UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 15.3s

Alternatives: 19

Speedup: 1.2×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (fma
    (* (* zi (- 1.0 ux)) maxCos)
    ux
    (*
     (sqrt (fma (* (- ux 1.0) maxCos) (* (* (* maxCos (- 1.0 ux)) ux) ux) 1.0))
     (fma yi (sin t_0) (* xi (cos t_0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf(((zi * (1.0f - ux)) * maxCos), ux, (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(yi, sinf(t_0), (xi * cosf(t_0)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(Float32(zi * Float32(Float32(1.0) - ux)) * maxCos), ux, Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(yi, sin(t_0), Float32(xi * cos(t_0)))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]
4. Add Preprocessing

Alternative 2: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (fma
    (* zi (- 1.0 ux))
    (* maxCos ux)
    (*
     (sqrt (fma (* (- ux 1.0) maxCos) (* (* (* maxCos (- 1.0 ux)) ux) ux) 1.0))
     (fma yi (sin t_0) (* xi (cos t_0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf((zi * (1.0f - ux)), (maxCos * ux), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(yi, sinf(t_0), (xi * cosf(t_0)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(zi * Float32(Float32(1.0) - ux)), Float32(maxCos * ux), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(yi, sin(t_0), Float32(xi * cos(t_0)))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ t_1 := \left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\\ \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, t\_1 \cdot ux, 1\right)}, \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right), zi \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))) (t_1 (* (* maxCos (- 1.0 ux)) ux)))
   (fma
    (sqrt (fma (* (- ux 1.0) maxCos) (* t_1 ux) 1.0))
    (fma xi (cos t_0) (* yi (sin t_0)))
    (* zi t_1))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float t_1 = (maxCos * (1.0f - ux)) * ux;
	return fmaf(sqrtf(fmaf(((ux - 1.0f) * maxCos), (t_1 * ux), 1.0f)), fmaf(xi, cosf(t_0), (yi * sinf(t_0))), (zi * t_1));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	t_1 = Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux)
	return fma(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(t_1 * ux), Float32(1.0))), fma(xi, cos(t_0), Float32(yi * sin(t_0))), Float32(zi * t_1))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \mathsf{fma}\left(xi, \cos \left(\pi \cdot \left(uy + uy\right)\right), yi \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\right), zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)} \]
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (fma maxCos (* ux (* zi (- 1.0 ux))) (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * (zi * (1.0f - ux))), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.028999999165534973:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(ux \cdot zi\right) \cdot maxCos, \mathsf{fma}\left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right), uy, xi\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= uy 0.028999999165534973)
     (fma
      (- 1.0 ux)
      (* (* ux zi) maxCos)
      (*
       (fma
        (fma
         (+ PI PI)
         yi
         (*
          (fma
           (* (* PI PI) xi)
           -2.0
           (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
          uy))
        uy
        xi)
       (sqrt
        (fma (* (* (* (- ux 1.0) maxCos) ux) maxCos) (* (- 1.0 ux) ux) 1.0))))
     (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if (uy <= 0.028999999165534973f) {
		tmp = fmaf((1.0f - ux), ((ux * zi) * maxCos), (fmaf(fmaf((((float) M_PI) + ((float) M_PI)), yi, (fmaf(((((float) M_PI) * ((float) M_PI)) * xi), -2.0f, (((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy) * -1.3333333333333333f)) * uy)), uy, xi) * sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), ((1.0f - ux) * ux), 1.0f))));
	} else {
		tmp = fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.028999999165534973))
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(ux * zi) * maxCos), Float32(fma(fma(Float32(Float32(pi) + Float32(pi)), yi, Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))) * uy)), uy, xi) * sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), Float32(Float32(Float32(1.0) - ux) * ux), Float32(1.0)))));
	else
		tmp = fma(xi, cos(t_0), Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0289999992

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, \color{blue}{yi \cdot \mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \color{blue}{\mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

      5. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   6. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)}\right) \]

   8. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(ux \cdot zi\right) \cdot maxCos, \mathsf{fma}\left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right), uy, xi\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}\right)} \]

   if 0.0289999992 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f32 90.9

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy + uy\right) \cdot \pi\\ \mathsf{fma}\left(\sin t\_0, yi, \mathsf{fma}\left(\cos t\_0, xi, \left(ux \cdot zi\right) \cdot maxCos\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ uy uy) PI)))
   (fma (sin t_0) yi (fma (cos t_0) xi (* (* ux zi) maxCos)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (uy + uy) * ((float) M_PI);
	return fmaf(sinf(t_0), yi, fmaf(cosf(t_0), xi, ((ux * zi) * maxCos)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy + uy) * Float32(pi))
	return fma(sin(t_0), yi, fma(cos(t_0), xi, Float32(Float32(ux * zi) * maxCos)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - \left|uy\right| \cdot \left(\pi + \pi\right)\right)}\right)\right) \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

8. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

10. Applied rewrites 95.9%

    \[\leadsto \mathsf{fma}\left(\sin \left(\left(uy + uy\right) \cdot \pi\right), \color{blue}{yi}, \mathsf{fma}\left(\cos \left(\left(uy + uy\right) \cdot \pi\right), xi, \left(ux \cdot zi\right) \cdot maxCos\right)\right) \]
11. Add Preprocessing

Alternative 7: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 - ux, \left(ux \cdot zi\right) \cdot maxCos, \mathsf{fma}\left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right), uy, xi\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (- 1.0 ux)
  (* (* ux zi) maxCos)
  (*
   (fma
    (fma
     (+ PI PI)
     yi
     (*
      (fma
       (* (* PI PI) xi)
       -2.0
       (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
      uy))
    uy
    xi)
   (sqrt
    (fma (* (* (* (- ux 1.0) maxCos) ux) maxCos) (* (- 1.0 ux) ux) 1.0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((1.0f - ux), ((ux * zi) * maxCos), (fmaf(fmaf((((float) M_PI) + ((float) M_PI)), yi, (fmaf(((((float) M_PI) * ((float) M_PI)) * xi), -2.0f, (((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy) * -1.3333333333333333f)) * uy)), uy, xi) * sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), ((1.0f - ux) * ux), 1.0f))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(ux * zi) * maxCos), Float32(fma(fma(Float32(Float32(pi) + Float32(pi)), yi, Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))) * uy)), uy, xi) * sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), Float32(Float32(Float32(1.0) - ux) * ux), Float32(1.0)))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

4. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, \color{blue}{yi \cdot \mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \color{blue}{\mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

6. Applied rewrites 89.3%

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)}\right) \]

8. Applied rewrites 89.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(ux \cdot zi\right) \cdot maxCos, \mathsf{fma}\left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right), uy, xi\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}\right)} \]
9. Add Preprocessing

Alternative 8: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right), uy, xi\right), \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (sqrt (fma (* (* (* (- ux 1.0) maxCos) ux) maxCos) (* (- 1.0 ux) ux) 1.0))
  (fma
   (fma
    (+ PI PI)
    yi
    (*
     (fma
      (* (* PI PI) xi)
      -2.0
      (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
     uy))
   uy
   xi)
  (* (* (* (- 1.0 ux) zi) ux) maxCos)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), ((1.0f - ux) * ux), 1.0f)), fmaf(fmaf((((float) M_PI) + ((float) M_PI)), yi, (fmaf(((((float) M_PI) * ((float) M_PI)) * xi), -2.0f, (((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy) * -1.3333333333333333f)) * uy)), uy, xi), ((((1.0f - ux) * zi) * ux) * maxCos));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), Float32(Float32(Float32(1.0) - ux) * ux), Float32(1.0))), fma(fma(Float32(Float32(pi) + Float32(pi)), yi, Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))) * uy)), uy, xi), Float32(Float32(Float32(Float32(Float32(1.0) - ux) * zi) * ux) * maxCos))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

4. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, \color{blue}{yi \cdot \mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \color{blue}{\mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

6. Applied rewrites 89.3%

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)}\right) \]

8. Applied rewrites 89.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right), uy, xi\right), \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)} \]
9. Add Preprocessing

Alternative 9: 87.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot zi, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right) \cdot uy + xi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (* maxCos zi)
  ux
  (*
   (sqrt (fma (* (- ux 1.0) maxCos) (* (* (* maxCos (- 1.0 ux)) ux) ux) 1.0))
   (+
    (*
     (fma
      (+ PI PI)
      yi
      (*
       (fma
        (* (* PI PI) xi)
        -2.0
        (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
       uy))
     uy)
    xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((maxCos * zi), ux, (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * ((fmaf((((float) M_PI) + ((float) M_PI)), yi, (fmaf(((((float) M_PI) * ((float) M_PI)) * xi), -2.0f, (((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy) * -1.3333333333333333f)) * uy)) * uy) + xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * zi), ux, Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(Float32(fma(Float32(Float32(pi) + Float32(pi)), yi, Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))) * uy)) * uy) + xi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

4. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, \color{blue}{yi \cdot \mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \color{blue}{\mathsf{PI}\left(\right)}, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

6. Applied rewrites 89.3%

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right) + \color{blue}{xi}\right)\right) \]

   3. lower-+.f32 89.3

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right) + \color{blue}{xi}\right)\right) \]

8. Applied rewrites 89.4%

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right) \cdot uy + \color{blue}{xi}\right)\right) \]

9. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos \cdot zi}, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot \frac{-4}{3}\right) \cdot uy\right) \cdot uy + xi\right)\right) \]
10. Step-by-step derivation

    1. lower-*.f32 86.6

       \[\leadsto \mathsf{fma}\left(maxCos \cdot \color{blue}{zi}, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right) \cdot uy + xi\right)\right) \]

11. Applied rewrites 86.6%

    \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos \cdot zi}, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right) \cdot uy + xi\right)\right) \]
12. Add Preprocessing

Alternative 10: 86.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  maxCos
  (* ux zi)
  (fma
   2.0
   (* uy (* yi PI))
   (* xi (sin (- (* 0.5 PI) (* 2.0 (* PI (fabs uy)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * zi), fmaf(2.0f, (uy * (yi * ((float) M_PI))), (xi * sinf(((0.5f * ((float) M_PI)) - (2.0f * (((float) M_PI) * fabsf(uy))))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * zi), fma(Float32(2.0), Float32(uy * Float32(yi * Float32(pi))), Float32(xi * sin(Float32(Float32(Float32(0.5) * Float32(pi)) - Float32(Float32(2.0) * Float32(Float32(pi) * abs(uy))))))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - \left|uy\right| \cdot \left(\pi + \pi\right)\right)}\right)\right) \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

8. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

9. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right) \]
10. Step-by-step derivation

    1. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    2. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    4. lower-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    5. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    6. lower-sin.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    7. lower--.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    8. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    9. lower-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    10. lower-*.f32 N/A

        \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    11. lower-*.f32 N/A

        \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)\right)\right) \]

    12. lower-PI.f32 N/A

        \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)\right)\right) \]

    13. lower-fabs.f32 87.1

        \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)\right)\right) \]

11. Applied rewrites 87.1%

    \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), xi \cdot \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 83.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.006500000134110451:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.006500000134110451)
   (fma
    (* zi (- 1.0 ux))
    (* maxCos ux)
    (*
     (sqrt (fma (* (- ux 1.0) maxCos) (* (* (* maxCos (- 1.0 ux)) ux) ux) 1.0))
     (+ xi (* 2.0 (* uy (* yi PI))))))
   (fma maxCos (* ux zi) (* yi (sin (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.006500000134110451f) {
		tmp = fmaf((zi * (1.0f - ux)), (maxCos * ux), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * (xi + (2.0f * (uy * (yi * ((float) M_PI)))))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), (yi * sinf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.006500000134110451))
		tmp = fma(Float32(zi * Float32(Float32(1.0) - ux)), Float32(maxCos * ux), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, Float32(ux * zi), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00650000013

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

      5. lower-PI.f32 81.7

         \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 81.7%

      \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]

   if 0.00650000013 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - \left|uy\right| \cdot \left(\pi + \pi\right)\right)}\right)\right) \]

   6. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   9. Taylor expanded in xi around 0

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       3. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       4. lower-sin.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       5. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. lower-PI.f32 42.3

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   11. Applied rewrites 42.3%

       \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := xi \cdot \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)\\ \mathbf{if}\;xi \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;xi \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* xi (sin (- (* 0.5 PI) (* 2.0 (* PI (fabs uy))))))))
   (if (<= xi -4.999999918875795e-18)
     t_0
     (if (<= xi 4.99999991225835e-15)
       (fma maxCos (* ux zi) (* yi (sin (* 2.0 (* uy PI)))))
       t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = xi * sinf(((0.5f * ((float) M_PI)) - (2.0f * (((float) M_PI) * fabsf(uy)))));
	float tmp;
	if (xi <= -4.999999918875795e-18f) {
		tmp = t_0;
	} else if (xi <= 4.99999991225835e-15f) {
		tmp = fmaf(maxCos, (ux * zi), (yi * sinf((2.0f * (uy * ((float) M_PI))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(xi * sin(Float32(Float32(Float32(0.5) * Float32(pi)) - Float32(Float32(2.0) * Float32(Float32(pi) * abs(uy))))))
	tmp = Float32(0.0)
	if (xi <= Float32(-4.999999918875795e-18))
		tmp = t_0;
	elseif (xi <= Float32(4.99999991225835e-15))
		tmp = fma(maxCos, Float32(ux * zi), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if xi < -4.99999992e-18 or 4.99999991e-15 < xi

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - \left|uy\right| \cdot \left(\pi + \pi\right)\right)}\right)\right) \]

   6. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   9. Taylor expanded in xi around inf

      \[\leadsto xi \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       2. lower-sin.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       3. lower--.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       4. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       5. lower-PI.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       6. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       7. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       8. lower-PI.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right) \]

       9. lower-fabs.f32 53.0

          \[\leadsto xi \cdot \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right) \]

   11. Applied rewrites 53.0%

       \[\leadsto xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)} \]

   if -4.99999992e-18 < xi < 4.99999991e-15

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - \left|uy\right| \cdot \left(\pi + \pi\right)\right)}\right)\right) \]

   6. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   9. Taylor expanded in xi around 0

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       3. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       4. lower-sin.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       5. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. lower-PI.f32 42.3

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   11. Applied rewrites 42.3%

       \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0001500000071246177:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0001500000071246177)
   (fma
    (sqrt (fma (* (* (* (- ux 1.0) maxCos) ux) maxCos) (* (- 1.0 ux) ux) 1.0))
    xi
    (* (* (- 1.0 ux) (* maxCos ux)) zi))
   (* xi (sin (- (* 0.5 PI) (* 2.0 (* PI (fabs uy))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0001500000071246177f) {
		tmp = fmaf(sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), ((1.0f - ux) * ux), 1.0f)), xi, (((1.0f - ux) * (maxCos * ux)) * zi));
	} else {
		tmp = xi * sinf(((0.5f * ((float) M_PI)) - (2.0f * (((float) M_PI) * fabsf(uy)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0001500000071246177))
		tmp = fma(sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), Float32(Float32(Float32(1.0) - ux) * ux), Float32(1.0))), xi, Float32(Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * ux)) * zi));
	else
		tmp = Float32(xi * sin(Float32(Float32(Float32(0.5) * Float32(pi)) - Float32(Float32(2.0) * Float32(Float32(pi) * abs(uy))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.50000007e-4

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   6. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot zi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot zi\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot zi\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]

      6. lower-*.f32 51.6

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]

   8. Applied rewrites 51.6%

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]

   if 1.50000007e-4 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - \left|uy\right| \cdot \left(\pi + \pi\right)\right)}\right)\right) \]

   6. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   9. Taylor expanded in xi around inf

      \[\leadsto xi \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       2. lower-sin.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       3. lower--.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       4. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       5. lower-PI.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       6. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       7. lower-*.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left|uy\right|\right)\right) \]

       8. lower-PI.f32 N/A

          \[\leadsto xi \cdot \sin \left(\frac{1}{2} \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right) \]

       9. lower-fabs.f32 53.0

          \[\leadsto xi \cdot \sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right) \]

   11. Applied rewrites 53.0%

       \[\leadsto xi \cdot \color{blue}{\sin \left(0.5 \cdot \pi - 2 \cdot \left(\pi \cdot \left|uy\right|\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (sqrt (fma (* (* (* (- ux 1.0) maxCos) ux) maxCos) (* (- 1.0 ux) ux) 1.0))
  xi
  (* (* (- 1.0 ux) (* maxCos ux)) zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), ((1.0f - ux) * ux), 1.0f)), xi, (((1.0f - ux) * (maxCos * ux)) * zi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), Float32(Float32(Float32(1.0) - ux) * ux), Float32(1.0))), xi, Float32(Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * ux)) * zi))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

6. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot zi\right)} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot zi\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot zi\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]

   6. lower-*.f32 51.6

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]

8. Applied rewrites 51.6%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}, xi, \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi\right) \]
9. Add Preprocessing

Alternative 15: 51.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (* maxCos (* ux (* zi (- 1.0 ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * (ux * (zi * (1.0f - ux))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + (maxcos * (ux * (zi * (1.0e0 - ux))))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in maxCos around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right) \]

   5. lower--.f32 51.5

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

7. Applied rewrites 51.5%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
8. Add Preprocessing

Alternative 16: 49.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot zi, ux, xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (fma (* maxCos zi) ux xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((maxCos * zi), ux, xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * zi), ux, xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.5

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.5%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. +-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   3. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   4. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   5. *-commutative N/A

      \[\leadsto maxCos \cdot \left(zi \cdot ux\right) + xi \]

   6. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot zi\right) \cdot ux + xi \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot zi, ux, xi\right) \]

   8. lower-*.f32 49.5

      \[\leadsto \mathsf{fma}\left(maxCos \cdot zi, ux, xi\right) \]

9. Applied rewrites 49.5%

   \[\leadsto \mathsf{fma}\left(maxCos \cdot zi, ux, xi\right) \]
10. Add Preprocessing

Alternative 17: 49.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (fma (* maxCos ux) zi xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((maxCos * ux), zi, xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), zi, xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.5

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.5%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. +-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   3. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   4. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   5. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   6. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

   7. lower-*.f32 49.5

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

9. Applied rewrites 49.5%

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]
10. Add Preprocessing

Alternative 18: 11.8% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \left(maxCos \cdot zi\right) \cdot ux \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* (* maxCos zi) ux))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (maxCos * zi) * ux;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (maxcos * zi) * ux
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(maxCos * zi) * ux)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (maxCos * zi) * ux;
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.5

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.5%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 11.8

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 11.8%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

    2. lift-*.f32 N/A

       \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

    3. *-commutative N/A

       \[\leadsto maxCos \cdot \left(zi \cdot ux\right) \]

    4. associate-*r* N/A

       \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]

    5. lower-*.f32 N/A

       \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]

    6. lower-*.f32 11.8

       \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]

12. Applied rewrites 11.8%

    \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]
13. Add Preprocessing

Alternative 19: 11.8% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.5

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.5%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 11.8

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 11.8%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025151 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))