\lfloor w\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right) \cdot 0.5}\right) \]

   11. Simplified 55.1%

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

4. Final simplification 48.4%

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

Alternative 14: 47.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor h) dX.v) 2.0)))
   (if (<= dY.u 1000000.0)
     (log2
      (sqrt
       (fmax t_0 (pow (hypot (* (floor d) dY.w) (* (floor h) dY.v)) 2.0))))
     (log2 (sqrt (fmax t_0 (* (pow (floor w) 2.0) (pow dY.u 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(h) * dX_46_v), 2.0f);
	float tmp;
	if (dY_46_u <= 1000000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf((floorf(d) * dY_46_w), (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf(floorf(w), 2.0f) * powf(dY_46_u, 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(h) * dX_46_v) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(1000000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? t_0 : max(t_0, (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))) : ((Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))) != Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) ? t_0 : max(t_0, Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ 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(h) * dX_46_v) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_u <= single(1000000.0))
		tmp = log2(sqrt(max(t_0, (hypot((floor(d) * dY_46_w), (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, ((floor(w) ^ single(2.0)) * (dY_46_u ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 1e6

   1. Initial program 68.1%

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

   3. Taylor expanded in w around 0 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in dY.u around 0 61.9%

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

      1. *-commutative 61.9%

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

   8. Simplified 61.9%

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

   9. Taylor expanded in dX.v around inf 45.7%

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

       1. unpow2 45.7%

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

       2. unpow2 45.7%

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

       3. swap-sqr 45.8%

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

       4. unpow2 45.8%

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

   11. Simplified 45.8%

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

   if 1e6 < dY.u

   1. Initial program 68.8%

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

   3. Taylor expanded in w around 0 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in dX.v around inf 60.9%

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

      1. unpow2 37.6%

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

      2. unpow2 37.6%

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

      3. swap-sqr 37.6%

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

      4. unpow2 37.6%

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

   8. Simplified 60.9%

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

      1. pow-to-exp 60.7%

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

   10. Applied egg-rr 60.7%

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

   11. Taylor expanded in dY.u around inf 54.7%

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

       1. *-commutative 54.7%

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

   13. Simplified 54.7%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 1000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\\ \mathbf{if}\;dY.u \leq 60000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor h) dX.v) 2.0)))
   (if (<= dY.u 60000.0)
     (log2 (sqrt (fmax t_0 (* (pow (floor d) 2.0) (pow dY.w 2.0)))))
     (log2 (sqrt (fmax t_0 (* (pow (floor w) 2.0) (pow dY.u 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(h) * dX_46_v), 2.0f);
	float tmp;
	if (dY_46_u <= 60000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf(floorf(d), 2.0f) * powf(dY_46_w, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf(floorf(w), 2.0f) * powf(dY_46_u, 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(h) * dX_46_v) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(60000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) : ((Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) != Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0)))) ? t_0 : max(t_0, Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))) : ((Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))) != Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) ? t_0 : max(t_0, Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ 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(h) * dX_46_v) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_u <= single(60000.0))
		tmp = log2(sqrt(max(t_0, ((floor(d) ^ single(2.0)) * (dY_46_w ^ single(2.0))))));
	else
		tmp = log2(sqrt(max(t_0, ((floor(w) ^ single(2.0)) * (dY_46_u ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 6e4

   1. Initial program 67.9%

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

   3. Taylor expanded in w around 0 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in dX.v around inf 52.3%

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

      1. unpow2 45.5%

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

      2. unpow2 45.5%

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

      3. swap-sqr 45.5%

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

      4. unpow2 45.5%

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

   8. Simplified 52.3%

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

      1. pow-to-exp 52.1%

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

   10. Applied egg-rr 52.1%

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

   11. Taylor expanded in dY.w around inf 39.0%

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

       1. *-commutative 39.0%

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

   13. Simplified 39.0%

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

   if 6e4 < dY.u

   1. Initial program 69.5%

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

   3. Taylor expanded in w around 0 69.5%

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

   5. Simplified 69.5%

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

   6. Taylor expanded in dX.v around inf 61.9%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. swap-sqr 39.0%

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

      4. unpow2 39.0%

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

   8. Simplified 61.9%

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

      1. pow-to-exp 61.5%

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

   10. Applied egg-rr 61.5%

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

   11. Taylor expanded in dY.u around inf 54.2%

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

       1. *-commutative 54.2%

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

   13. Simplified 54.2%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 60000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\\ \mathbf{if}\;dY.w \leq 1000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor h) dX.v) 2.0)))
   (if (<= dY.w 1000.0)
     (log2 (sqrt (fmax t_0 (pow (* (floor h) dY.v) 2.0))))
     (log2 (sqrt (fmax t_0 (* (pow (floor d) 2.0) (pow 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(h) * dX_46_v), 2.0f);
	float tmp;
	if (dY_46_w <= 1000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((floorf(h) * dY_46_v), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf(floorf(d), 2.0f) * powf(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 = Float32(floor(h) * dX_46_v) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(1000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(h) * dY_46_v) ^ Float32(2.0)) : (((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) : ((Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) != Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0)))) ? t_0 : max(t_0, Float32((floor(d) ^ Float32(2.0)) * (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(h) * dX_46_v) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_w <= single(1000.0))
		tmp = log2(sqrt(max(t_0, ((floor(h) * dY_46_v) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, ((floor(d) ^ single(2.0)) * (dY_46_w ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 1e3

   1. Initial program 68.2%

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

   3. Taylor expanded in w around 0 68.3%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in dY.u around 0 58.5%

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

      1. *-commutative 58.5%

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

   8. Simplified 58.5%

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

   9. Taylor expanded in dX.v around inf 41.0%

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

       1. unpow2 41.0%

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

       2. unpow2 41.0%

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

       3. swap-sqr 41.0%

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

       4. unpow2 41.0%

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

   11. Simplified 41.0%

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

   12. Taylor expanded in dY.w around 0 34.9%

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

       1. *-commutative 34.9%

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

       2. unpow2 34.9%

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

       3. unpow2 34.9%

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

       4. swap-sqr 34.9%

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

       5. unpow2 34.9%

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

   14. Simplified 34.9%

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

   if 1e3 < dY.w

   1. Initial program 68.0%

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

   3. Taylor expanded in w around 0 68.0%

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

   5. Simplified 68.0%

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

   6. Taylor expanded in dX.v around inf 65.1%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. swap-sqr 64.0%

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

      4. unpow2 64.0%

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

   8. Simplified 65.2%

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

      1. pow-to-exp 64.9%

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

   10. Applied egg-rr 64.9%

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

   11. Taylor expanded in dY.w around inf 57.4%

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

       1. *-commutative 57.4%

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

   13. Simplified 57.4%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 1000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\\ \mathbf{if}\;dY.w \leq 1000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor h) dX.v) 2.0)))
   (if (<= dY.w 1000.0)
     (log2 (sqrt (fmax t_0 (pow (* (floor h) dY.v) 2.0))))
     (log2 (sqrt (fmax 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(h) * dX_46_v), 2.0f);
	float tmp;
	if (dY_46_w <= 1000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((floorf(h) * dY_46_v), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(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 = Float32(floor(h) * dX_46_v) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(1000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(h) * dY_46_v) ^ Float32(2.0)) : (((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != 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))) ? t_0 : max(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(h) * dX_46_v) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_w <= single(1000.0))
		tmp = log2(sqrt(max(t_0, ((floor(h) * dY_46_v) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(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 < 1e3

   1. Initial program 68.2%

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

   3. Taylor expanded in w around 0 68.3%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in dY.u around 0 58.5%

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

      1. *-commutative 58.5%

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

   8. Simplified 58.5%

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

   9. Taylor expanded in dX.v around inf 41.0%

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

       1. unpow2 41.0%

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

       2. unpow2 41.0%

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

       3. swap-sqr 41.0%

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

       4. unpow2 41.0%

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

   11. Simplified 41.0%

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

   12. Taylor expanded in dY.w around 0 34.9%

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

       1. *-commutative 34.9%

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

       2. unpow2 34.9%

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

       3. unpow2 34.9%

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

       4. swap-sqr 34.9%

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

       5. unpow2 34.9%

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

   14. Simplified 34.9%

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

   if 1e3 < dY.w

   1. Initial program 68.0%

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

   3. Taylor expanded in w around 0 68.0%

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

   5. Simplified 68.0%

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

   6. Taylor expanded in dY.u around 0 66.6%

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

      1. *-commutative 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in dX.v around inf 63.9%

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

       1. unpow2 63.9%

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

       2. unpow2 63.9%

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

       3. swap-sqr 64.0%

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

       4. unpow2 64.0%

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

   11. Simplified 64.0%

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

   12. Taylor expanded in dY.w around inf 57.4%

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

       1. *-commutative 57.4%

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

   14. Simplified 57.4%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 1000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.1% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor h) dX.v) 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(h) * dX_46_v), 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(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ 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(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ 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(h) * dX_46_v) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
end

Derivation

1. Initial program 68.2%

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

3. Taylor expanded in w around 0 68.2%

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

5. Simplified 68.2%

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

6. Taylor expanded in dY.u around 0 59.7%

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

   1. *-commutative 59.7%

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

8. Simplified 59.7%

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

9. Taylor expanded in dX.v around inf 44.4%

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

    1. unpow2 44.4%

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

    2. unpow2 44.4%

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

    3. swap-sqr 44.4%

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

    4. unpow2 44.4%

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

11. Simplified 44.4%

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

12. Taylor expanded in dY.w around inf 38.0%

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

    1. *-commutative 38.0%

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

14. Simplified 38.0%

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

15. Final simplification 38.0%

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

Reproduce

herbie shell --seed 2024169 
(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)))))))