Isotropic LOD (LOD)

Percentage Accurate: 68.3% → 70.8%

Time: 17.9s

Alternatives: 5

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 70.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_2 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_3 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_4 := t\_3 \cdot t\_3\\ t_5 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_6 := dY.w \cdot \left\lfloor d\right\rfloor \\ t_7 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_8 := t\_7 \cdot t\_7\\ t_9 := t\_8 + \left(t\_2 \cdot t\_2 + t\_5 \cdot t\_5\right)\\ t_10 := t\_6 \cdot t\_6\\ \mathbf{if}\;\mathsf{max}\left(t\_9, t\_10 + \left(t\_1 \cdot t\_1 + t\_4\right)\right) \leq 2.6000000588511473 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_9, \left(\left(dY.v \cdot dY.v\right) \cdot t\_0 + t\_4\right) + t\_10\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {t\_2}^{2}\right) + t\_8, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left(t\_0 \cdot dY.v\right) \cdot dY.v\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* dY.v (floor h)))
        (t_2 (* dX.v (floor h)))
        (t_3 (* dY.u (floor w)))
        (t_4 (* t_3 t_3))
        (t_5 (* dX.u (floor w)))
        (t_6 (* dY.w (floor d)))
        (t_7 (* dX.w (floor d)))
        (t_8 (* t_7 t_7))
        (t_9 (+ t_8 (+ (* t_2 t_2) (* t_5 t_5))))
        (t_10 (* t_6 t_6)))
   (if (<= (fmax t_9 (+ t_10 (+ (* t_1 t_1) t_4))) 2.6000000588511473e+38)
     (log2 (sqrt (fmax t_9 (+ (+ (* (* dY.v dY.v) t_0) t_4) t_10))))
     (log2
      (sqrt
       (fmax
        (+ (fma (floor w) (* (* dX.u dX.u) (floor w)) (pow t_2 2.0)) t_8)
        (fma (* (pow (floor d) 2.0) dY.w) dY.w (* (* t_0 dY.v) dY.v))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = dY_46_v * floorf(h);
	float t_2 = dX_46_v * floorf(h);
	float t_3 = dY_46_u * floorf(w);
	float t_4 = t_3 * t_3;
	float t_5 = dX_46_u * floorf(w);
	float t_6 = dY_46_w * floorf(d);
	float t_7 = dX_46_w * floorf(d);
	float t_8 = t_7 * t_7;
	float t_9 = t_8 + ((t_2 * t_2) + (t_5 * t_5));
	float t_10 = t_6 * t_6;
	float tmp;
	if (fmaxf(t_9, (t_10 + ((t_1 * t_1) + t_4))) <= 2.6000000588511473e+38f) {
		tmp = log2f(sqrtf(fmaxf(t_9, ((((dY_46_v * dY_46_v) * t_0) + t_4) + t_10))));
	} else {
		tmp = log2f(sqrtf(fmaxf((fmaf(floorf(w), ((dX_46_u * dX_46_u) * floorf(w)), powf(t_2, 2.0f)) + t_8), fmaf((powf(floorf(d), 2.0f) * dY_46_w), dY_46_w, ((t_0 * dY_46_v) * dY_46_v)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(dY_46_v * floor(h))
	t_2 = Float32(dX_46_v * floor(h))
	t_3 = Float32(dY_46_u * floor(w))
	t_4 = Float32(t_3 * t_3)
	t_5 = Float32(dX_46_u * floor(w))
	t_6 = Float32(dY_46_w * floor(d))
	t_7 = Float32(dX_46_w * floor(d))
	t_8 = Float32(t_7 * t_7)
	t_9 = Float32(t_8 + Float32(Float32(t_2 * t_2) + Float32(t_5 * t_5)))
	t_10 = Float32(t_6 * t_6)
	tmp = Float32(0.0)
	if (((t_9 != t_9) ? Float32(t_10 + Float32(Float32(t_1 * t_1) + t_4)) : ((Float32(t_10 + Float32(Float32(t_1 * t_1) + t_4)) != Float32(t_10 + Float32(Float32(t_1 * t_1) + t_4))) ? t_9 : max(t_9, Float32(t_10 + Float32(Float32(t_1 * t_1) + t_4))))) <= Float32(2.6000000588511473e+38))
		tmp = log2(sqrt(((t_9 != t_9) ? Float32(Float32(Float32(Float32(dY_46_v * dY_46_v) * t_0) + t_4) + t_10) : ((Float32(Float32(Float32(Float32(dY_46_v * dY_46_v) * t_0) + t_4) + t_10) != Float32(Float32(Float32(Float32(dY_46_v * dY_46_v) * t_0) + t_4) + t_10)) ? t_9 : max(t_9, Float32(Float32(Float32(Float32(dY_46_v * dY_46_v) * t_0) + t_4) + t_10))))));
	else
		tmp = log2(sqrt(((Float32(fma(floor(w), Float32(Float32(dX_46_u * dX_46_u) * floor(w)), (t_2 ^ Float32(2.0))) + t_8) != Float32(fma(floor(w), Float32(Float32(dX_46_u * dX_46_u) * floor(w)), (t_2 ^ Float32(2.0))) + t_8)) ? fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32(t_0 * dY_46_v) * dY_46_v)) : ((fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32(t_0 * dY_46_v) * dY_46_v)) != fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32(t_0 * dY_46_v) * dY_46_v))) ? Float32(fma(floor(w), Float32(Float32(dX_46_u * dX_46_u) * floor(w)), (t_2 ^ Float32(2.0))) + t_8) : max(Float32(fma(floor(w), Float32(Float32(dX_46_u * dX_46_u) * floor(w)), (t_2 ^ Float32(2.0))) + t_8), fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32(t_0 * dY_46_v) * dY_46_v)))))));
	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)))) < 2.60000006e38

   1. Initial program 99.9%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. swap-sqr N/A

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

      5. pow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-*.f32 100.0

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

   4. Applied rewrites 100.0%

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

   if 2.60000006e38 < (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 7.5%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. swap-sqr N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f32 N/A

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

   4. Applied rewrites 12.5%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. exp-prod N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. unpow1 N/A

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

      13. metadata-eval N/A

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

      14. pow-to-exp N/A

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

      15. rem-log-exp N/A

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

      16. lower-pow.f32 N/A

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

      17. lower-exp.f32 N/A

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

      18. rem-log-exp N/A

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

      19. pow-to-exp N/A

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

      20. metadata-eval N/A

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

      21. unpow1 N/A

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

      22. lower-log.f32 10.5

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

   6. Applied rewrites 10.5%

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

      1. lift-+.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-exp.f32 N/A

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

      4. pow-exp N/A

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

      5. *-commutative N/A

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

      6. lift-log.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. pow-to-exp N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. lift-*.f32 N/A

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

      14. unpow-prod-down N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

   8. Applied rewrites 11.2%

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

   9. Taylor expanded in dY.u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-fma.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lower-pow.f32 N/A

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

       8. lower-floor.f32 N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f32 N/A

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

       13. lower-*.f32 N/A

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

       14. lower-pow.f32 N/A

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

       15. lower-floor.f32 13.1

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

   11. Applied rewrites 13.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right), \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)\right) \leq 2.6000000588511473 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right), \left(\left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right) + \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right) + \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 50000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), t\_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w (floor d)) 2.0))
        (t_1 (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (floor h)) 2.0))))
   (if (<= dY.w 50000000.0)
     (log2
      (sqrt
       (fmax
        (+ t_0 (+ (pow (* dX.u (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
        t_1)))
     (log2 (sqrt (fmax t_0 (+ (pow (* dY.w (floor d)) 2.0) t_1)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dX_46_w * floorf(d)), 2.0f);
	float t_1 = powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dY_46_w <= 50000000.0f) {
		tmp = log2f(sqrtf(fmaxf((t_0 + (powf((dX_46_u * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f))), t_1)));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf((dY_46_w * floorf(d)), 2.0f) + t_1))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dX_46_w * floor(d)) ^ Float32(2.0)
	t_1 = Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(50000000.0))
		tmp = log2(sqrt(((Float32(t_0 + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) != Float32(t_0 + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))))) ? t_1 : ((t_1 != t_1) ? Float32(t_0 + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) : max(Float32(t_0 + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), t_1)))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1) : ((Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1) != Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1)) ? t_0 : max(t_0, Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1))))));
	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 = (dX_46_w * floor(d)) ^ single(2.0);
	t_1 = ((dY_46_u * floor(w)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0));
	tmp = single(0.0);
	if (dY_46_w <= single(50000000.0))
		tmp = log2(sqrt(max((t_0 + (((dX_46_u * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0)))), t_1)));
	else
		tmp = log2(sqrt(max(t_0, (((dY_46_w * floor(d)) ^ single(2.0)) + t_1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 5e7

   1. Initial program 70.4%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. swap-sqr N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f32 N/A

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

   4. Applied rewrites 46.6%

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

   5. Taylor expanded in dY.w around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. lower-floor.f32 46.7

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

   7. Applied rewrites 46.5%

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

   9. Applied rewrites 63.7%

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

   if 5e7 < dY.w

   1. Initial program 55.6%

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

   3. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 56.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 50000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_1 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ \mathbf{if}\;dY.w \leq 50000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 \cdot t\_0 + \left(t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* dX.w (floor d)))
        (t_1 (* dX.v (floor h)))
        (t_2 (* dX.u (floor w))))
   (if (<= dY.w 50000000.0)
     (log2
      (sqrt
       (fmax
        (+ (* t_0 t_0) (+ (* t_1 t_1) (* t_2 t_2)))
        (* (* (pow (floor w) 2.0) dY.u) dY.u))))
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (+
         (pow (* dY.w (floor d)) 2.0)
         (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (floor h)) 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 = dX_46_w * floorf(d);
	float t_1 = dX_46_v * floorf(h);
	float t_2 = dX_46_u * floorf(w);
	float tmp;
	if (dY_46_w <= 50000000.0f) {
		tmp = log2f(sqrtf(fmaxf(((t_0 * t_0) + ((t_1 * t_1) + (t_2 * t_2))), ((powf(floorf(w), 2.0f) * dY_46_u) * dY_46_u))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), (powf((dY_46_w * floorf(d)), 2.0f) + (powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * floorf(h)), 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(dX_46_w * floor(d))
	t_1 = Float32(dX_46_v * floor(h))
	t_2 = Float32(dX_46_u * floor(w))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(50000000.0))
		tmp = log2(sqrt(((Float32(Float32(t_0 * t_0) + Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))) != Float32(Float32(t_0 * t_0) + Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)))) ? Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) : ((Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) != Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)) ? Float32(Float32(t_0 * t_0) + Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))) : max(Float32(Float32(t_0 * t_0) + Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))), Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u))))));
	else
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) : ((Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) != Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ 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 = dX_46_w * floor(d);
	t_1 = dX_46_v * floor(h);
	t_2 = dX_46_u * floor(w);
	tmp = single(0.0);
	if (dY_46_w <= single(50000000.0))
		tmp = log2(sqrt(max(((t_0 * t_0) + ((t_1 * t_1) + (t_2 * t_2))), (((floor(w) ^ single(2.0)) * dY_46_u) * dY_46_u))));
	else
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (((dY_46_w * floor(d)) ^ single(2.0)) + (((dY_46_u * floor(w)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 5e7

   1. Initial program 70.4%

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

   3. Taylor expanded in dY.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.8

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

   5. Applied rewrites 56.8%

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

   if 5e7 < dY.w

   1. Initial program 55.6%

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

   3. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 56.3%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 50000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(((powf(floorf(d), 2.0f) * dX_46_w) * dX_46_w), ((powf((dY_46_u * floorf(w)), 2.0f) + ((dY_46_v * dY_46_v) * powf(floorf(h), 2.0f))) + powf((dY_46_w * floorf(d)), 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(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) != Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w)) ? Float32(Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ Float32(2.0)))) + (Float32(dY_46_w * floor(d)) ^ Float32(2.0))) : ((Float32(Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ Float32(2.0)))) + (Float32(dY_46_w * floor(d)) ^ Float32(2.0))) != Float32(Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ Float32(2.0)))) + (Float32(dY_46_w * floor(d)) ^ Float32(2.0)))) ? Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) : max(Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w), Float32(Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ Float32(2.0)))) + (Float32(dY_46_w * floor(d)) ^ Float32(2.0))))))))
end

MATLAB

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

Derivation

1. Initial program 67.1%

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

3. Taylor expanded in dX.w around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-floor.f32 53.5

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

5. Applied rewrites 53.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. swap-sqr N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 53.5

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

7. Applied rewrites 53.5%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-pow.f32 53.5

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

9. Applied rewrites 53.5%

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

    1. lift-*.f32 N/A

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

    2. pow2 N/A

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

    3. lift-*.f32 N/A

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

    4. *-commutative N/A

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

    5. lift-*.f32 N/A

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

    6. lift-pow.f32 53.5

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

11. Applied rewrites 53.5%

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

12. Final simplification 53.5%

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

Alternative 5: 53.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (pow (* dX.w (floor d)) 2.0)
    (+
     (pow (* dY.w (floor d)) 2.0)
     (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (floor h)) 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((dX_46_w * floorf(d)), 2.0f), (powf((dY_46_w * floorf(d)), 2.0f) + (powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * floorf(h)), 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(dX_46_w * floor(d)) ^ Float32(2.0)) != (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) ? Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) : ((Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) != Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))))) ? (Float32(dX_46_w * floor(d)) ^ Float32(2.0)) : max((Float32(dX_46_w * floor(d)) ^ Float32(2.0)), Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ 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(((dX_46_w * floor(d)) ^ single(2.0)), (((dY_46_w * floor(d)) ^ single(2.0)) + (((dY_46_u * floor(w)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0)))))));
end

Derivation

1. Initial program 67.1%

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

3. Taylor expanded in dX.w around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-floor.f32 53.5

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

5. Applied rewrites 53.5%

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

7. Applied rewrites 53.5%

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

8. Final simplification 53.5%

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

Reproduce

herbie shell --seed 2024304 
(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)))))))