UniformSampleCone, y

Percentage Accurate: 57.2% → 98.3%

Time: 7.3s

Alternatives: 14

Speedup: 3.2×

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

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

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: 57.2% accurate, 1.0× speedup

Math

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

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 57.2%

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

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. lower-fma.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift--.f32 N/A

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

   8. --rgt-identity N/A

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

   9. lift--.f32 N/A

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

   10. sub-negate-rev N/A

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

   11. lower--.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. lift--.f32 N/A

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

   15. --rgt-identity N/A

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

   16. lift--.f32 N/A

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

   17. sub-negate-rev N/A

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

   18. lower--.f32 N/A

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

   19. metadata-eval 98.3%

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

5. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3%

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

4. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lower--.f32 N/A

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

   6. lower--.f32 98.3%

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

   10. lower--.f32 N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity 98.3%

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

   13. lift-*.f32 N/A

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

   14. lift-pow.f32 N/A

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

   15. unpow2 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f32 N/A

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

   18. lower-*.f32 98.3%

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

6. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 57.2%

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

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. --rgt-identity N/A

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

   4. lift--.f32 N/A

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

   5. sub-negate-rev N/A

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

   6. sub-to-mult N/A

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

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

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

   8. lower-unsound--.f32 N/A

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

   9. lower-unsound-/.f32 97.4%

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

5. Applied rewrites 97.4%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. sub-to-fraction N/A

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

   4. frac-2neg N/A

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

   5. *-lft-identity N/A

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

   6. sub-negate-rev N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. distribute-rgt1-in N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-out N/A

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

   13. times-frac N/A

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

   14. lift-/.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lower-+.f32 N/A

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

   18. lower-neg.f32 N/A

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

   19. lower-neg.f32 94.7%

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

7. Applied rewrites 94.7%

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

8. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 98.3%

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

10. Applied rewrites 98.3%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 57.2%

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

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. --rgt-identity N/A

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

   4. lift--.f32 N/A

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

   5. sub-negate-rev N/A

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

   6. lower--.f32 98.3%

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate--l- N/A

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

   10. lower--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-fma.f32 98.3%

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

5. Applied rewrites 98.3%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 57.2%

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

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. lower-fma.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift--.f32 N/A

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

   8. --rgt-identity N/A

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

   9. lift--.f32 N/A

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

   10. sub-negate-rev N/A

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

   11. lower--.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. lift--.f32 N/A

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

   15. --rgt-identity N/A

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

   16. lift--.f32 N/A

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

   17. sub-negate-rev N/A

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

   18. lower--.f32 N/A

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

   19. metadata-eval 98.3%

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

5. Applied rewrites 98.3%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift--.f32 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f32 N/A

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

   9. distribute-lft-out-- N/A

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

   10. sqrt-prod N/A

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

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

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

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

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-unsound-sqrt.f32 N/A

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.3%

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

Alternative 6: 97.1% accurate, 1.1× speedup

Math

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

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 57.2%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 57.2%

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

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 97.1%

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

6. Applied rewrites 97.1%

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

Alternative 7: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 57.2%

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

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. lower-fma.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift--.f32 N/A

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

   8. --rgt-identity N/A

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

   9. lift--.f32 N/A

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

   10. sub-negate-rev N/A

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

   11. lower--.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. lift--.f32 N/A

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

   15. --rgt-identity N/A

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

   16. lift--.f32 N/A

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

   17. sub-negate-rev N/A

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

   18. lower--.f32 N/A

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

   19. metadata-eval 98.3%

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

5. Applied rewrites 98.3%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift--.f32 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f32 N/A

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

   9. distribute-lft-out-- N/A

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

   10. sqrt-prod N/A

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

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

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

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

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-unsound-sqrt.f32 N/A

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 96.9%

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

10. Applied rewrites 96.9%

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

Alternative 8: 96.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.00031999999191612005:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00031999999191612005)
   (*
    (* 2.0 (* uy PI))
    (sqrt
     (*
      ux
      (- (+ 2.0 (* -1.0 (* ux (pow (- maxCos 1.0) 2.0)))) (* 2.0 maxCos)))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00031999999191612005f) {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (2.0f * maxCos))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00031999999191612005))
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.00031999999191612005))
		tmp = (single(2.0) * (uy * single(pi))) * sqrt((ux * ((single(2.0) + (single(-1.0) * (ux * ((maxCos - single(1.0)) ^ single(2.0))))) - (single(2.0) * maxCos))));
	else
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.19999992e-4

   1. Initial program 57.2%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 98.3%

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.4%

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

   7. Applied rewrites 81.4%

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

   if 3.19999992e-4 < uy

   1. Initial program 57.2%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 98.3%

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

   4. Applied rewrites 98.3%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. associate--r+ N/A

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

      5. lower--.f32 N/A

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

      6. lower--.f32 98.3%

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

      7. lift-+.f32 N/A

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

      8. lift-*.f32 N/A

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

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

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

      10. lower--.f32 N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity 98.3%

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

      13. lift-*.f32 N/A

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

      14. lift-pow.f32 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 98.3%

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in maxCos around 0

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

      1. lower--.f32 92.4%

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

   9. Applied rewrites 92.4%

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

Alternative 9: 81.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3%

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.4%

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

7. Applied rewrites 81.4%

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

Alternative 10: 81.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3%

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

4. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lower--.f32 N/A

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

   6. lower--.f32 98.3%

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

   10. lower--.f32 N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity 98.3%

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

   13. lift-*.f32 N/A

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

   14. lift-pow.f32 N/A

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

   15. unpow2 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f32 N/A

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

   18. lower-*.f32 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.4%

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

9. Applied rewrites 81.4%

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

Alternative 11: 81.4% accurate, 2.3× speedup

Math

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

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 57.2%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 57.2%

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

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

   \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - \left(2 + maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot ux - ux\right)}\right)\right)} \]
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 81.4%

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

6. Applied rewrites 81.4%

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

Alternative 12: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996318817138672:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* 2.0 (* uy PI))))
   (if (<= (* t_0 t_0) 0.9996318817138672)
     (* t_1 (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))
     (* t_1 (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if ((t_0 * t_0) <= 0.9996318817138672f) {
		tmp = t_1 * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	} else {
		tmp = t_1 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9996318817138672))
		tmp = Float32(t_1 * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(t_1 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	t_1 = single(2.0) * (uy * single(pi));
	tmp = single(0.0);
	if ((t_0 * t_0) <= single(0.9996318817138672))
		tmp = t_1 * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	else
		tmp = t_1 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99963188

   1. Initial program 57.2%

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

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 50.2%

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

   4. Applied rewrites 50.2%

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

   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 48.8%

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

   7. Applied rewrites 48.8%

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

   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 48.7%

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

   10. Applied rewrites 48.7%

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

   if 0.99963188 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.2%

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

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 50.2%

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

   4. Applied rewrites 50.2%

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

   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. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 65.9%

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

   10. Applied rewrites 65.9%

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

Alternative 13: 65.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.2%

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

4. Applied rewrites 50.2%

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

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. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 65.9%

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

10. Applied rewrites 65.9%

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

Alternative 14: 7.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.2%

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

4. Applied rewrites 50.2%

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

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. associate-*r* N/A

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

   4. count-2 N/A

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

   5. lift-+.f32 N/A

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

   6. lower-*.f32 7.1%

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

9. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025183 
(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)))))))