Isotropic LOD (LOD)

Percentage Accurate: 66.7% → 70.3%

Time: 9.4s

Alternatives: 6

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

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

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: 66.7% accurate, 1.0× speedup

Math

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

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: 70.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_2 := \left\lfloor d\right\rfloor \cdot \left|dY.w\right|\\ t_3 := \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_7 := t\_6 \cdot dY.u\\ t_8 := t\_1 \cdot t\_1\\ t_9 := \left\lfloor h\right\rfloor \cdot dY.v\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_8\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_9 \cdot t\_9\right) + t\_2 \cdot t\_2\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_6 \cdot dX.u, dX.u, \mathsf{fma}\left(dX.w \cdot dX.w, \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right), \mathsf{fma}\left(t\_3, dY.v, \mathsf{fma}\left(t\_7, dY.u, \left(\left(\left|dY.w\right| \cdot \left\lfloor d\right\rfloor \right) \cdot \left|dY.w\right|\right) \cdot \left\lfloor d\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_4 \cdot dX.w, \left\lfloor d\right\rfloor , t\_8\right), \mathsf{fma}\left(t\_3, dY.v, \mathsf{fma}\left(t\_7, dY.u, e^{\log \left(\left(-\left|dY.w\right|\right) \cdot \left\lfloor d\right\rfloor \right) \cdot 2}\right)\right)\right)}\right)\\ \end{array} \]

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) dX.v))
        (t_2 (* (floor d) (fabs dY.w)))
        (t_3 (* (* (floor h) (floor h)) dY.v))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u))
        (t_6 (* (floor w) (floor w)))
        (t_7 (* t_6 dY.u))
        (t_8 (* t_1 t_1))
        (t_9 (* (floor h) dY.v)))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_5 t_5) t_8) (* t_4 t_4))
           (+ (+ (* t_0 t_0) (* t_9 t_9)) (* t_2 t_2)))))
        100.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* t_6 dX.u)
         dX.u
         (fma
          (* dX.w dX.w)
          (* (floor d) (floor d))
          (* (* (* dX.v (floor h)) dX.v) (floor h))))
        (fma
         t_3
         dY.v
         (fma
          t_7
          dY.u
          (* (* (* (fabs dY.w) (floor d)) (fabs dY.w)) (floor d)))))))
     (log2
      (sqrt
       (fmax
        (fma (* t_4 dX.w) (floor d) t_8)
        (fma
         t_3
         dY.v
         (fma
          t_7
          dY.u
          (exp (* (log (* (- (fabs dY.w)) (floor d))) 2.0))))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = floorf(d) * fabsf(dY_46_w);
	float t_3 = (floorf(h) * floorf(h)) * dY_46_v;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = floorf(w) * floorf(w);
	float t_7 = t_6 * dY_46_u;
	float t_8 = t_1 * t_1;
	float t_9 = floorf(h) * dY_46_v;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_5 * t_5) + t_8) + (t_4 * t_4)), (((t_0 * t_0) + (t_9 * t_9)) + (t_2 * t_2))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_6 * dX_46_u), dX_46_u, fmaf((dX_46_w * dX_46_w), (floorf(d) * floorf(d)), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))), fmaf(t_3, dY_46_v, fmaf(t_7, dY_46_u, (((fabsf(dY_46_w) * floorf(d)) * fabsf(dY_46_w)) * floorf(d)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_4 * dX_46_w), floorf(d), t_8), fmaf(t_3, dY_46_v, fmaf(t_7, dY_46_u, expf((logf((-fabsf(dY_46_w) * floorf(d))) * 2.0f)))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(floor(d) * abs(dY_46_w))
	t_3 = Float32(Float32(floor(h) * floor(h)) * dY_46_v)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	t_6 = Float32(floor(w) * floor(w))
	t_7 = Float32(t_6 * dY_46_u)
	t_8 = Float32(t_1 * t_1)
	t_9 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_5 * t_5) + t_8) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_9 * t_9)) + Float32(t_2 * t_2))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_6 * dX_46_u), dX_46_u, fma(Float32(dX_46_w * dX_46_w), Float32(floor(d) * floor(d)), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))), fma(t_3, dY_46_v, fma(t_7, dY_46_u, Float32(Float32(Float32(abs(dY_46_w) * floor(d)) * abs(dY_46_w)) * floor(d)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_4 * dX_46_w), floor(d), t_8), fma(t_3, dY_46_v, fma(t_7, dY_46_u, exp(Float32(log(Float32(Float32(-abs(dY_46_w)) * floor(d))) * Float32(2.0))))))));
	end
	return 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 66.7%

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

   3. Applied rewrites 66.7%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. swap-sqr N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f32 N/A

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

   5. Applied rewrites 66.7%

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

   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 66.7%

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

   3. Applied rewrites 66.7%

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

   4. Taylor expanded in dX.u around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.3

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

   6. Applied rewrites 60.3%

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

   8. Applied rewrites 60.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. lift-floor.f32 N/A

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

      7. associate-*l* N/A

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

      8. lift-floor.f32 N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-unsound-exp.f32 N/A

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

      13. lower-unsound-*.f32 N/A

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

      14. lower-unsound-log.f32 N/A

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

      15. lift-floor.f32 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 47.8

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

   10. Applied rewrites 47.8%

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

Alternative 2: 70.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * abs(dY_46_w))
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(t_2 * t_2)
	t_6 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + t_5) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6)) + Float32(t_3 * t_3))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(abs(dY_46_w) * floor(d)) * abs(dY_46_w)), floor(d), fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))), fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_4 * dX_46_w), floor(d), t_5), fma(Float32(Float32(floor(h) * floor(h)) * dY_46_v), dY_46_v, fma(Float32(Float32(floor(w) * floor(w)) * dY_46_u), dY_46_u, exp(Float32(log(Float32(Float32(-abs(dY_46_w)) * floor(d))) * Float32(2.0))))))));
	end
	return 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 66.7%

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

   3. Applied rewrites 66.7%

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

   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 66.7%

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

   3. Applied rewrites 66.7%

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

   4. Taylor expanded in dX.u around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.3

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

   6. Applied rewrites 60.3%

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

   8. Applied rewrites 60.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. lift-floor.f32 N/A

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

      7. associate-*l* N/A

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

      8. lift-floor.f32 N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-unsound-exp.f32 N/A

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

      13. lower-unsound-*.f32 N/A

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

      14. lower-unsound-log.f32 N/A

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

      15. lift-floor.f32 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 47.8

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

   10. Applied rewrites 47.8%

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

Alternative 3: 64.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|dX.v\right| \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_4 := dY.v \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;\left|dX.v\right| \leq 60:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.w \cdot dY.w, t\_1, \mathsf{fma}\left(dY.u \cdot dY.u, t\_3, t\_4 \cdot t\_4\right)\right), \mathsf{fma}\left(dX.w \cdot dX.w, t\_1, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_3 \cdot dX.u, dX.u, \mathsf{fma}\left(t\_1 \cdot dX.w, dX.w, t\_0 \cdot t\_0\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot dY.u, \left\lfloor w\right\rfloor , t\_2 \cdot t\_2\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (fabs dX.v) (floor h)))
        (t_1 (* (floor d) (floor d)))
        (t_2 (* (floor d) dY.w))
        (t_3 (* (floor w) (floor w)))
        (t_4 (* dY.v (floor h))))
   (if (<= (fabs dX.v) 60.0)
     (log2
      (sqrt
       (fmax
        (fma (* dY.w dY.w) t_1 (fma (* dY.u dY.u) t_3 (* t_4 t_4)))
        (fma (* dX.w dX.w) t_1 (* (* (* dX.u (floor w)) dX.u) (floor w))))))
     (log2
      (sqrt
       (fmax
        (fma (* t_3 dX.u) dX.u (fma (* t_1 dX.w) dX.w (* t_0 t_0)))
        (fma (* (* (floor w) dY.u) dY.u) (floor w) (* t_2 t_2))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = fabsf(dX_46_v) * floorf(h);
	float t_1 = floorf(d) * floorf(d);
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = floorf(w) * floorf(w);
	float t_4 = dY_46_v * floorf(h);
	float tmp;
	if (fabsf(dX_46_v) <= 60.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dY_46_w * dY_46_w), t_1, fmaf((dY_46_u * dY_46_u), t_3, (t_4 * t_4))), fmaf((dX_46_w * dX_46_w), t_1, (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_3 * dX_46_u), dX_46_u, fmaf((t_1 * dX_46_w), dX_46_w, (t_0 * t_0))), fmaf(((floorf(w) * dY_46_u) * dY_46_u), floorf(w), (t_2 * t_2)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(abs(dX_46_v) * floor(h))
	t_1 = Float32(floor(d) * floor(d))
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = Float32(floor(w) * floor(w))
	t_4 = Float32(dY_46_v * floor(h))
	tmp = Float32(0.0)
	if (abs(dX_46_v) <= Float32(60.0))
		tmp = log2(sqrt(fmax(fma(Float32(dY_46_w * dY_46_w), t_1, fma(Float32(dY_46_u * dY_46_u), t_3, Float32(t_4 * t_4))), fma(Float32(dX_46_w * dX_46_w), t_1, Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_3 * dX_46_u), dX_46_u, fma(Float32(t_1 * dX_46_w), dX_46_w, Float32(t_0 * t_0))), fma(Float32(Float32(floor(w) * dY_46_u) * dY_46_u), floor(w), Float32(t_2 * t_2)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 60

   1. Initial program 66.7%

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

   2. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 59.4

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

   4. Applied rewrites 59.4%

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

   6. Applied rewrites 59.4%

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

   if 60 < dX.v

   1. Initial program 66.7%

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

   2. Taylor expanded in dY.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.7%

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

Alternative 4: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot \left|dX.v\right|\\ t_1 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_3 := dY.v \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;\left|dX.v\right| \leq 13.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.w \cdot dY.w, t\_1, \mathsf{fma}\left(dY.u \cdot dY.u, t\_2, t\_3 \cdot t\_3\right)\right), \mathsf{fma}\left(dX.w \cdot dX.w, t\_1, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot dX.w, \left\lfloor d\right\rfloor , t\_0 \cdot t\_0\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \mathsf{fma}\left(t\_2 \cdot dY.u, dY.u, \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) (fabs dX.v)))
        (t_1 (* (floor d) (floor d)))
        (t_2 (* (floor w) (floor w)))
        (t_3 (* dY.v (floor h))))
   (if (<= (fabs dX.v) 13.5)
     (log2
      (sqrt
       (fmax
        (fma (* dY.w dY.w) t_1 (fma (* dY.u dY.u) t_2 (* t_3 t_3)))
        (fma (* dX.w dX.w) t_1 (* (* (* dX.u (floor w)) dX.u) (floor w))))))
     (log2
      (sqrt
       (fmax
        (fma (* (* (floor d) dX.w) dX.w) (floor d) (* t_0 t_0))
        (fma
         (* (* (floor h) (floor h)) dY.v)
         dY.v
         (fma
          (* t_2 dY.u)
          dY.u
          (* (* (* dY.w (floor d)) dY.w) (floor d))))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * fabsf(dX_46_v);
	float t_1 = floorf(d) * floorf(d);
	float t_2 = floorf(w) * floorf(w);
	float t_3 = dY_46_v * floorf(h);
	float tmp;
	if (fabsf(dX_46_v) <= 13.5f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dY_46_w * dY_46_w), t_1, fmaf((dY_46_u * dY_46_u), t_2, (t_3 * t_3))), fmaf((dX_46_w * dX_46_w), t_1, (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(((floorf(d) * dX_46_w) * dX_46_w), floorf(d), (t_0 * t_0)), fmaf(((floorf(h) * floorf(h)) * dY_46_v), dY_46_v, fmaf((t_2 * dY_46_u), dY_46_u, (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * abs(dX_46_v))
	t_1 = Float32(floor(d) * floor(d))
	t_2 = Float32(floor(w) * floor(w))
	t_3 = Float32(dY_46_v * floor(h))
	tmp = Float32(0.0)
	if (abs(dX_46_v) <= Float32(13.5))
		tmp = log2(sqrt(fmax(fma(Float32(dY_46_w * dY_46_w), t_1, fma(Float32(dY_46_u * dY_46_u), t_2, Float32(t_3 * t_3))), fma(Float32(dX_46_w * dX_46_w), t_1, Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(Float32(floor(d) * dX_46_w) * dX_46_w), floor(d), Float32(t_0 * t_0)), fma(Float32(Float32(floor(h) * floor(h)) * dY_46_v), dY_46_v, fma(Float32(t_2 * dY_46_u), dY_46_u, Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 13.5

   1. Initial program 66.7%

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

   2. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 59.4

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

   4. Applied rewrites 59.4%

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

   6. Applied rewrites 59.4%

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

   if 13.5 < dX.v

   1. Initial program 66.7%

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

   3. Applied rewrites 66.7%

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

   4. Taylor expanded in dX.u around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.3

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

   6. Applied rewrites 60.3%

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

   8. Applied rewrites 60.3%

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

Alternative 5: 60.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v)))
   (log2
    (sqrt
     (fmax
      (fma (* (* (floor d) dX.w) dX.w) (floor d) (* t_0 t_0))
      (fma
       (* (* (floor h) (floor h)) dY.v)
       dY.v
       (fma
        (* (* (floor w) (floor w)) dY.u)
        dY.u
        (* (* (* dY.w (floor d)) dY.w) (floor d)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	return log2f(sqrtf(fmaxf(fmaf(((floorf(d) * dX_46_w) * dX_46_w), floorf(d), (t_0 * t_0)), fmaf(((floorf(h) * floorf(h)) * dY_46_v), dY_46_v, fmaf(((floorf(w) * floorf(w)) * dY_46_u), dY_46_u, (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	return log2(sqrt(fmax(fma(Float32(Float32(floor(d) * dX_46_w) * dX_46_w), floor(d), Float32(t_0 * t_0)), fma(Float32(Float32(floor(h) * floor(h)) * dY_46_v), dY_46_v, fma(Float32(Float32(floor(w) * floor(w)) * dY_46_u), dY_46_u, Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))))))
end

Derivation

1. Initial program 66.7%

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

3. Applied rewrites 66.7%

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

4. Taylor expanded in dX.u around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-floor.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-floor.f32 60.3

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

6. Applied rewrites 60.3%

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

8. Applied rewrites 60.3%

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

Alternative 6: 52.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf((fabsf(dY_46_u) * (fabsf((fabsf(dY_46_u) * floorf(w))) * floorf(w))), fmaf(((dX_46_w * floorf(d)) * dX_46_w), floorf(d), fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(fmax(Float32(abs(dY_46_u) * Float32(abs(Float32(abs(dY_46_u) * floor(w))) * floor(w))), fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))))))
end

Derivation

1. Initial program 66.7%

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

3. Applied rewrites 66.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 66.7

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

   4. rem-exp-log N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. exp-fabs N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. rem-exp-log N/A

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

   15. lower-fabs.f32 62.8

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 62.8

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

5. Applied rewrites 62.8%

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

6. Taylor expanded in dY.u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fabs.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-floor.f32 N/A

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

   6. lower-floor.f32 48.8

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

8. Applied rewrites 48.8%

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

Reproduce

herbie shell --seed 2025175 
(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)))))))