UniformSampleCone, y

Percentage Accurate: 58.1% → 98.3%

Time: 6.3s

Alternatives: 12

Speedup: 2.3×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 98.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (*
    (/ (fma (* (- 1.0 maxCos) (- maxCos 1.0)) ux (fma -2.0 maxCos 2.0)) ux)
    (* ux ux)))
  (sin (* (+ uy uy) PI))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((fmaf(((1.0f - maxCos) * (maxCos - 1.0f)), ux, fmaf(-2.0f, maxCos, 2.0f)) / ux) * (ux * ux))) * sinf(((uy + uy) * ((float) M_PI)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(fma(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos - Float32(1.0))), ux, fma(Float32(-2.0), maxCos, Float32(2.0))) / ux) * Float32(ux * ux))) * sin(Float32(Float32(uy + uy) * Float32(pi))))
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 98.2

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

4. Applied rewrites 98.2%

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

6. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. add-to-fraction N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 98.2

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 98.2

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

8. Applied rewrites 98.2%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (*
    (fma (- maxCos 1.0) (- 1.0 maxCos) (/ (fma -2.0 maxCos 2.0) ux))
    (* ux ux)))
  (sin (* (+ uy uy) PI))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((fmaf((maxCos - 1.0f), (1.0f - maxCos), (fmaf(-2.0f, maxCos, 2.0f) / ux)) * (ux * ux))) * sinf(((uy + uy) * ((float) M_PI)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(fma(Float32(maxCos - Float32(1.0)), Float32(Float32(1.0) - maxCos), Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux)) * Float32(ux * ux))) * sin(Float32(Float32(uy + uy) * Float32(pi))))
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 98.2

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

4. Applied rewrites 98.2%

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

6. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)} \]
7. Add Preprocessing

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (fma (- 1.0 maxCos) ux -2.0) (- (* maxCos ux) ux)))
  (sin (* (+ uy uy) PI))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(fma(Float32(Float32(1.0) - maxCos), ux, Float32(-2.0)) * Float32(Float32(maxCos * ux) - ux))) * sin(Float32(Float32(uy + uy) * Float32(pi))))
end

Derivation

1. Initial program 58.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 58.1

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

3. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]
4. Step-by-step derivation

5. Applied rewrites 98.3%

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

Alternative 4: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(ux - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (- (- (- ux (* maxCos ux)) 0.0)) (- ux 2.0)))
  (sin (* PI (+ uy uy)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((-((ux - (maxCos * ux)) - 0.0f) * (ux - 2.0f))) * sinf((((float) M_PI) * (uy + uy)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(-Float32(Float32(ux - Float32(maxCos * ux)) - Float32(0.0))) * Float32(ux - Float32(2.0)))) * sin(Float32(Float32(pi) * Float32(uy + uy))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((-((ux - (maxCos * ux)) - single(0.0)) * (ux - single(2.0)))) * sin((single(pi) * (uy + uy)));
end

Derivation

1. Initial program 58.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 58.1

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

3. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

4. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \color{blue}{\left(ux - 2\right)}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
5. Step-by-step derivation

   1. lower--.f32 97.0

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(ux - \color{blue}{2}\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

6. Applied rewrites 97.0%

   \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \color{blue}{\left(ux - 2\right)}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
7. Add Preprocessing

Alternative 5: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999873689376e-5)
   (* (sqrt (* -1.0 (* ux (- ux 2.0)))) (sin (* PI (+ uy uy))))
   (*
    (sqrt
     (*
      (fma (- maxCos 1.0) (- 1.0 maxCos) (/ (fma -2.0 maxCos 2.0) ux))
      (* ux ux)))
    (* 2.0 (* uy PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999873689376e-5f) {
		tmp = sqrtf((-1.0f * (ux * (ux - 2.0f)))) * sinf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = sqrtf((fmaf((maxCos - 1.0f), (1.0f - maxCos), (fmaf(-2.0f, maxCos, 2.0f) / ux)) * (ux * ux))) * (2.0f * (uy * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999873689376e-5))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(maxCos - Float32(1.0)), Float32(Float32(1.0) - maxCos), Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux)) * Float32(ux * ux))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-5

   1. Initial program 58.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 58.1

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

   3. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

   4. Taylor expanded in maxCos around 0

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 92.6

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

   6. Applied rewrites 92.6%

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

   if 4.99999987e-5 < maxCos

   1. Initial program 58.1%

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

   2. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 98.2

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

   4. Applied rewrites 98.2%

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

   6. Applied rewrites 98.2%

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

   7. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.5

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

   9. Applied rewrites 81.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.000750000006519258:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right), ux, \mathsf{fma}\left(-2, maxCos, 2\right)\right)}{ux} \cdot \left(ux \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.000750000006519258)
   (*
    (sqrt
     (*
      (/ (fma (* (- 1.0 maxCos) (- maxCos 1.0)) ux (fma -2.0 maxCos 2.0)) ux)
      (* ux ux)))
    (* 2.0 (* uy PI)))
   (* (sqrt (* -2.0 (* ux (- maxCos 1.0)))) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.000750000006519258f) {
		tmp = sqrtf(((fmaf(((1.0f - maxCos) * (maxCos - 1.0f)), ux, fmaf(-2.0f, maxCos, 2.0f)) / ux) * (ux * ux))) * (2.0f * (uy * ((float) M_PI)));
	} else {
		tmp = sqrtf((-2.0f * (ux * (maxCos - 1.0f)))) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.000750000006519258))
		tmp = Float32(sqrt(Float32(Float32(fma(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos - Float32(1.0))), ux, fma(Float32(-2.0), maxCos, Float32(2.0))) / ux) * Float32(ux * ux))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(Float32(-2.0) * Float32(ux * Float32(maxCos - Float32(1.0))))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 7.50000007e-4

   1. Initial program 58.1%

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

   2. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 98.2

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

   4. Applied rewrites 98.2%

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

   6. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. add-to-fraction N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 98.2

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 98.2

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-PI.f32 81.5

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

   11. Applied rewrites 81.5%

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

   if 7.50000007e-4 < uy

   1. Initial program 58.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 58.1

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

   3. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

   4. Taylor expanded in ux around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 76.3

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

   6. Applied rewrites 76.3%

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

Alternative 7: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.000750000006519258:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right), ux, \mathsf{fma}\left(-2, maxCos, 2\right)\right)}{ux} \cdot \left(ux \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.000750000006519258)
   (*
    (sqrt
     (*
      (/ (fma (* (- 1.0 maxCos) (- maxCos 1.0)) ux (fma -2.0 maxCos 2.0)) ux)
      (* ux ux)))
    (* 2.0 (* uy PI)))
   (* (sqrt (* 2.0 ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.000750000006519258f) {
		tmp = sqrtf(((fmaf(((1.0f - maxCos) * (maxCos - 1.0f)), ux, fmaf(-2.0f, maxCos, 2.0f)) / ux) * (ux * ux))) * (2.0f * (uy * ((float) M_PI)));
	} else {
		tmp = sqrtf((2.0f * ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.000750000006519258))
		tmp = Float32(sqrt(Float32(Float32(fma(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos - Float32(1.0))), ux, fma(Float32(-2.0), maxCos, Float32(2.0))) / ux) * Float32(ux * ux))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(Float32(2.0) * ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 7.50000007e-4

   1. Initial program 58.1%

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

   2. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 98.2

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

   4. Applied rewrites 98.2%

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

   6. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. add-to-fraction N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 98.2

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 98.2

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-PI.f32 81.5

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

   11. Applied rewrites 81.5%

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

   if 7.50000007e-4 < uy

   1. Initial program 58.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 58.1

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

   3. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

   4. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-pow.f32 N/A

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in ux around 0

      \[\leadsto \sqrt{2 \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
   8. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \sqrt{2 \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      2. lower-*.f32 73.0

         \[\leadsto \sqrt{2 \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   9. Applied rewrites 73.0%

      \[\leadsto \sqrt{2 \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (*
    (/ (fma (* (- 1.0 maxCos) (- maxCos 1.0)) ux (fma -2.0 maxCos 2.0)) ux)
    (* ux ux)))
  (* 2.0 (* uy PI))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((fmaf(((1.0f - maxCos) * (maxCos - 1.0f)), ux, fmaf(-2.0f, maxCos, 2.0f)) / ux) * (ux * ux))) * (2.0f * (uy * ((float) M_PI)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(fma(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos - Float32(1.0))), ux, fma(Float32(-2.0), maxCos, Float32(2.0))) / ux) * Float32(ux * ux))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))))
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 98.2

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

4. Applied rewrites 98.2%

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

6. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. add-to-fraction N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 98.2

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 98.2

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

8. Applied rewrites 98.2%

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

9. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 81.5

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

11. Applied rewrites 81.5%

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

Alternative 9: 81.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (*
    (fma (- maxCos 1.0) (- 1.0 maxCos) (/ (fma -2.0 maxCos 2.0) ux))
    (* ux ux)))
  (* 2.0 (* uy PI))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((fmaf((maxCos - 1.0f), (1.0f - maxCos), (fmaf(-2.0f, maxCos, 2.0f) / ux)) * (ux * ux))) * (2.0f * (uy * ((float) M_PI)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(fma(Float32(maxCos - Float32(1.0)), Float32(Float32(1.0) - maxCos), Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux)) * Float32(ux * ux))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))))
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 98.2

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

4. Applied rewrites 98.2%

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.5

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

9. Applied rewrites 81.5%

   \[\leadsto \sqrt{\mathsf{fma}\left(maxCos - 1, 1 - maxCos, \frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
10. Add Preprocessing

Alternative 10: 81.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - \left(2 + maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot ux - ux\right)}\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  2.0
  (* uy (* PI (sqrt (* (- ux (+ 2.0 (* maxCos ux))) (- (* maxCos ux) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	return 2.0f * (uy * (((float) M_PI) * sqrtf(((ux - (2.0f + (maxCos * ux))) * ((maxCos * ux) - ux)))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux - Float32(Float32(2.0) + Float32(maxCos * ux))) * Float32(Float32(maxCos * ux) - ux))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(2.0) * (uy * (single(pi) * sqrt(((ux - (single(2.0) + (maxCos * ux))) * ((maxCos * ux) - ux)))));
end

Derivation

1. Initial program 58.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 58.1

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

3. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 81.6

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

6. Applied rewrites 81.6%

   \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - \left(2 + maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot ux - ux\right)}\right)\right)} \]
7. Add Preprocessing

Alternative 11: 49.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* 2.0 (* uy PI)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	return (2.0f * (uy * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(2.0) * (uy * single(pi))) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 51.3

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

4. Applied rewrites 51.3%

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

5. Taylor expanded in maxCos around 0

   \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. lower--.f32 49.9

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

7. Applied rewrites 49.9%

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

8. Taylor expanded in maxCos around 0

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

   1. lower--.f32 49.7

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

10. Applied rewrites 49.7%

    \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
11. Add Preprocessing

Alternative 12: 7.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(\pi + \pi\right) \cdot uy\right) \cdot \sqrt{1 - 1} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (+ PI PI) uy) (sqrt (- 1.0 1.0))))

C

float code(float ux, float uy, float maxCos) {
	return ((((float) M_PI) + ((float) M_PI)) * uy) * sqrtf((1.0f - 1.0f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(Float32(pi) + Float32(pi)) * uy) * sqrt(Float32(Float32(1.0) - Float32(1.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ((single(pi) + single(pi)) * uy) * sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 51.3

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

4. Applied rewrites 51.3%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

      \[\leadsto \left(\left(\pi + \pi\right) \cdot uy\right) \cdot \sqrt{1 - 1} \]

   7. lower-+.f32 7.1

      \[\leadsto \left(\left(\pi + \pi\right) \cdot uy\right) \cdot \sqrt{1 - 1} \]

9. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\left(\left(\pi + \pi\right) \cdot uy\right)} \cdot \sqrt{1 - 1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))