UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 8.2s

Alternatives: 15

Speedup: 0.5×

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, 0.5× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (* PI (* 2.0 uy)))
        (t_2 (sin (acos t_0))))
   (+ (fma (* (sin t_1) t_2) yi (* (cos t_1) (* t_2 xi))) (* 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 = ((float) M_PI) * (2.0f * uy);
	float t_2 = sinf(acosf(t_0));
	return fmaf((sinf(t_1) * t_2), yi, (cosf(t_1) * (t_2 * xi))) + (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 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	t_2 = sin(acos(t_0))
	return Float32(fma(Float32(sin(t_1) * t_2), yi, Float32(cos(t_1) * Float32(t_2 * xi))) + Float32(t_0 * zi))
end

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 99.0%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := yi \cdot \sin t\_0\\ \mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, \mathsf{fma}\left(-0.5, \left(maxCos \cdot maxCos\right) \cdot \left(xi \cdot t\_1\right), -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot t\_2\right)\right)\right)\right), \mathsf{fma}\left(xi, t\_1, t\_2\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))) (t_1 (cos t_0)) (t_2 (* yi (sin t_0))))
   (fma
    ux
    (fma
     maxCos
     zi
     (*
      ux
      (fma
       -1.0
       (* maxCos zi)
       (fma
        -0.5
        (* (* maxCos maxCos) (* xi t_1))
        (* -0.5 (* (* maxCos maxCos) t_2))))))
    (fma xi t_1 t_2))))

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 t_1 = cosf(t_0);
	float t_2 = yi * sinf(t_0);
	return fmaf(ux, fmaf(maxCos, zi, (ux * fmaf(-1.0f, (maxCos * zi), fmaf(-0.5f, ((maxCos * maxCos) * (xi * t_1)), (-0.5f * ((maxCos * maxCos) * t_2)))))), fmaf(xi, t_1, t_2));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = cos(t_0)
	t_2 = Float32(yi * sin(t_0))
	return fma(ux, fma(maxCos, zi, Float32(ux * fma(Float32(-1.0), Float32(maxCos * zi), fma(Float32(-0.5), Float32(Float32(maxCos * maxCos) * Float32(xi * t_1)), Float32(Float32(-0.5) * Float32(Float32(maxCos * maxCos) * t_2)))))), fma(xi, t_1, t_2))
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 zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift--.f32 13.7

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

4. Applied rewrites 13.7%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, \mathsf{fma}\left(-0.5, \left(maxCos \cdot maxCos\right) \cdot \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right), -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\right)\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right), ux \cdot ux, 1\right)\right) \cdot xi + \left(\sin t\_1 \cdot \sqrt{1 - t\_0 \cdot t\_0}\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)) (t_1 (* (* uy 2.0) PI)))
   (+
    (+
     (*
      (*
       (cos t_1)
       (fma
        (fma -0.5 (* maxCos maxCos) (* (* maxCos maxCos) ux))
        (* ux ux)
        1.0))
      xi)
     (* (* (sin t_1) (sqrt (- 1.0 (* t_0 t_0)))) 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 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_1) * fmaf(fmaf(-0.5f, (maxCos * maxCos), ((maxCos * maxCos) * ux)), (ux * ux), 1.0f)) * xi) + ((sinf(t_1) * sqrtf((1.0f - (t_0 * t_0)))) * 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 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_1) * fma(fma(Float32(-0.5), Float32(maxCos * maxCos), Float32(Float32(maxCos * maxCos) * ux)), Float32(ux * ux), Float32(1.0))) * xi) + Float32(Float32(sin(t_1) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) * yi)) + Float32(t_0 * 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 ux around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 98.9

       \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right), ux \cdot \color{blue}{ux}, 1\right)\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied rewrites 98.9%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right), ux \cdot ux, 1\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	return fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), fma(cos(t_0), xi, Float32(sin(t_0) * yi)))
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. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

Alternative 5: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	return fma(Float32(maxCos * ux), zi, fma(cos(t_0), xi, Float32(sin(t_0) * yi)))
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 ux around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

4. Applied rewrites 95.7%

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

Alternative 6: 95.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := xi \cdot \left(\pi \cdot \pi\right)\\ t_1 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;uy \leq 0.0035000001080334187:\\ \;\;\;\;xi + \mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right), uy \cdot \mathsf{fma}\left(-1, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(yi \cdot \pi\right)\right), \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, t\_0, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot t\_0\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_1, xi, \sin t\_1 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* xi (* PI PI))) (t_1 (* PI (* 2.0 uy))))
   (if (<= uy 0.0035000001080334187)
     (+
      xi
      (fma
       ux
       (fma
        maxCos
        zi
        (* ux (fma -1.0 (* maxCos zi) (* -0.5 (* (* maxCos maxCos) xi)))))
       (*
        uy
        (fma
         -1.0
         (* (* maxCos maxCos) (* (* ux ux) (* yi PI)))
         (fma
          2.0
          (* yi PI)
          (* uy (fma -2.0 t_0 (* (* maxCos maxCos) (* (* ux ux) t_0)))))))))
     (fma (cos t_1) xi (* (sin t_1) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = xi * (((float) M_PI) * ((float) M_PI));
	float t_1 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if (uy <= 0.0035000001080334187f) {
		tmp = xi + fmaf(ux, fmaf(maxCos, zi, (ux * fmaf(-1.0f, (maxCos * zi), (-0.5f * ((maxCos * maxCos) * xi))))), (uy * fmaf(-1.0f, ((maxCos * maxCos) * ((ux * ux) * (yi * ((float) M_PI)))), fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, t_0, ((maxCos * maxCos) * ((ux * ux) * t_0))))))));
	} else {
		tmp = fmaf(cosf(t_1), xi, (sinf(t_1) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(xi * Float32(Float32(pi) * Float32(pi)))
	t_1 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0035000001080334187))
		tmp = Float32(xi + fma(ux, fma(maxCos, zi, Float32(ux * fma(Float32(-1.0), Float32(maxCos * zi), Float32(Float32(-0.5) * Float32(Float32(maxCos * maxCos) * xi))))), Float32(uy * fma(Float32(-1.0), Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(yi * Float32(pi)))), fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), t_0, Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * t_0)))))))));
	else
		tmp = fma(cos(t_1), xi, Float32(sin(t_1) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00350000011

   1. Initial program 99.3%

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

   2. Taylor expanded in zi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 14.1

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

   4. Applied rewrites 14.1%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, \mathsf{fma}\left(-0.5, \left(maxCos \cdot maxCos\right) \cdot \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right), -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\right)\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 97.8%

       \[\leadsto xi + \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right), uy \cdot \mathsf{fma}\left(-1, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(yi \cdot \pi\right)\right), \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \]

   if 0.00350000011 < uy

   1. Initial program 98.0%

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

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.7%

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

Alternative 7: 85.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := xi \cdot \left(\pi \cdot \pi\right)\\ xi + \mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right), uy \cdot \mathsf{fma}\left(-1, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(yi \cdot \pi\right)\right), \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, t\_0, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot t\_0\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* xi (* PI PI))))
   (+
    xi
    (fma
     ux
     (fma
      maxCos
      zi
      (* ux (fma -1.0 (* maxCos zi) (* -0.5 (* (* maxCos maxCos) xi)))))
     (*
      uy
      (fma
       -1.0
       (* (* maxCos maxCos) (* (* ux ux) (* yi PI)))
       (fma
        2.0
        (* yi PI)
        (* uy (fma -2.0 t_0 (* (* maxCos maxCos) (* (* ux ux) t_0)))))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = xi * (((float) M_PI) * ((float) M_PI));
	return xi + fmaf(ux, fmaf(maxCos, zi, (ux * fmaf(-1.0f, (maxCos * zi), (-0.5f * ((maxCos * maxCos) * xi))))), (uy * fmaf(-1.0f, ((maxCos * maxCos) * ((ux * ux) * (yi * ((float) M_PI)))), fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, t_0, ((maxCos * maxCos) * ((ux * ux) * t_0))))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(xi * Float32(Float32(pi) * Float32(pi)))
	return Float32(xi + fma(ux, fma(maxCos, zi, Float32(ux * fma(Float32(-1.0), Float32(maxCos * zi), Float32(Float32(-0.5) * Float32(Float32(maxCos * maxCos) * xi))))), Float32(uy * fma(Float32(-1.0), Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(yi * Float32(pi)))), fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), t_0, Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * 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 zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift--.f32 13.7

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

4. Applied rewrites 13.7%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, \mathsf{fma}\left(-0.5, \left(maxCos \cdot maxCos\right) \cdot \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right), -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\right)\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

8. Taylor expanded in uy around 0

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

10. Applied rewrites 85.6%

    \[\leadsto xi + \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right), uy \cdot \mathsf{fma}\left(-1, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(yi \cdot \pi\right)\right), \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \]
11. Add Preprocessing

Alternative 8: 81.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ xi + \mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right), uy \cdot \mathsf{fma}\left(-1, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(yi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (fma
   ux
   (fma
    maxCos
    zi
    (* ux (fma -1.0 (* maxCos zi) (* -0.5 (* (* maxCos maxCos) xi)))))
   (*
    uy
    (fma
     -1.0
     (* (* maxCos maxCos) (* (* ux ux) (* yi PI)))
     (* 2.0 (* yi PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + fmaf(ux, fmaf(maxCos, zi, (ux * fmaf(-1.0f, (maxCos * zi), (-0.5f * ((maxCos * maxCos) * xi))))), (uy * fmaf(-1.0f, ((maxCos * maxCos) * ((ux * ux) * (yi * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(ux, fma(maxCos, zi, Float32(ux * fma(Float32(-1.0), Float32(maxCos * zi), Float32(Float32(-0.5) * Float32(Float32(maxCos * maxCos) * xi))))), Float32(uy * fma(Float32(-1.0), Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(yi * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))))
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 zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift--.f32 13.7

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

4. Applied rewrites 13.7%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, \mathsf{fma}\left(-0.5, \left(maxCos \cdot maxCos\right) \cdot \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right), -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\right)\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

8. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

10. Applied rewrites 81.4%

    \[\leadsto xi + \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right), uy \cdot \mathsf{fma}\left(-1, \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(yi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)} \]
11. Add Preprocessing

Alternative 9: 60.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \mathbf{if}\;yi \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot t\_0\\ \mathbf{elif}\;yi \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(1 + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right), xi, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (* uy PI)) yi)))
   (if (<= yi -4.99999991225835e-14)
     (* (sqrt (- 1.0 (* (* ux ux) (* maxCos maxCos)))) t_0)
     (if (<= yi 3.5000000934815034e-5)
       (fma
        (+
         1.0
         (* (* ux ux) (fma -0.5 (* maxCos maxCos) (* (* maxCos maxCos) ux))))
        xi
        (* (* (* (- 1.0 ux) zi) ux) maxCos))
       (* 1.0 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))) * yi;
	float tmp;
	if (yi <= -4.99999991225835e-14f) {
		tmp = sqrtf((1.0f - ((ux * ux) * (maxCos * maxCos)))) * t_0;
	} else if (yi <= 3.5000000934815034e-5f) {
		tmp = fmaf((1.0f + ((ux * ux) * fmaf(-0.5f, (maxCos * maxCos), ((maxCos * maxCos) * ux)))), xi, ((((1.0f - ux) * zi) * ux) * maxCos));
	} else {
		tmp = 1.0f * t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi)
	tmp = Float32(0.0)
	if (yi <= Float32(-4.99999991225835e-14))
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(ux * ux) * Float32(maxCos * maxCos)))) * t_0);
	elseif (yi <= Float32(3.5000000934815034e-5))
		tmp = fma(Float32(Float32(1.0) + Float32(Float32(ux * ux) * fma(Float32(-0.5), Float32(maxCos * maxCos), Float32(Float32(maxCos * maxCos) * ux)))), xi, Float32(Float32(Float32(Float32(Float32(1.0) - ux) * zi) * ux) * maxCos));
	else
		tmp = Float32(Float32(1.0) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if yi < -4.99999991e-14

   1. Initial program 98.8%

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

   2. Taylor expanded in yi around inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      3. lift-PI.f32 51.8

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   7. Applied rewrites 51.8%

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   8. Taylor expanded in ux around 0

      \[\leadsto \sqrt{1 - {ux}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

      2. lift-*.f32 51.8

         \[\leadsto \sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   10. Applied rewrites 51.8%

       \[\leadsto \sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   if -4.99999991e-14 < yi < 3.50000009e-5

   1. Initial program 99.0%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(1 + {ux}^{2} \cdot \left(\frac{-1}{2} \cdot {maxCos}^{2} + {maxCos}^{2} \cdot ux\right), xi, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right) \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-fma.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 63.3

          \[\leadsto \mathsf{fma}\left(1 + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right), xi, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right) \]

   7. Applied rewrites 63.3%

      \[\leadsto \mathsf{fma}\left(1 + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right), xi, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right) \]

   if 3.50000009e-5 < yi

   1. Initial program 98.7%

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

   2. Taylor expanded in yi around inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      3. lift-PI.f32 61.4

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   7. Applied rewrites 61.4%

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   8. Taylor expanded in ux around 0

      \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.2%

       \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \mathbf{if}\;yi \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot t\_0\\ \mathbf{elif}\;yi \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;xi + ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (* uy PI)) yi)))
   (if (<= yi -4.99999991225835e-14)
     (* (sqrt (- 1.0 (* (* ux ux) (* maxCos maxCos)))) t_0)
     (if (<= yi 3.5000000934815034e-5)
       (+
        xi
        (*
         ux
         (fma
          maxCos
          zi
          (* ux (fma -1.0 (* maxCos zi) (* -0.5 (* (* maxCos maxCos) xi)))))))
       (* 1.0 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))) * yi;
	float tmp;
	if (yi <= -4.99999991225835e-14f) {
		tmp = sqrtf((1.0f - ((ux * ux) * (maxCos * maxCos)))) * t_0;
	} else if (yi <= 3.5000000934815034e-5f) {
		tmp = xi + (ux * fmaf(maxCos, zi, (ux * fmaf(-1.0f, (maxCos * zi), (-0.5f * ((maxCos * maxCos) * xi))))));
	} else {
		tmp = 1.0f * t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi)
	tmp = Float32(0.0)
	if (yi <= Float32(-4.99999991225835e-14))
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(ux * ux) * Float32(maxCos * maxCos)))) * t_0);
	elseif (yi <= Float32(3.5000000934815034e-5))
		tmp = Float32(xi + Float32(ux * fma(maxCos, zi, Float32(ux * fma(Float32(-1.0), Float32(maxCos * zi), Float32(Float32(-0.5) * Float32(Float32(maxCos * maxCos) * xi)))))));
	else
		tmp = Float32(Float32(1.0) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if yi < -4.99999991e-14

   1. Initial program 98.8%

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

   2. Taylor expanded in yi around inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      3. lift-PI.f32 51.8

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   7. Applied rewrites 51.8%

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   8. Taylor expanded in ux around 0

      \[\leadsto \sqrt{1 - {ux}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

      2. lift-*.f32 51.8

         \[\leadsto \sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   10. Applied rewrites 51.8%

       \[\leadsto \sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   if -4.99999991e-14 < yi < 3.50000009e-5

   1. Initial program 99.0%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

         \[\leadsto xi + ux \cdot \color{blue}{\left(maxCos \cdot zi + ux \cdot \left(-1 \cdot \left(maxCos \cdot zi\right) + \frac{-1}{2} \cdot \left({maxCos}^{2} \cdot xi\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto xi + ux \cdot \left(maxCos \cdot zi + \color{blue}{ux \cdot \left(-1 \cdot \left(maxCos \cdot zi\right) + \frac{-1}{2} \cdot \left({maxCos}^{2} \cdot xi\right)\right)}\right) \]

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 63.3

          \[\leadsto xi + ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right) \]

   7. Applied rewrites 63.3%

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

   if 3.50000009e-5 < yi

   1. Initial program 98.7%

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

   2. Taylor expanded in yi around inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      3. lift-PI.f32 61.4

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   7. Applied rewrites 61.4%

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   8. Taylor expanded in ux around 0

      \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.2%

       \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\\ \mathbf{if}\;yi \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;xi + ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 1.0 (* (* 2.0 (* uy PI)) yi))))
   (if (<= yi -4.99999991225835e-14)
     t_0
     (if (<= yi 3.5000000934815034e-5)
       (+
        xi
        (*
         ux
         (fma
          maxCos
          zi
          (* ux (fma -1.0 (* maxCos zi) (* -0.5 (* (* maxCos maxCos) xi)))))))
       t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 1.0f * ((2.0f * (uy * ((float) M_PI))) * yi);
	float tmp;
	if (yi <= -4.99999991225835e-14f) {
		tmp = t_0;
	} else if (yi <= 3.5000000934815034e-5f) {
		tmp = xi + (ux * fmaf(maxCos, zi, (ux * fmaf(-1.0f, (maxCos * zi), (-0.5f * ((maxCos * maxCos) * xi))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) * Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi))
	tmp = Float32(0.0)
	if (yi <= Float32(-4.99999991225835e-14))
		tmp = t_0;
	elseif (yi <= Float32(3.5000000934815034e-5))
		tmp = Float32(xi + Float32(ux * fma(maxCos, zi, Float32(ux * fma(Float32(-1.0), Float32(maxCos * zi), Float32(Float32(-0.5) * Float32(Float32(maxCos * maxCos) * xi)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if yi < -4.99999991e-14 or 3.50000009e-5 < yi

   1. Initial program 98.7%

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

   2. Taylor expanded in yi around inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      3. lift-PI.f32 54.9

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   7. Applied rewrites 54.9%

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   8. Taylor expanded in ux around 0

      \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

       \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]

   if -4.99999991e-14 < yi < 3.50000009e-5

   1. Initial program 99.0%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

         \[\leadsto xi + ux \cdot \color{blue}{\left(maxCos \cdot zi + ux \cdot \left(-1 \cdot \left(maxCos \cdot zi\right) + \frac{-1}{2} \cdot \left({maxCos}^{2} \cdot xi\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto xi + ux \cdot \left(maxCos \cdot zi + \color{blue}{ux \cdot \left(-1 \cdot \left(maxCos \cdot zi\right) + \frac{-1}{2} \cdot \left({maxCos}^{2} \cdot xi\right)\right)}\right) \]

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 63.3

          \[\leadsto xi + ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right) \]

   7. Applied rewrites 63.3%

      \[\leadsto xi + \color{blue}{ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.7% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\\ \mathbf{if}\;yi \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 1.0 (* (* 2.0 (* uy PI)) yi))))
   (if (<= yi -4.99999991225835e-14)
     t_0
     (if (<= yi 3.5000000934815034e-5)
       (+ xi (* maxCos (* ux (* zi (- 1.0 ux)))))
       t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 1.0f * ((2.0f * (uy * ((float) M_PI))) * yi);
	float tmp;
	if (yi <= -4.99999991225835e-14f) {
		tmp = t_0;
	} else if (yi <= 3.5000000934815034e-5f) {
		tmp = xi + (maxCos * (ux * (zi * (1.0f - ux))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) * Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi))
	tmp = Float32(0.0)
	if (yi <= Float32(-4.99999991225835e-14))
		tmp = t_0;
	elseif (yi <= Float32(3.5000000934815034e-5))
		tmp = Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(1.0) * ((single(2.0) * (uy * single(pi))) * yi);
	tmp = single(0.0);
	if (yi <= single(-4.99999991225835e-14))
		tmp = t_0;
	elseif (yi <= single(3.5000000934815034e-5))
		tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -4.99999991e-14 or 3.50000009e-5 < yi

   1. Initial program 98.7%

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

   2. Taylor expanded in yi around inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      3. lift-PI.f32 54.9

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   7. Applied rewrites 54.9%

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   8. Taylor expanded in ux around 0

      \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

       \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]

   if -4.99999991e-14 < yi < 3.50000009e-5

   1. Initial program 99.0%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 63.3%

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

   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. lift--.f32 63.3

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

   7. Applied rewrites 63.3%

      \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\\ \mathbf{if}\;yi \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;xi + maxCos \cdot \left(ux \cdot zi\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 1.0 (* (* 2.0 (* uy PI)) yi))))
   (if (<= yi -4.99999991225835e-14)
     t_0
     (if (<= yi 3.5000000934815034e-5) (+ xi (* maxCos (* ux zi))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 1.0f * ((2.0f * (uy * ((float) M_PI))) * yi);
	float tmp;
	if (yi <= -4.99999991225835e-14f) {
		tmp = t_0;
	} else if (yi <= 3.5000000934815034e-5f) {
		tmp = xi + (maxCos * (ux * zi));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) * Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi))
	tmp = Float32(0.0)
	if (yi <= Float32(-4.99999991225835e-14))
		tmp = t_0;
	elseif (yi <= Float32(3.5000000934815034e-5))
		tmp = Float32(xi + Float32(maxCos * Float32(ux * zi)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(1.0) * ((single(2.0) * (uy * single(pi))) * yi);
	tmp = single(0.0);
	if (yi <= single(-4.99999991225835e-14))
		tmp = t_0;
	elseif (yi <= single(3.5000000934815034e-5))
		tmp = xi + (maxCos * (ux * zi));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -4.99999991e-14 or 3.50000009e-5 < yi

   1. Initial program 98.7%

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

   2. Taylor expanded in yi around inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]

      3. lift-PI.f32 54.9

         \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   7. Applied rewrites 54.9%

      \[\leadsto \sqrt{1 - {\left(\left(1 - ux\right) \cdot ux\right)}^{2} \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \]

   8. Taylor expanded in ux around 0

      \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

       \[\leadsto 1 \cdot \left(\color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) \]

   if -4.99999991e-14 < yi < 3.50000009e-5

   1. Initial program 99.0%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 63.3%

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

   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 60.4

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

   7. Applied rewrites 60.4%

      \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 49.8% accurate, 25.2× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (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 = xi + (maxcos * (ux * zi))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 52.0%

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

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 49.8

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

7. Applied rewrites 49.8%

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

Alternative 15: 45.6% accurate, 353.0× speedup

Math

\[\begin{array}{l} \\ xi \end{array} \]

FPCore

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

C

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

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
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = 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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 52.0%

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

5. Taylor expanded in ux around 0

   \[\leadsto xi \]
6. Step-by-step derivation

7. Applied rewrites 45.6%

   \[\leadsto xi \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(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)))