Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 71.2%

Time: 9.3s

Alternatives: 12

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 67.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 71.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = (t_0 * t_0) + (t_3 * t_3);
	float t_6 = floorf(d) * dX_46_w_m;
	float t_7 = log2f(sqrtf(fmaxf((t_5 + (t_6 * t_6)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4)))));
	float tmp;
	if (t_7 <= 100.0f) {
		tmp = t_7;
	} else {
		tmp = log2f(sqrtf(fmaxf((t_5 + expf((logf((-dX_46_w_m * floorf(d))) * 2.0f))), (powf((dY_46_u * floorf(w)), 2.0f) - -powf((dY_46_w * floorf(d)), 2.0f)))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = Float32(Float32(t_0 * t_0) + Float32(t_3 * t_3))
	t_6 = Float32(floor(d) * dX_46_w_m)
	t_7 = log2(sqrt(fmax(Float32(t_5 + Float32(t_6 * t_6)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)))))
	tmp = Float32(0.0)
	if (t_7 <= Float32(100.0))
		tmp = t_7;
	else
		tmp = log2(sqrt(fmax(Float32(t_5 + exp(Float32(log(Float32(Float32(-dX_46_w_m) * floor(d))) * Float32(2.0)))), Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) - Float32(-(Float32(dY_46_w * floor(d)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(h) * dX_46_v;
	t_4 = floor(d) * dY_46_w;
	t_5 = (t_0 * t_0) + (t_3 * t_3);
	t_6 = floor(d) * dX_46_w_m;
	t_7 = log2(sqrt(max((t_5 + (t_6 * t_6)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4)))));
	tmp = single(0.0);
	if (t_7 <= single(100.0))
		tmp = t_7;
	else
		tmp = log2(sqrt(max((t_5 + exp((log((-dX_46_w_m * floor(d))) * single(2.0)))), (((dY_46_u * floor(w)) ^ single(2.0)) - -((dY_46_w * floor(d)) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 100

   1. Initial program 100.0%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 6.5%

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

   3. Applied rewrites 9.7%

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

      1. lift-*.f32 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 10.2

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

   5. Applied rewrites 10.2%

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

      1. rem-exp-log N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. log-prod N/A

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

      5. exp-sum N/A

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

      6. rem-exp-log N/A

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

      7. lower-*.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-log.f32 5.1

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

   7. Applied rewrites 5.1%

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

   8. Taylor expanded in dY.v around 0

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

      1. mul-1-neg N/A

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

      2. lift-floor.f32 N/A

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

      3. unpow-prod-down N/A

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

      4. lift-*.f32 N/A

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

      5. lift-pow.f32 N/A

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

      6. lift-neg.f32 16.5

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

   10. Applied rewrites 16.5%

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

Alternative 2: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_2 := {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ t_4 := {\left(dX.w\_m \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.w\_m \leq 150:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, t\_2 - t\_4\right), t\_0 + \left(t\_1 + t\_3\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_4 + \left(t\_2 + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), t\_3 + \left(t\_1 - t\_0\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.w (floor d)) 2.0))
        (t_1 (pow (* dY.u (floor w)) 2.0))
        (t_2 (pow (* dX.u (floor w)) 2.0))
        (t_3 (pow (* dY.v (floor h)) 2.0))
        (t_4 (pow (* dX.w_m (floor d)) 2.0)))
   (if (<= dX.w_m 150.0)
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor h) 2.0) dX.v) dX.v (- t_2 t_4))
        (+ t_0 (+ t_1 t_3)))))
     (log2
      (sqrt
       (fmax
        (+ t_4 (+ t_2 (pow (* dX.v (floor h)) 2.0)))
        (+ t_3 (- t_1 t_0))))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dY_46_w * floorf(d)), 2.0f);
	float t_1 = powf((dY_46_u * floorf(w)), 2.0f);
	float t_2 = powf((dX_46_u * floorf(w)), 2.0f);
	float t_3 = powf((dY_46_v * floorf(h)), 2.0f);
	float t_4 = powf((dX_46_w_m * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_w_m <= 150.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(h), 2.0f) * dX_46_v), dX_46_v, (t_2 - t_4)), (t_0 + (t_1 + t_3)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_4 + (t_2 + powf((dX_46_v * floorf(h)), 2.0f))), (t_3 + (t_1 - t_0)))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	t_1 = Float32(dY_46_u * floor(w)) ^ Float32(2.0)
	t_2 = Float32(dX_46_u * floor(w)) ^ Float32(2.0)
	t_3 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	t_4 = Float32(dX_46_w_m * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w_m <= Float32(150.0))
		tmp = log2(sqrt(fmax(fma(Float32((floor(h) ^ Float32(2.0)) * dX_46_v), dX_46_v, Float32(t_2 - t_4)), Float32(t_0 + Float32(t_1 + t_3)))));
	else
		tmp = log2(sqrt(fmax(Float32(t_4 + Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), Float32(t_3 + Float32(t_1 - t_0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 150

   1. Initial program 72.0%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lower-+.f32 71.3

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f32 71.3

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 71.3

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

      16. lift-*.f32 N/A

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

   5. Applied rewrites 71.3%

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

   if 150 < dX.w

   1. Initial program 62.6%

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

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

      3. flip-+ N/A

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

   3. Applied rewrites 58.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lower-+.f32 58.6

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. lift-pow.f32 58.6

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

      16. lift-*.f32 N/A

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

   5. Applied rewrites 58.6%

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

Alternative 3: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_2 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_4 := {\left(dX.w\_m \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.w\_m \leq 150:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 - t\_4, \left(t\_1 + t\_2\right) + t\_3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_4 + t\_0, t\_2 + \left(t\_1 - t\_3\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (+ (pow (* dX.u (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
        (t_1 (pow (* dY.u (floor w)) 2.0))
        (t_2 (pow (* dY.v (floor h)) 2.0))
        (t_3 (pow (* dY.w (floor d)) 2.0))
        (t_4 (pow (* dX.w_m (floor d)) 2.0)))
   (if (<= dX.w_m 150.0)
     (log2 (sqrt (fmax (- t_0 t_4) (+ (+ t_1 t_2) t_3))))
     (log2 (sqrt (fmax (+ t_4 t_0) (+ t_2 (- t_1 t_3))))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dX_46_u * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f);
	float t_1 = powf((dY_46_u * floorf(w)), 2.0f);
	float t_2 = powf((dY_46_v * floorf(h)), 2.0f);
	float t_3 = powf((dY_46_w * floorf(d)), 2.0f);
	float t_4 = powf((dX_46_w_m * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_w_m <= 150.0f) {
		tmp = log2f(sqrtf(fmaxf((t_0 - t_4), ((t_1 + t_2) + t_3))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_4 + t_0), (t_2 + (t_1 - t_3)))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))
	t_1 = Float32(dY_46_u * floor(w)) ^ Float32(2.0)
	t_2 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	t_3 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	t_4 = Float32(dX_46_w_m * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w_m <= Float32(150.0))
		tmp = log2(sqrt(fmax(Float32(t_0 - t_4), Float32(Float32(t_1 + t_2) + t_3))));
	else
		tmp = log2(sqrt(fmax(Float32(t_4 + t_0), Float32(t_2 + Float32(t_1 - t_3)))));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = ((dX_46_u * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0));
	t_1 = (dY_46_u * floor(w)) ^ single(2.0);
	t_2 = (dY_46_v * floor(h)) ^ single(2.0);
	t_3 = (dY_46_w * floor(d)) ^ single(2.0);
	t_4 = (dX_46_w_m * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_w_m <= single(150.0))
		tmp = log2(sqrt(max((t_0 - t_4), ((t_1 + t_2) + t_3))));
	else
		tmp = log2(sqrt(max((t_4 + t_0), (t_2 + (t_1 - t_3)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 150

   1. Initial program 72.0%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

   3. Applied rewrites 71.3%

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

   5. Applied rewrites 71.3%

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

   if 150 < dX.w

   1. Initial program 62.6%

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

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

      3. flip-+ N/A

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

   3. Applied rewrites 58.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lower-+.f32 58.6

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. lift-pow.f32 58.6

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

      16. lift-*.f32 N/A

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

   5. Applied rewrites 58.6%

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

Alternative 4: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := {\left(dX.w\_m \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_2 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_4 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_5 := {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.v \leq 30000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 + \left(t\_3 + t\_5\right), t\_1 - \left(t\_2 - t\_4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 + \left(t\_5 - t\_3\right), t\_1 + \left(t\_4 + t\_2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w_m (floor d)) 2.0))
        (t_1 (pow (* dY.w (floor d)) 2.0))
        (t_2 (pow (* dY.v (floor h)) 2.0))
        (t_3 (pow (* dX.u (floor w)) 2.0))
        (t_4 (pow (* dY.u (floor w)) 2.0))
        (t_5 (pow (* dX.v (floor h)) 2.0)))
   (if (<= dY.v 30000.0)
     (log2 (sqrt (fmax (+ t_0 (+ t_3 t_5)) (- t_1 (- t_2 t_4)))))
     (log2 (sqrt (fmax (+ t_0 (- t_5 t_3)) (+ t_1 (+ t_4 t_2))))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dX_46_w_m * floorf(d)), 2.0f);
	float t_1 = powf((dY_46_w * floorf(d)), 2.0f);
	float t_2 = powf((dY_46_v * floorf(h)), 2.0f);
	float t_3 = powf((dX_46_u * floorf(w)), 2.0f);
	float t_4 = powf((dY_46_u * floorf(w)), 2.0f);
	float t_5 = powf((dX_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dY_46_v <= 30000.0f) {
		tmp = log2f(sqrtf(fmaxf((t_0 + (t_3 + t_5)), (t_1 - (t_2 - t_4)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 + (t_5 - t_3)), (t_1 + (t_4 + t_2)))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dX_46_w_m * floor(d)) ^ Float32(2.0)
	t_1 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	t_2 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	t_3 = Float32(dX_46_u * floor(w)) ^ Float32(2.0)
	t_4 = Float32(dY_46_u * floor(w)) ^ Float32(2.0)
	t_5 = Float32(dX_46_v * floor(h)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(30000.0))
		tmp = log2(sqrt(fmax(Float32(t_0 + Float32(t_3 + t_5)), Float32(t_1 - Float32(t_2 - t_4)))));
	else
		tmp = log2(sqrt(fmax(Float32(t_0 + Float32(t_5 - t_3)), Float32(t_1 + Float32(t_4 + t_2)))));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dX_46_w_m * floor(d)) ^ single(2.0);
	t_1 = (dY_46_w * floor(d)) ^ single(2.0);
	t_2 = (dY_46_v * floor(h)) ^ single(2.0);
	t_3 = (dX_46_u * floor(w)) ^ single(2.0);
	t_4 = (dY_46_u * floor(w)) ^ single(2.0);
	t_5 = (dX_46_v * floor(h)) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_v <= single(30000.0))
		tmp = log2(sqrt(max((t_0 + (t_3 + t_5)), (t_1 - (t_2 - t_4)))));
	else
		tmp = log2(sqrt(max((t_0 + (t_5 - t_3)), (t_1 + (t_4 + t_2)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 3e4

   1. Initial program 69.6%

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

   3. Applied rewrites 64.5%

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

   5. Applied rewrites 64.5%

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

   if 3e4 < dY.v

   1. Initial program 60.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 60.1

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

      4. lift-*.f32 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f32 60.1

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 60.1

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. sqr-neg-rev N/A

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

      14. fp-cancel-sub-sign N/A

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

   3. Applied rewrites 56.7%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lower-+.f32 56.7

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f32 56.7

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 56.7

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

      16. lift-*.f32 N/A

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

   5. Applied rewrites 56.7%

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

Alternative 5: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := dX.w\_m \cdot \left\lfloor d\right\rfloor \\ t_4 := \left\lfloor w\right\rfloor \cdot dX.u\\ \mathbf{if}\;dY.v \leq 400:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_3}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} - \left(t\_1 - t\_2\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_4 \cdot t\_4 + t\_0 \cdot t\_0\right) + e^{\left(\log t\_3 \cdot 1\right) \cdot 2}, t\_1 + t\_2\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (pow (* dY.v (floor h)) 2.0))
        (t_2 (pow (* dY.u (floor w)) 2.0))
        (t_3 (* dX.w_m (floor d)))
        (t_4 (* (floor w) dX.u)))
   (if (<= dY.v 400.0)
     (log2
      (sqrt
       (fmax
        (+
         (pow t_3 2.0)
         (+ (pow (* dX.u (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
        (- (pow (* dY.w (floor d)) 2.0) (- t_1 t_2)))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_4 t_4) (* t_0 t_0)) (exp (* (* (log t_3) 1.0) 2.0)))
        (+ t_1 t_2)))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = powf((dY_46_v * floorf(h)), 2.0f);
	float t_2 = powf((dY_46_u * floorf(w)), 2.0f);
	float t_3 = dX_46_w_m * floorf(d);
	float t_4 = floorf(w) * dX_46_u;
	float tmp;
	if (dY_46_v <= 400.0f) {
		tmp = log2f(sqrtf(fmaxf((powf(t_3, 2.0f) + (powf((dX_46_u * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f))), (powf((dY_46_w * floorf(d)), 2.0f) - (t_1 - t_2)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_4 * t_4) + (t_0 * t_0)) + expf(((logf(t_3) * 1.0f) * 2.0f))), (t_1 + t_2))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	t_2 = Float32(dY_46_u * floor(w)) ^ Float32(2.0)
	t_3 = Float32(dX_46_w_m * floor(d))
	t_4 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(400.0))
		tmp = log2(sqrt(fmax(Float32((t_3 ^ Float32(2.0)) + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) - Float32(t_1 - t_2)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_4 * t_4) + Float32(t_0 * t_0)) + exp(Float32(Float32(log(t_3) * Float32(1.0)) * Float32(2.0)))), Float32(t_1 + t_2))));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) * dX_46_v;
	t_1 = (dY_46_v * floor(h)) ^ single(2.0);
	t_2 = (dY_46_u * floor(w)) ^ single(2.0);
	t_3 = dX_46_w_m * floor(d);
	t_4 = floor(w) * dX_46_u;
	tmp = single(0.0);
	if (dY_46_v <= single(400.0))
		tmp = log2(sqrt(max(((t_3 ^ single(2.0)) + (((dX_46_u * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0)))), (((dY_46_w * floor(d)) ^ single(2.0)) - (t_1 - t_2)))));
	else
		tmp = log2(sqrt(max((((t_4 * t_4) + (t_0 * t_0)) + exp(((log(t_3) * single(1.0)) * single(2.0)))), (t_1 + t_2))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 400

   1. Initial program 69.5%

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

   3. Applied rewrites 64.5%

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

   5. Applied rewrites 64.5%

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

   if 400 < dY.v

   1. Initial program 61.4%

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

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

      3. flip-+ N/A

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

   3. Applied rewrites 56.0%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. sqr-pow N/A

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

      4. pow-to-exp N/A

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

      5. pow-to-exp N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. lower-log.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f32 N/A

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

      20. metadata-eval 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in dY.u around inf

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

      1. lift-floor.f32 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 56.6

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

   8. Applied rewrites 56.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + e^{\left(\log \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot 1\right) \cdot 2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := dX.w\_m \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.v \leq 400:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_1}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot e^{\log dX.v}\right)}^{2}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + t\_3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right) + e^{\left(\log t\_1 \cdot 1\right) \cdot 2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + t\_3\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* dX.w_m (floor d)))
        (t_2 (* (floor h) dX.v))
        (t_3 (pow (* dY.u (floor w)) 2.0)))
   (if (<= dY.v 400.0)
     (log2
      (sqrt
       (fmax
        (+
         (pow t_1 2.0)
         (+
          (pow (* dX.u (floor w)) 2.0)
          (pow (* (floor h) (exp (log dX.v))) 2.0)))
        (+ (pow (* dY.w (floor d)) 2.0) t_3))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_0 t_0) (* t_2 t_2)) (exp (* (* (log t_1) 1.0) 2.0)))
        (+ (pow (* dY.v (floor h)) 2.0) t_3)))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = dX_46_w_m * floorf(d);
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = powf((dY_46_u * floorf(w)), 2.0f);
	float tmp;
	if (dY_46_v <= 400.0f) {
		tmp = log2f(sqrtf(fmaxf((powf(t_1, 2.0f) + (powf((dX_46_u * floorf(w)), 2.0f) + powf((floorf(h) * expf(logf(dX_46_v))), 2.0f))), (powf((dY_46_w * floorf(d)), 2.0f) + t_3))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_2 * t_2)) + expf(((logf(t_1) * 1.0f) * 2.0f))), (powf((dY_46_v * floorf(h)), 2.0f) + t_3))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(dX_46_w_m * floor(d))
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(dY_46_u * floor(w)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(400.0))
		tmp = log2(sqrt(fmax(Float32((t_1 ^ Float32(2.0)) + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * exp(log(dX_46_v))) ^ Float32(2.0)))), Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_3))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + exp(Float32(Float32(log(t_1) * Float32(1.0)) * Float32(2.0)))), Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + t_3))));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dX_46_u;
	t_1 = dX_46_w_m * floor(d);
	t_2 = floor(h) * dX_46_v;
	t_3 = (dY_46_u * floor(w)) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_v <= single(400.0))
		tmp = log2(sqrt(max(((t_1 ^ single(2.0)) + (((dX_46_u * floor(w)) ^ single(2.0)) + ((floor(h) * exp(log(dX_46_v))) ^ single(2.0)))), (((dY_46_w * floor(d)) ^ single(2.0)) + t_3))));
	else
		tmp = log2(sqrt(max((((t_0 * t_0) + (t_2 * t_2)) + exp(((log(t_1) * single(1.0)) * single(2.0)))), (((dY_46_v * floor(h)) ^ single(2.0)) + t_3))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 400

   1. Initial program 69.5%

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

   3. Applied rewrites 64.5%

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 64.5

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

      4. rem-exp-log N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. log-prod N/A

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

      11. exp-sum N/A

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

      12. rem-exp-log N/A

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

      13. lower-*.f32 N/A

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

      14. lower-exp.f32 N/A

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

      15. lower-log.f32 48.7

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in dY.v around 0

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

   10. Applied rewrites 50.8%

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

   if 400 < dY.v

   1. Initial program 61.4%

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

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

      3. flip-+ N/A

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

   3. Applied rewrites 56.0%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. sqr-pow N/A

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

      4. pow-to-exp N/A

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

      5. pow-to-exp N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. lower-log.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f32 N/A

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

      20. metadata-eval 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in dY.u around inf

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

      1. lift-floor.f32 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 56.6

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

   8. Applied rewrites 56.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + e^{\left(\log \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot 1\right) \cdot 2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_1 \cdot t\_1 + t\_0 \cdot t\_0\right) + e^{\left(\log \left(dX.w\_m \cdot \left\lfloor d\right\rfloor \right) \cdot 1\right) \cdot 2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v)) (t_1 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+
       (+ (* t_1 t_1) (* t_0 t_0))
       (exp (* (* (log (* dX.w_m (floor d))) 1.0) 2.0)))
      (+ (pow (* dY.v (floor h)) 2.0) (pow (* dY.u (floor w)) 2.0)))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_1 * t_1) + (t_0 * t_0)) + expf(((logf((dX_46_w_m * floorf(d))) * 1.0f) * 2.0f))), (powf((dY_46_v * floorf(h)), 2.0f) + powf((dY_46_u * floorf(w)), 2.0f)))));
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(fmax(Float32(Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0)) + exp(Float32(Float32(log(Float32(dX_46_w_m * floor(d))) * Float32(1.0)) * Float32(2.0)))), Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_1 * t_1) + (t_0 * t_0)) + exp(((log((dX_46_w_m * floor(d))) * single(1.0)) * single(2.0)))), (((dY_46_v * floor(h)) ^ single(2.0)) + ((dY_46_u * floor(w)) ^ single(2.0))))));
end

Derivation

1. Initial program 67.7%

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

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

   3. flip-+ N/A

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

3. Applied rewrites 60.2%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. sqr-pow N/A

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

   4. pow-to-exp N/A

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

   5. pow-to-exp N/A

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

   6. exp-lft-sqr-rev N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lower-log.f32 N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

   20. metadata-eval 60.1

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in dY.u around inf

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

   1. lift-floor.f32 N/A

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

   2. unpow-prod-down N/A

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

   3. lift-*.f32 N/A

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

   4. lift-pow.f32 60.7

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

8. Applied rewrites 60.7%

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

Alternative 8: 56.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.v \leq 107.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + e^{\left(\log \left(dX.w\_m \cdot \left\lfloor d\right\rfloor \right) \cdot 1\right) \cdot 2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({t\_1}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\_m\right)}^{2}\right) + {t\_0}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u)) (t_1 (* (floor h) dX.v)))
   (if (<= dY.v 107.5)
     (log2
      (sqrt
       (fmax
        (+
         (+ (* t_0 t_0) (* t_1 t_1))
         (exp (* (* (log (* dX.w_m (floor d))) 1.0) 2.0)))
        (pow (* dY.u (floor w)) 2.0))))
     (log2
      (sqrt
       (fmax
        (+ (+ (pow t_1 2.0) (pow (* (floor d) dX.w_m) 2.0)) (pow t_0 2.0))
        (pow (* dY.v (floor h)) 2.0)))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_v <= 107.5f) {
		tmp = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_1 * t_1)) + expf(((logf((dX_46_w_m * floorf(d))) * 1.0f) * 2.0f))), powf((dY_46_u * floorf(w)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(t_1, 2.0f) + powf((floorf(d) * dX_46_w_m), 2.0f)) + powf(t_0, 2.0f)), powf((dY_46_v * floorf(h)), 2.0f))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(107.5))
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + exp(Float32(Float32(log(Float32(dX_46_w_m * floor(d))) * Float32(1.0)) * Float32(2.0)))), (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32((t_1 ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w_m) ^ Float32(2.0))) + (t_0 ^ Float32(2.0))), (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(h) * dX_46_v;
	tmp = single(0.0);
	if (dY_46_v <= single(107.5))
		tmp = log2(sqrt(max((((t_0 * t_0) + (t_1 * t_1)) + exp(((log((dX_46_w_m * floor(d))) * single(1.0)) * single(2.0)))), ((dY_46_u * floor(w)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((((t_1 ^ single(2.0)) + ((floor(d) * dX_46_w_m) ^ single(2.0))) + (t_0 ^ 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.v < 107.5

   1. Initial program 69.5%

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

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

      3. flip-+ N/A

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

   3. Applied rewrites 61.3%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. sqr-pow N/A

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

      4. pow-to-exp N/A

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

      5. pow-to-exp N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. lower-log.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f32 N/A

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

      20. metadata-eval 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in dY.u around inf

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

      1. lift-floor.f32 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 57.2

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

   8. Applied rewrites 57.2%

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

   if 107.5 < dY.v

   1. Initial program 61.7%

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

   3. Applied rewrites 56.9%

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in dY.v around inf

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

      1. lift-floor.f32 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 52.9

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

   8. Applied rewrites 52.9%

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

Alternative 9: 56.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := \left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\_m\right)}^{2}\right) + {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\\ \mathbf{if}\;dY.v \leq 400:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0
         (+
          (+ (pow (* (floor h) dX.v) 2.0) (pow (* (floor d) dX.w_m) 2.0))
          (pow (* (floor w) dX.u) 2.0))))
   (if (<= dY.v 400.0)
     (log2 (sqrt (fmax t_0 (pow (* dY.w (floor d)) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (* dY.v (floor h)) 2.0)))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = (powf((floorf(h) * dX_46_v), 2.0f) + powf((floorf(d) * dX_46_w_m), 2.0f)) + powf((floorf(w) * dX_46_u), 2.0f);
	float tmp;
	if (dY_46_v <= 400.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((dY_46_w * floorf(d)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((dY_46_v * floorf(h)), 2.0f))));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w_m) ^ Float32(2.0))) + (Float32(floor(w) * dX_46_u) ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(400.0))
		tmp = log2(sqrt(fmax(t_0, (Float32(dY_46_w * floor(d)) ^ Float32(2.0)))));
	else
		tmp = log2(sqrt(fmax(t_0, (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (((floor(h) * dX_46_v) ^ single(2.0)) + ((floor(d) * dX_46_w_m) ^ single(2.0))) + ((floor(w) * dX_46_u) ^ single(2.0));
	tmp = single(0.0);
	if (dY_46_v <= single(400.0))
		tmp = log2(sqrt(max(t_0, ((dY_46_w * floor(d)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, ((dY_46_v * floor(h)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 400

   1. Initial program 69.5%

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

   3. Applied rewrites 60.9%

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in dY.w around inf

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

      1. lift-floor.f32 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 56.9

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

   8. Applied rewrites 56.9%

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

   if 400 < dY.v

   1. Initial program 61.4%

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

   3. Applied rewrites 56.8%

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in dY.v around inf

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

      1. lift-floor.f32 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-*.f32 N/A

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

      4. lift-pow.f32 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 10: 46.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.w_m = \left|dX.w\right| \\ \begin{array}{l} t_0 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} - {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)\\ \mathbf{if}\;dX.v \leq 6200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0
         (+
          (pow (* dY.v (floor h)) 2.0)
          (- (pow (* dY.u (floor w)) 2.0) (pow (* dY.w (floor d)) 2.0)))))
   (if (<= dX.v 6200.0)
     (log2 (sqrt (fmax (pow (* dX.u (floor w)) 2.0) t_0)))
     (log2 (sqrt (fmax (pow (* dX.v (floor h)) 2.0) t_0))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dY_46_v * floorf(h)), 2.0f) + (powf((dY_46_u * floorf(w)), 2.0f) - powf((dY_46_w * floorf(d)), 2.0f));
	float tmp;
	if (dX_46_v <= 6200.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_u * floorf(w)), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_v * floorf(h)), 2.0f), t_0)));
	}
	return tmp;
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) - (Float32(dY_46_w * floor(d)) ^ Float32(2.0))))
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(6200.0))
		tmp = log2(sqrt(fmax((Float32(dX_46_u * floor(w)) ^ Float32(2.0)), t_0)));
	else
		tmp = log2(sqrt(fmax((Float32(dX_46_v * floor(h)) ^ Float32(2.0)), t_0)));
	end
	return tmp
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	t_0 = ((dY_46_v * floor(h)) ^ single(2.0)) + (((dY_46_u * floor(w)) ^ single(2.0)) - ((dY_46_w * floor(d)) ^ single(2.0)));
	tmp = single(0.0);
	if (dX_46_v <= single(6200.0))
		tmp = log2(sqrt(max(((dX_46_u * floor(w)) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((dX_46_v * floor(h)) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 6200

   1. Initial program 69.6%

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

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

      3. flip-+ N/A

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

   3. Applied rewrites 60.8%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. sqr-pow N/A

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

      4. pow-to-exp N/A

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

      5. pow-to-exp N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. lower-log.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f32 N/A

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

      20. metadata-eval 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in dX.u around inf

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

   8. Applied rewrites 45.5%

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

   if 6200 < dX.v

   1. Initial program 60.4%

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

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

      3. flip-+ N/A

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

   3. Applied rewrites 57.8%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. sqr-pow N/A

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

      4. pow-to-exp N/A

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

      5. pow-to-exp N/A

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

      6. exp-lft-sqr-rev N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. lower-log.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f32 N/A

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

      20. metadata-eval 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in dX.v around inf

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

   8. Applied rewrites 50.1%

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

Alternative 11: 53.7% accurate, 1.5× speedup

Math

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

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (+
     (+ (pow (* (floor h) dX.v) 2.0) (pow (* (floor d) dX.w_m) 2.0))
     (pow (* (floor w) dX.u) 2.0))
    (pow (* dY.v (floor h)) 2.0)))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(((powf((floorf(h) * dX_46_v), 2.0f) + powf((floorf(d) * dX_46_w_m), 2.0f)) + powf((floorf(w) * dX_46_u), 2.0f)), powf((dY_46_v * floorf(h)), 2.0f))));
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(fmax(Float32(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w_m) ^ Float32(2.0))) + (Float32(floor(w) * dX_46_u) ^ Float32(2.0))), (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))))
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((((floor(h) * dX_46_v) ^ single(2.0)) + ((floor(d) * dX_46_w_m) ^ single(2.0))) + ((floor(w) * dX_46_u) ^ single(2.0))), ((dY_46_v * floor(h)) ^ single(2.0)))));
end

Derivation

1. Initial program 67.7%

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

3. Applied rewrites 60.0%

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

5. Applied rewrites 60.0%

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

6. Taylor expanded in dY.v around inf

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

   1. lift-floor.f32 N/A

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

   2. unpow-prod-down N/A

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

   3. lift-*.f32 N/A

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

   4. lift-pow.f32 53.7

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

8. Applied rewrites 53.7%

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

Alternative 12: 42.9% accurate, 1.5× speedup

Math

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

FPCore

dX.w_m = (fabs.f32 dX.w)
(FPCore (w h d dX.u dX.v dX.w_m dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (pow (* dX.u (floor w)) 2.0)
    (+
     (pow (* dY.v (floor h)) 2.0)
     (- (pow (* dY.u (floor w)) 2.0) (pow (* dY.w (floor d)) 2.0)))))))

C

dX.w_m = fabs(dX_46_w);
float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w_m, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(powf((dX_46_u * floorf(w)), 2.0f), (powf((dY_46_v * floorf(h)), 2.0f) + (powf((dY_46_u * floorf(w)), 2.0f) - powf((dY_46_w * floorf(d)), 2.0f))))));
}

Julia

dX.w_m = abs(dX_46_w)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(fmax((Float32(dX_46_u * floor(w)) ^ Float32(2.0)), Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) - (Float32(dY_46_w * floor(d)) ^ Float32(2.0)))))))
end

MATLAB

dX.w_m = abs(dX_46_w);
function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w_m, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((dX_46_u * floor(w)) ^ single(2.0)), (((dY_46_v * floor(h)) ^ single(2.0)) + (((dY_46_u * floor(w)) ^ single(2.0)) - ((dY_46_w * floor(d)) ^ single(2.0)))))));
end

Derivation

1. Initial program 67.7%

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

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

   3. flip-+ N/A

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

3. Applied rewrites 60.2%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. sqr-pow N/A

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

   4. pow-to-exp N/A

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

   5. pow-to-exp N/A

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

   6. exp-lft-sqr-rev N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lower-log.f32 N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

   20. metadata-eval 60.1

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in dX.u around inf

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

8. Applied rewrites 42.9%

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

Reproduce

herbie shell --seed 2025107 
(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)))))))