Isotropic LOD (LOD)

Percentage Accurate: 68.0% → 71.3%

Time: 44.4s

Alternatives: 17

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right)}\right) \end{array} \end{array} \]

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right)}\right) \end{array} \end{array} \]

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: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloord\right\rfloor \cdot \left\lfloord\right\rfloor\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_4 := \left\lfloord\right\rfloor \cdot dY.w\\ t_5 := \left\lfloord\right\rfloor \cdot dX.w\\ t_6 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_7 := \left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\\ \mathbf{if}\;\mathsf{max}\left(\left(t_6 \cdot t_6 + t_3 \cdot t_3\right) + t_5 \cdot t_5, \left(t_0 \cdot t_0 + t_2 \cdot t_2\right) + t_4 \cdot t_4\right) \leq 3.0000000054977558 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t_6, t_6, t_7 \cdot \left(dX.v \cdot dX.v\right)\right) + t_1 \cdot \left(dX.w \cdot dX.w\right), \mathsf{fma}\left(t_0, t_0, t_7 \cdot \left(dY.v \cdot dY.v\right)\right) + t_1 \cdot \left(dY.w \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({t_3}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

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)))
        3.0000000054977558e+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
      (exp
       (*
        (*
         3.0
         (log
          (cbrt (fmax (pow t_3 2.0) (* (pow (floor w) 2.0) (pow 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) {
	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))) <= 3.0000000054977558e+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(expf(((3.0f * logf(cbrtf(fmaxf(powf(t_3, 2.0f), (powf(floorf(w), 2.0f) * powf(dY_46_u, 2.0f)))))) * 0.5f)));
	}
	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(3.0000000054977558e+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(exp(Float32(Float32(Float32(3.0) * log(cbrt((((t_3 ^ Float32(2.0)) != (t_3 ^ 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))) != Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) ? (t_3 ^ Float32(2.0)) : max((t_3 ^ Float32(2.0)), Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))))))))) * Float32(0.5))));
	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)))) < 3.00000001e38

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   if 3.00000001e38 < (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 7.4%

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

   4. Applied egg-rr 7.4%

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

   5. Taylor expanded in dX.v around inf 14.3%

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

      1. unpow2 14.3%

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

      2. unpow2 14.3%

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

      3. swap-sqr 14.3%

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

      4. unpow2 14.3%

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

   7. Simplified 14.3%

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

      1. add-cube-cbrt 14.3%

         \[\leadsto \log_{2} \left(e^{\log \color{blue}{\left(\left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \cdot \sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} \cdot 0.5}\right) \]

      2. log-prod 14.3%

         \[\leadsto \log_{2} \left(e^{\color{blue}{\left(\log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) + \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]

   9. Applied egg-rr 14.3%

      \[\leadsto \log_{2} \left(e^{\color{blue}{\left(\log \left({\left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)}^{2}\right) + \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]
   10. Step-by-step derivation

       1. log-pow 14.3%

          \[\leadsto \log_{2} \left(e^{\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} + \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right) \cdot 0.5}\right) \]

       2. distribute-lft1-in 14.3%

          \[\leadsto \log_{2} \left(e^{\color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]

       3. metadata-eval 14.3%

          \[\leadsto \log_{2} \left(e^{\left(\color{blue}{3} \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right) \cdot 0.5}\right) \]

   11. Simplified 14.3%

       \[\leadsto \log_{2} \left(e^{\color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]

   12. Taylor expanded in dY.u around inf 19.5%

       \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]
   13. Step-by-step derivation

       1. *-commutative 19.5%

          \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]

   14. Simplified 19.5%

       \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right) \leq 3.0000000054977558 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dX.v \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot \left\lfloord\right\rfloor\right) \cdot \left(dX.w \cdot dX.w\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dY.v \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot \left\lfloord\right\rfloor\right) \cdot \left(dY.w \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\right) \cdot 0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right) \leq 3.0000000054977558 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_4, \mathsf{hypot}\left(t_5, t_2\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t_3, \mathsf{hypot}\left(t_0, t_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({t_2}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

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)))
        3.0000000054977558e+38)
     (log2
      (sqrt
       (fmax
        (pow (hypot t_4 (hypot t_5 t_2)) 2.0)
        (pow (hypot t_3 (hypot t_0 t_1)) 2.0))))
     (log2
      (exp
       (*
        (*
         3.0
         (log
          (cbrt (fmax (pow t_2 2.0) (* (pow (floor w) 2.0) (pow 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) {
	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))) <= 3.0000000054977558e+38f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_4, hypotf(t_5, t_2)), 2.0f), powf(hypotf(t_3, hypotf(t_0, t_1)), 2.0f))));
	} else {
		tmp = log2f(expf(((3.0f * logf(cbrtf(fmaxf(powf(t_2, 2.0f), (powf(floorf(w), 2.0f) * powf(dY_46_u, 2.0f)))))) * 0.5f)));
	}
	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(3.0000000054977558e+38))
		tmp = log2(sqrt((((hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)) != (hypot(t_4, hypot(t_5, t_2)) ^ 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(t_4, hypot(t_5, t_2)) ^ Float32(2.0)) : max((hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)), (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)))))));
	else
		tmp = log2(exp(Float32(Float32(Float32(3.0) * log(cbrt((((t_2 ^ Float32(2.0)) != (t_2 ^ 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))) != Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) ? (t_2 ^ Float32(2.0)) : max((t_2 ^ Float32(2.0)), Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))))))))) * Float32(0.5))));
	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)))) < 3.00000001e38

   1. Initial program 100.0%

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

   4. Applied egg-rr 98.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 99.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 3.00000001e38 < (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 7.4%

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

   4. Applied egg-rr 7.4%

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

   5. Taylor expanded in dX.v around inf 14.3%

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

      1. unpow2 14.3%

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

      2. unpow2 14.3%

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

      3. swap-sqr 14.3%

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

      4. unpow2 14.3%

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

   7. Simplified 14.3%

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

      1. add-cube-cbrt 14.3%

         \[\leadsto \log_{2} \left(e^{\log \color{blue}{\left(\left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \cdot \sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} \cdot 0.5}\right) \]

      2. log-prod 14.3%

         \[\leadsto \log_{2} \left(e^{\color{blue}{\left(\log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) + \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]

   9. Applied egg-rr 14.3%

      \[\leadsto \log_{2} \left(e^{\color{blue}{\left(\log \left({\left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)}^{2}\right) + \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]
   10. Step-by-step derivation

       1. log-pow 14.3%

          \[\leadsto \log_{2} \left(e^{\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} + \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right) \cdot 0.5}\right) \]

       2. distribute-lft1-in 14.3%

          \[\leadsto \log_{2} \left(e^{\color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]

       3. metadata-eval 14.3%

          \[\leadsto \log_{2} \left(e^{\left(\color{blue}{3} \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right) \cdot 0.5}\right) \]

   11. Simplified 14.3%

       \[\leadsto \log_{2} \left(e^{\color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)} \cdot 0.5}\right) \]

   12. Taylor expanded in dY.u around inf 19.5%

       \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]
   13. Step-by-step derivation

       1. *-commutative 19.5%

          \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]

   14. Simplified 19.5%

       \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right) \leq 3.0000000054977558 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\right) \cdot 0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dX.w\\ t_1 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_0, t_1\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_0, \mathsf{hypot}\left(t_1, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)) (t_1 (* (floor w) dX.u)))
   (if (<= dX.v 150000.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot t_0 t_1) 2.0)
        (pow
         (hypot
          (* (floor d) dY.w)
          (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot t_0 (hypot t_1 (* (floor h) dX.v))) 2.0)
        (* (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 = floorf(d) * dX_46_w;
	float t_1 = floorf(w) * dX_46_u;
	float tmp;
	if (dX_46_v <= 150000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, t_1), 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, hypotf(t_1, (floorf(h) * dX_46_v))), 2.0f), (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(d) * dX_46_w)
	t_1 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(150000.0))
		tmp = log2(sqrt((((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (hypot(t_0, t_1) ^ Float32(2.0)) : max((hypot(t_0, t_1) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_0, hypot(t_1, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(t_1, Float32(floor(h) * dX_46_v))) ^ 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))) != Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) ? (hypot(t_0, hypot(t_1, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(t_0, hypot(t_1, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), 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(d) * dX_46_w;
	t_1 = floor(w) * dX_46_u;
	tmp = single(0.0);
	if (dX_46_v <= single(150000.0))
		tmp = log2(sqrt(max((hypot(t_0, t_1) ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, hypot(t_1, (floor(h) * dX_46_v))) ^ single(2.0)), ((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 dX.v < 1.5e5

   1. Initial program 66.7%

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

   4. Applied egg-rr 66.0%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 66.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 66.7%

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

      3. *-commutative 66.7%

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

      4. *-commutative 66.7%

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

      5. *-commutative 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in dX.u around inf 62.8%

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

   if 1.5e5 < dX.v

   1. Initial program 59.7%

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

   4. Applied egg-rr 59.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.2%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 59.7%

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

      3. *-commutative 59.7%

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

      4. *-commutative 59.7%

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

      5. *-commutative 59.7%

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

   6. Simplified 59.7%

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

   7. Taylor expanded in dY.u around inf 54.4%

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

      1. *-commutative 46.1%

         \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]

   9. Simplified 54.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \left\lfloorw\right\rfloor \cdot dX.u\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({t_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(t_0, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u)))
   (if (<= dX.v 150000.0)
     (log2
      (exp
       (*
        0.5
        (log
         (fmax
          (pow t_0 2.0)
          (pow
           (hypot
            (* (floor d) dY.w)
            (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
           2.0))))))
     (log2
      (sqrt
       (fmax
        (pow (hypot (* (floor d) dX.w) (hypot t_0 (* (floor h) dX.v))) 2.0)
        (* (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 = floorf(w) * dX_46_u;
	float tmp;
	if (dX_46_v <= 150000.0f) {
		tmp = log2f(expf((0.5f * logf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf(t_0, (floorf(h) * dX_46_v))), 2.0f), (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(w) * dX_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(150000.0))
		tmp = log2(exp(Float32(Float32(0.5) * log((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ 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))) != Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), 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(w) * dX_46_u;
	tmp = single(0.0);
	if (dX_46_v <= single(150000.0))
		tmp = log2(exp((single(0.5) * log(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot(t_0, (floor(h) * dX_46_v))) ^ single(2.0)), ((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 dX.v < 1.5e5

   1. Initial program 66.7%

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

   4. Applied egg-rr 66.1%

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

   5. Taylor expanded in dX.u around inf 55.5%

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

      1. unpow2 55.5%

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

      2. unpow2 55.5%

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

      3. swap-sqr 55.5%

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

      4. unpow2 55.5%

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

   7. Simplified 55.5%

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

   if 1.5e5 < dX.v

   1. Initial program 59.7%

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

   4. Applied egg-rr 59.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.2%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 59.7%

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

      3. *-commutative 59.7%

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

      4. *-commutative 59.7%

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

      5. *-commutative 59.7%

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

   6. Simplified 59.7%

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

   7. Taylor expanded in dY.u around inf 54.4%

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

      1. *-commutative 46.1%

         \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]

   9. Simplified 54.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dX.w\\ \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_0, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)))
   (if (<= dX.v 150000.0)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow
         (hypot
          (* (floor d) dY.w)
          (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot t_0 (hypot (* (floor w) dX.u) (* (floor h) dX.v))) 2.0)
        (* (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 = floorf(d) * dX_46_w;
	float tmp;
	if (dX_46_v <= 150000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), (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(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(150000.0))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ 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))) != Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) ? (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), 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(d) * dX_46_w;
	tmp = single(0.0);
	if (dX_46_v <= single(150000.0))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), ((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 dX.v < 1.5e5

   1. Initial program 66.7%

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

   4. Applied egg-rr 66.1%

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

   5. Taylor expanded in dX.w around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. swap-sqr 55.1%

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

      5. unpow2 55.1%

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

      6. *-commutative 55.1%

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

   7. Simplified 55.1%

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

      1. exp-to-pow 55.5%

         \[\leadsto \log_{2} \color{blue}{\left({\left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{0.5}\right)} \]

      2. pow1/2 55.5%

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

   9. Applied egg-rr 55.5%

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

   if 1.5e5 < dX.v

   1. Initial program 59.7%

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

   4. Applied egg-rr 59.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.2%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 59.7%

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

      3. *-commutative 59.7%

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

      4. *-commutative 59.7%

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

      5. *-commutative 59.7%

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

   6. Simplified 59.7%

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

   7. Taylor expanded in dY.u around inf 54.4%

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

      1. *-commutative 46.1%

         \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]

   9. Simplified 54.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloord\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.u \leq 10000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_0}^{2}, {\left(\mathsf{hypot}\left(t_1, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, t_0\right)\right)}^{2}, {t_1}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v)) (t_1 (* (floor d) dY.w)))
   (if (<= dX.u 10000000.0)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow (hypot t_1 (hypot (* (floor w) dY.u) (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt (fmax (pow (hypot (* (floor w) dX.u) t_0) 2.0) (pow 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(h) * dX_46_v;
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_u <= 10000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf(t_1, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), t_0), 2.0f), powf(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(h) * dX_46_v)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(10000000.0))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ 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))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0)), (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(h) * dX_46_v;
	t_1 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_u <= single(10000000.0))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot(t_1, hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), t_0) ^ single(2.0)), (t_1 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1e7

   1. Initial program 66.7%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in dX.v around inf 55.1%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. swap-sqr 55.1%

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

      4. unpow2 55.1%

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

   7. Simplified 55.1%

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

   8. Taylor expanded in dX.v around 0 55.6%

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

      1. *-commutative 55.6%

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

      2. *-commutative 55.6%

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

      3. *-commutative 55.6%

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

   10. Simplified 55.6%

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

   if 1e7 < dX.u

   1. Initial program 58.4%

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

   4. Applied egg-rr 57.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 57.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 58.4%

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

      3. *-commutative 58.4%

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

      4. *-commutative 58.4%

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

      5. *-commutative 58.4%

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

   6. Simplified 58.4%

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

   7. Taylor expanded in dY.w around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. swap-sqr 55.1%

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

      5. unpow2 55.1%

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

   9. Simplified 55.1%

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

   10. Taylor expanded in dX.w around 0 52.3%

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

       1. unpow2 52.3%

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

       2. unpow2 52.3%

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

       3. swap-sqr 52.3%

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

       4. unpow2 52.3%

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

       5. unpow2 52.3%

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

       6. unpow2 52.3%

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

       7. swap-sqr 52.3%

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

       8. unpow2 52.3%

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

   12. Simplified 52.3%

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

       1. expm1-log1p-u 51.8%

          \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\right)\right)} \]

       2. expm1-udef 51.8%

          \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} - 1\right)} \]

   14. Applied egg-rr 51.8%

       \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 51.8%

          \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\right)\right)} \]

       2. expm1-log1p 52.3%

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

   16. Simplified 52.3%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 10000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.u \leq 10000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(t_0, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {t_0}^{2}\right)}\right)\\ \end{array} \end{array} \]

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.u 10000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (pow (hypot t_0 (hypot (* (floor w) dY.u) (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
        (pow 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(d) * dY_46_w;
	float tmp;
	if (dX_46_u <= 10000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf(t_0, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), powf(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(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(10000000.0))
		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(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))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), (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(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_u <= single(10000000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot(t_0, hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), (t_0 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1e7

   1. Initial program 66.7%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in dX.w around inf 56.4%

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

      1. *-commutative 56.4%

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

      2. unpow2 56.4%

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

      3. unpow2 56.4%

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

      4. swap-sqr 56.4%

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

      5. unpow2 56.4%

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

      6. *-commutative 56.4%

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

   7. Simplified 56.4%

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

      1. exp-to-pow 56.9%

         \[\leadsto \log_{2} \color{blue}{\left({\left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{0.5}\right)} \]

      2. pow1/2 56.9%

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

   9. Applied egg-rr 56.9%

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

   if 1e7 < dX.u

   1. Initial program 58.4%

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

   4. Applied egg-rr 57.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 57.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 58.4%

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

      3. *-commutative 58.4%

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

      4. *-commutative 58.4%

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

      5. *-commutative 58.4%

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

   6. Simplified 58.4%

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

   7. Taylor expanded in dY.w around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. swap-sqr 55.1%

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

      5. unpow2 55.1%

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

   9. Simplified 55.1%

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

   10. Taylor expanded in dX.w around 0 52.3%

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

       1. unpow2 52.3%

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

       2. unpow2 52.3%

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

       3. swap-sqr 52.3%

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

       4. unpow2 52.3%

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

       5. unpow2 52.3%

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

       6. unpow2 52.3%

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

       7. swap-sqr 52.3%

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

       8. unpow2 52.3%

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

   12. Simplified 52.3%

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

       1. expm1-log1p-u 51.8%

          \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\right)\right)} \]

       2. expm1-udef 51.8%

          \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} - 1\right)} \]

   14. Applied egg-rr 51.8%

       \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 51.8%

          \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\right)\right)} \]

       2. expm1-log1p 52.3%

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

   16. Simplified 52.3%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 10000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dX.w\\ t_1 := \left\lfloord\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.u \leq 10000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_0}^{2}, {\left(\mathsf{hypot}\left(t_1, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_0, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {t_1}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)) (t_1 (* (floor d) dY.w)))
   (if (<= dX.u 10000000.0)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow (hypot t_1 (hypot (* (floor w) dY.u) (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot t_0 (hypot (* (floor w) dX.u) (* (floor h) dX.v))) 2.0)
        (pow 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) * dX_46_w;
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_u <= 10000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf(t_1, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), powf(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) * dX_46_w)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(10000000.0))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ 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))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), (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) * dX_46_w;
	t_1 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_u <= single(10000000.0))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot(t_1, hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), (t_1 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1e7

   1. Initial program 66.7%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in dX.w around inf 56.4%

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

      1. *-commutative 56.4%

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

      2. unpow2 56.4%

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

      3. unpow2 56.4%

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

      4. swap-sqr 56.4%

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

      5. unpow2 56.4%

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

      6. *-commutative 56.4%

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

   7. Simplified 56.4%

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

      1. exp-to-pow 56.9%

         \[\leadsto \log_{2} \color{blue}{\left({\left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{0.5}\right)} \]

      2. pow1/2 56.9%

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

   9. Applied egg-rr 56.9%

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

   if 1e7 < dX.u

   1. Initial program 58.4%

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

   4. Applied egg-rr 57.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 57.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 58.4%

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

      3. *-commutative 58.4%

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

      4. *-commutative 58.4%

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

      5. *-commutative 58.4%

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

   6. Simplified 58.4%

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

   7. Taylor expanded in dY.w around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. swap-sqr 55.1%

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

      5. unpow2 55.1%

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

   9. Simplified 55.1%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 10000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dX.w\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(t_1, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_0, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {t_1}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)) (t_1 (* (floor w) dY.u)))
   (if (<= dX.v 150000.0)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow (hypot (* (floor d) dY.w) (hypot t_1 (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot t_0 (hypot (* (floor w) dX.u) (* (floor h) dX.v))) 2.0)
        (pow 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) * dX_46_w;
	float t_1 = floorf(w) * dY_46_u;
	float tmp;
	if (dX_46_v <= 150000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf(t_1, (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), powf(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) * dX_46_w)
	t_1 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(150000.0))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ 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))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(t_1, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), (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) * dX_46_w;
	t_1 = floor(w) * dY_46_u;
	tmp = single(0.0);
	if (dX_46_v <= single(150000.0))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot(t_1, (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), (t_1 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.5e5

   1. Initial program 66.7%

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

   4. Applied egg-rr 66.1%

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

   5. Taylor expanded in dX.w around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. swap-sqr 55.1%

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

      5. unpow2 55.1%

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

      6. *-commutative 55.1%

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

   7. Simplified 55.1%

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

      1. exp-to-pow 55.5%

         \[\leadsto \log_{2} \color{blue}{\left({\left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{0.5}\right)} \]

      2. pow1/2 55.5%

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

   9. Applied egg-rr 55.5%

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

   if 1.5e5 < dX.v

   1. Initial program 59.7%

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

   4. Applied egg-rr 59.2%

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

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

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

      1. *-commutative 46.1%

         \[\leadsto \log_{2} \left(e^{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)}\right)\right) \cdot 0.5}\right) \]

   7. Simplified 54.2%

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

      1. exp-to-pow 54.4%

         \[\leadsto \log_{2} \color{blue}{\left({\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)\right)}^{0.5}\right)} \]

      2. pow1/2 54.4%

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

      3. *-commutative 54.4%

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

      4. *-commutative 54.4%

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

      5. pow-prod-down 54.4%

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

   9. Applied egg-rr 54.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 150000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.v \leq 1:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.v 1.0)
   (log2
    (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (* (floor d) dY.w) 2.0))))
   (log2
    (exp
     (*
      0.5
      (log
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (* (pow (floor h) 2.0) (pow 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 tmp;
	if (dY_46_v <= 1.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf((floorf(d) * dY_46_w), 2.0f))));
	} else {
		tmp = log2f(expf((0.5f * logf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), (powf(floorf(h), 2.0f) * powf(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)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(1.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	else
		tmp = log2(exp(Float32(Float32(0.5) * log((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (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)), Float32((floor(h) ^ Float32(2.0)) * (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)
	tmp = single(0.0);
	if (dY_46_v <= single(1.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
	else
		tmp = log2(exp((single(0.5) * log(max(((floor(h) * dX_46_v) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 1

   1. Initial program 67.3%

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

   4. Applied egg-rr 66.7%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 67.3%

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

      3. *-commutative 67.3%

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

      4. *-commutative 67.3%

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

      5. *-commutative 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in dY.w around inf 54.0%

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

      1. *-commutative 54.0%

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

      2. unpow2 54.0%

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

      3. unpow2 54.0%

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

      4. swap-sqr 54.0%

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

      5. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in dX.w around inf 36.9%

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

       1. *-commutative 36.9%

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

       2. unpow2 36.9%

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

       3. unpow2 36.9%

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

       4. swap-sqr 36.9%

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

       5. unpow2 36.9%

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

       6. *-commutative 36.9%

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

   12. Simplified 36.9%

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

   if 1 < dY.v

   1. Initial program 59.1%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in dX.v around inf 51.4%

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

      1. unpow2 51.4%

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

      2. unpow2 51.4%

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

      3. swap-sqr 51.4%

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

      4. unpow2 51.4%

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

   7. Simplified 51.4%

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

   8. Taylor expanded in dY.v around inf 43.6%

      \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}\right)\right) \cdot 0.5}\right) \]
   9. Step-by-step derivation

      1. *-commutative 43.6%

         \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]

   10. Simplified 43.6%

       \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 1:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.w \leq 15000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot {dY.w}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.w 15000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor w) dX.u) 2.0)
      (* (floor d) (* (floor d) (pow dY.w 2.0))))))
   (log2
    (exp
     (*
      0.5
      (log
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (* (pow (floor h) 2.0) (pow 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 tmp;
	if (dX_46_w <= 15000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (floorf(d) * (floorf(d) * powf(dY_46_w, 2.0f))))));
	} else {
		tmp = log2f(expf((0.5f * logf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), (powf(floorf(h), 2.0f) * powf(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)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(15000.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32(floor(d) * Float32(floor(d) * (dY_46_w ^ Float32(2.0)))) : ((Float32(floor(d) * Float32(floor(d) * (dY_46_w ^ Float32(2.0)))) != Float32(floor(d) * Float32(floor(d) * (dY_46_w ^ Float32(2.0))))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32(floor(d) * Float32(floor(d) * (dY_46_w ^ Float32(2.0)))))))));
	else
		tmp = log2(exp(Float32(Float32(0.5) * log((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (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)), Float32((floor(h) ^ Float32(2.0)) * (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)
	tmp = single(0.0);
	if (dX_46_w <= single(15000.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (floor(d) * (floor(d) * (dY_46_w ^ single(2.0)))))));
	else
		tmp = log2(exp((single(0.5) * log(max(((floor(d) * dX_46_w) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 15000

   1. Initial program 64.7%

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

   4. Applied egg-rr 64.1%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 64.2%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 64.7%

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

      3. *-commutative 64.7%

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

      4. *-commutative 64.7%

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

      5. *-commutative 64.7%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in dY.w around inf 50.5%

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

      1. *-commutative 50.5%

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

      2. unpow2 50.5%

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

      3. unpow2 50.5%

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

      4. swap-sqr 50.5%

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

      5. unpow2 50.5%

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

   9. Simplified 50.5%

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

   10. Taylor expanded in dX.u around inf 37.9%

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

       1. unpow2 37.9%

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

       2. unpow2 37.9%

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

       3. swap-sqr 37.9%

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

       4. unpow2 37.9%

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

   12. Simplified 37.9%

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

       1. unpow2 37.9%

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

       2. unswap-sqr 37.9%

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

       3. associate-*l* 37.9%

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

       4. pow2 37.9%

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

   14. Applied egg-rr 37.9%

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

   if 15000 < dX.w

   1. Initial program 67.6%

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

   4. Applied egg-rr 66.9%

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

   5. Taylor expanded in dX.w around inf 57.3%

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

      1. *-commutative 57.3%

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

      2. unpow2 57.3%

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

      3. unpow2 57.3%

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

      4. swap-sqr 57.3%

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

      5. unpow2 57.3%

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

      6. *-commutative 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in dY.v around inf 50.8%

      \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}\right)\right) \cdot 0.5}\right) \]
   9. Step-by-step derivation

      1. *-commutative 38.4%

         \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]

   10. Simplified 50.8%

       \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 15000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot {dY.w}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.v \leq 8000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \left\lfloorw\right\rfloor \cdot dX.u\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.v 8000000.0)
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor d) dX.w) (* (floor w) dX.u)) 2.0)
      (pow (* (floor d) dY.w) 2.0))))
   (log2
    (exp
     (*
      0.5
      (log
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (* (pow (floor h) 2.0) (pow 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 tmp;
	if (dY_46_v <= 8000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), (floorf(w) * dX_46_u)), 2.0f), powf((floorf(d) * dY_46_w), 2.0f))));
	} else {
		tmp = log2f(expf((0.5f * logf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), (powf(floorf(h), 2.0f) * powf(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)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(8000000.0))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), Float32(floor(w) * dX_46_u)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), Float32(floor(w) * dX_46_u)) ^ 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))) ? (hypot(Float32(floor(d) * dX_46_w), Float32(floor(w) * dX_46_u)) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), Float32(floor(w) * dX_46_u)) ^ Float32(2.0)), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	else
		tmp = log2(exp(Float32(Float32(0.5) * log((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (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)), Float32((floor(h) ^ Float32(2.0)) * (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)
	tmp = single(0.0);
	if (dY_46_v <= single(8000000.0))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), (floor(w) * dX_46_u)) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
	else
		tmp = log2(exp((single(0.5) * log(max(((floor(h) * dX_46_v) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 8e6

   1. Initial program 66.9%

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

   4. Applied egg-rr 66.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 66.3%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 66.9%

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

      3. *-commutative 66.9%

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

      4. *-commutative 66.9%

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

      5. *-commutative 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in dY.w around inf 53.7%

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

      1. *-commutative 53.7%

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

      2. unpow2 53.7%

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

      3. unpow2 53.7%

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

      4. swap-sqr 53.7%

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

      5. unpow2 53.7%

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

   9. Simplified 53.7%

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

   10. Taylor expanded in dX.u around inf 46.9%

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

   if 8e6 < dY.v

   1. Initial program 58.2%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in dX.v around inf 50.0%

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

      1. unpow2 50.0%

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

      2. unpow2 50.0%

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

      3. swap-sqr 50.0%

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

      4. unpow2 50.0%

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

   7. Simplified 50.0%

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

   8. Taylor expanded in dY.v around inf 47.0%

      \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}\right)\right) \cdot 0.5}\right) \]
   9. Step-by-step derivation

      1. *-commutative 47.0%

         \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]

   10. Simplified 47.0%

       \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 8000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \left\lfloorw\right\rfloor \cdot dX.u\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.w \leq 15780:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.w 15780.0)
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
      (pow (* (floor d) dY.w) 2.0))))
   (log2
    (exp
     (*
      0.5
      (log
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (* (pow (floor h) 2.0) (pow 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 tmp;
	if (dX_46_w <= 15780.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), powf((floorf(d) * dY_46_w), 2.0f))));
	} else {
		tmp = log2f(expf((0.5f * logf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), (powf(floorf(h), 2.0f) * powf(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)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(15780.0))
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), 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))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	else
		tmp = log2(exp(Float32(Float32(0.5) * log((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (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)), Float32((floor(h) ^ Float32(2.0)) * (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)
	tmp = single(0.0);
	if (dX_46_w <= single(15780.0))
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
	else
		tmp = log2(exp((single(0.5) * log(max(((floor(d) * dX_46_w) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 15780

   1. Initial program 64.9%

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

   4. Applied egg-rr 64.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 64.4%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 64.9%

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

      3. *-commutative 64.9%

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

      4. *-commutative 64.9%

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

      5. *-commutative 64.9%

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

   6. Simplified 64.9%

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

   7. Taylor expanded in dY.w around inf 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. swap-sqr 50.7%

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

      5. unpow2 50.7%

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

   9. Simplified 50.7%

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

   10. Taylor expanded in dX.w around 0 46.3%

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

       1. unpow2 46.3%

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

       2. unpow2 46.3%

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

       3. swap-sqr 46.3%

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

       4. unpow2 46.3%

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

       5. unpow2 46.3%

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

       6. unpow2 46.3%

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

       7. swap-sqr 46.3%

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

       8. unpow2 46.3%

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

   12. Simplified 46.3%

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

       1. expm1-log1p-u 45.9%

          \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\right)\right)} \]

       2. expm1-udef 45.9%

          \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} - 1\right)} \]

   14. Applied egg-rr 45.9%

       \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 45.9%

          \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\right)\right)} \]

       2. expm1-log1p 46.3%

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

   16. Simplified 46.3%

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

   if 15780 < dX.w

   1. Initial program 66.9%

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

   4. Applied egg-rr 66.3%

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

   5. Taylor expanded in dX.w around inf 57.3%

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

      1. *-commutative 57.3%

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

      2. unpow2 57.3%

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

      3. unpow2 57.3%

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

      4. swap-sqr 57.3%

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

      5. unpow2 57.3%

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

      6. *-commutative 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in dY.v around inf 50.6%

      \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}\right)\right) \cdot 0.5}\right) \]
   9. Step-by-step derivation

      1. *-commutative 37.3%

         \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]

   10. Simplified 50.6%

       \[\leadsto \log_{2} \left(e^{\log \left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}}\right)\right) \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 15780:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(e^{0.5 \cdot \log \left(\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.u \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot {dY.w}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.u 100.0)
   (log2
    (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (* (floor d) dY.w) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor w) dX.u) 2.0)
      (* (floor d) (* (floor d) (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 tmp;
	if (dX_46_u <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf((floorf(d) * dY_46_w), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (floorf(d) * (floorf(d) * 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)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(100.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32(floor(d) * Float32(floor(d) * (dY_46_w ^ Float32(2.0)))) : ((Float32(floor(d) * Float32(floor(d) * (dY_46_w ^ Float32(2.0)))) != Float32(floor(d) * Float32(floor(d) * (dY_46_w ^ Float32(2.0))))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32(floor(d) * 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)
	tmp = single(0.0);
	if (dX_46_u <= single(100.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (floor(d) * (floor(d) * (dY_46_w ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 100

   1. Initial program 66.8%

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

   4. Applied egg-rr 66.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 66.3%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. *-commutative 66.8%

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

      5. *-commutative 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in dY.w around inf 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. swap-sqr 50.7%

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

      5. unpow2 50.7%

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

   9. Simplified 50.7%

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

   10. Taylor expanded in dX.w around inf 36.9%

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

       1. *-commutative 36.9%

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

       2. unpow2 36.9%

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

       3. unpow2 36.9%

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

       4. swap-sqr 36.9%

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

       5. unpow2 36.9%

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

       6. *-commutative 36.9%

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

   12. Simplified 36.9%

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

   if 100 < dX.u

   1. Initial program 60.5%

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

   4. Applied egg-rr 59.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 60.5%

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

      3. *-commutative 60.5%

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

      4. *-commutative 60.5%

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

      5. *-commutative 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in dY.w around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. swap-sqr 53.9%

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

      5. unpow2 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in dX.u around inf 47.6%

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

       1. unpow2 47.6%

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

       2. unpow2 47.6%

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

       3. swap-sqr 47.6%

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

       4. unpow2 47.6%

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

   12. Simplified 47.6%

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

       1. unpow2 47.6%

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

       2. unswap-sqr 47.6%

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

       3. associate-*l* 47.6%

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

       4. pow2 47.6%

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

   14. Applied egg-rr 47.6%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot {dY.w}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.u \leq 0.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.u 0.5)
     (log2 (sqrt (fmax (pow (* (floor h) dX.v) 2.0) t_0)))
     (log2 (sqrt (fmax (pow (* (floor w) dX.u) 2.0) t_0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_u <= 0.5f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), t_0)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(0.5))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (floor(d) * dY_46_w) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(0.5))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.5

   1. Initial program 66.6%

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

   4. Applied egg-rr 65.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 66.0%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 66.6%

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

      3. *-commutative 66.6%

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

      4. *-commutative 66.6%

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

      5. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in dY.w around inf 50.5%

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

      1. *-commutative 50.5%

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

      2. unpow2 50.5%

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

      3. unpow2 50.5%

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

      4. swap-sqr 50.5%

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

      5. unpow2 50.5%

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

   9. Simplified 50.5%

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

   10. Taylor expanded in dX.v around inf 35.6%

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

       1. unpow2 35.6%

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

       2. unpow2 35.6%

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

       3. swap-sqr 35.6%

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

       4. unpow2 35.6%

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

   12. Simplified 35.6%

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

   if 0.5 < dX.u

   1. Initial program 62.1%

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

   4. Applied egg-rr 61.4%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 61.4%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 62.1%

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

      3. *-commutative 62.1%

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

      4. *-commutative 62.1%

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

      5. *-commutative 62.1%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in dY.w around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. swap-sqr 53.9%

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

      5. unpow2 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in dX.u around inf 45.2%

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

       1. unpow2 45.2%

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

       2. unpow2 45.2%

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

       3. swap-sqr 45.2%

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

       4. unpow2 45.2%

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

   12. Simplified 45.2%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 0.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.u \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.u 100.0)
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) t_0)))
     (log2 (sqrt (fmax (pow (* (floor w) dX.u) 2.0) t_0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_u <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), t_0)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(100.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (floor(d) * dY_46_w) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(100.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 100

   1. Initial program 66.8%

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

   4. Applied egg-rr 66.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 66.3%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. *-commutative 66.8%

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

      5. *-commutative 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in dY.w around inf 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. swap-sqr 50.7%

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

      5. unpow2 50.7%

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

   9. Simplified 50.7%

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

   10. Taylor expanded in dX.w around inf 36.9%

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

       1. *-commutative 36.9%

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

       2. unpow2 36.9%

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

       3. unpow2 36.9%

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

       4. swap-sqr 36.9%

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

       5. unpow2 36.9%

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

       6. *-commutative 36.9%

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

   12. Simplified 36.9%

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

   if 100 < dX.u

   1. Initial program 60.5%

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

   4. Applied egg-rr 59.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 60.5%

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

      3. *-commutative 60.5%

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

      4. *-commutative 60.5%

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

      5. *-commutative 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in dY.w around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. swap-sqr 53.9%

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

      5. unpow2 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in dX.u around inf 47.6%

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

       1. unpow2 47.6%

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

       2. unpow2 47.6%

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

       3. swap-sqr 47.6%

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

       4. unpow2 47.6%

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

   12. Simplified 47.6%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.4% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor w) dX.u) 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(w) * dX_46_u), 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(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ 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(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ 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(w) * dX_46_u) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
end

Derivation

1. Initial program 65.3%

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

4. Applied egg-rr 64.6%

   \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-def 64.7%

      \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

   2. expm1-log1p 65.3%

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

   3. *-commutative 65.3%

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

   4. *-commutative 65.3%

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

   5. *-commutative 65.3%

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

6. Simplified 65.3%

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

7. Taylor expanded in dY.w around inf 51.5%

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

   1. *-commutative 51.5%

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

   2. unpow2 51.5%

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

   3. unpow2 51.5%

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

   4. swap-sqr 51.5%

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

   5. unpow2 51.5%

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

9. Simplified 51.5%

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

10. Taylor expanded in dX.u around inf 36.4%

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

    1. unpow2 36.4%

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

    2. unpow2 36.4%

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

    3. swap-sqr 36.4%

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

    4. unpow2 36.4%

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

12. Simplified 36.4%

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

13. Final simplification 36.4%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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)))))))