UniformSampleCone, y

Percentage Accurate: 57.7% → 98.3%

Time: 6.6s

Alternatives: 18

Speedup: 4.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

Alternative 3: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. negate-sub N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. negate-sub N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 97.7

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

9. Applied rewrites 97.7%

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

Alternative 4: 96.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0017000000225380063:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 - ux\right) - maxCos\right) \cdot ux} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0017000000225380063)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sqrt (* (- (- 2.0 ux) maxCos) ux)) (sin (* (+ uy uy) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0017000000225380063f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sqrtf((((2.0f - ux) - maxCos) * ux)) * sinf(((uy + uy) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0017000000225380063))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(2.0) - ux) - maxCos) * ux)) * sin(Float32(Float32(uy + uy) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00170000002

   1. Initial program 57.5%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 98.5%

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

   if 0.00170000002 < uy

   1. Initial program 58.1%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 97.9

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

   4. Applied rewrites 97.9%

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in maxCos around 0

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

      1. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f32 92.7

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

   9. Applied rewrites 92.7%

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

Alternative 5: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0017000000225380063:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0017000000225380063)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sqrt (* (- 2.0 ux) ux)) (sin (* (+ uy uy) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0017000000225380063f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sqrtf(((2.0f - ux) * ux)) * sinf(((uy + uy) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0017000000225380063))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * sin(Float32(Float32(uy + uy) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00170000002

   1. Initial program 57.5%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 98.5%

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

   if 0.00170000002 < uy

   1. Initial program 58.1%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 97.9

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

   4. Applied rewrites 97.9%

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in maxCos around 0

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

      1. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f32 92.1

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

   9. Applied rewrites 92.1%

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

Alternative 6: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0017000000225380063:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(1 - maxCos\right) \cdot \left(1 - maxCos\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0017000000225380063)
   (*
    (sqrt
     (*
      (- (- (fma (- ux) (* (- 1.0 maxCos) (- 1.0 maxCos)) 2.0) maxCos) maxCos)
      ux))
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI))))
   (* (sqrt (* (- 2.0 ux) ux)) (sin (* (+ uy uy) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0017000000225380063f) {
		tmp = sqrtf((((fmaf(-ux, ((1.0f - maxCos) * (1.0f - maxCos)), 2.0f) - maxCos) - maxCos) * ux)) * (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = sqrtf(((2.0f - ux) * ux)) * sinf(((uy + uy) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0017000000225380063))
		tmp = Float32(sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos)), Float32(2.0)) - maxCos) - maxCos) * ux)) * Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * sin(Float32(Float32(uy + uy) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00170000002

   1. Initial program 57.5%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

   6. Applied rewrites 98.5%

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

   7. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow3 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lift-PI.f32 98.5

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

   9. Applied rewrites 98.5%

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

   if 0.00170000002 < uy

   1. Initial program 58.1%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 97.9

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

   4. Applied rewrites 97.9%

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in maxCos around 0

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

      1. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f32 92.1

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

   9. Applied rewrites 92.1%

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

Alternative 7: 96.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. mul-1-neg N/A

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

   2. negate-sub N/A

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

   3. lower--.f32 96.9

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

9. Applied rewrites 96.9%

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

Alternative 8: 96.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy + uy\right) \cdot \pi\\ \mathbf{if}\;uy \leq 0.0003499999875202775:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ uy uy) PI)))
   (if (<= uy 0.0003499999875202775)
     (*
      t_0
      (sqrt
       (*
        (-
         (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0)
         (+ maxCos maxCos))
        ux)))
     (* (sqrt (* (- 2.0 ux) ux)) (sin t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (uy + uy) * ((float) M_PI);
	float tmp;
	if (uy <= 0.0003499999875202775f) {
		tmp = t_0 * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sqrtf(((2.0f - ux) * ux)) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(uy + uy) * Float32(pi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0003499999875202775))
		tmp = Float32(t_0 * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.49999988e-4

   1. Initial program 57.7%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 98.1

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

   7. Applied rewrites 98.1%

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

   if 3.49999988e-4 < uy

   1. Initial program 57.7%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.0

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

   4. Applied rewrites 98.0%

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in maxCos around 0

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

      1. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f32 92.2

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

   9. Applied rewrites 92.2%

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

Alternative 9: 87.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (- (- (- 2.0 ux) maxCos) maxCos) ux))
  (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(Float32(Float32(Float32(2.0) - ux) - maxCos) - maxCos) * ux)) * Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. mul-1-neg N/A

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

   2. negate-sub N/A

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

   3. lower--.f32 96.9

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

9. Applied rewrites 96.9%

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

10. Taylor expanded in uy around 0

    \[\leadsto \sqrt{\left(\left(\left(2 - ux\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \color{blue}{\left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f32 N/A

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

    6. unpow3 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(2 - ux\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

    9. lift-PI.f32 N/A

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

    10. lift-PI.f32 N/A

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

    11. lift-PI.f32 N/A

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

    12. lower-*.f32 N/A

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

    13. lift-PI.f32 87.9

        \[\leadsto \sqrt{\left(\left(\left(2 - ux\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \]

12. Applied rewrites 87.9%

    \[\leadsto \sqrt{\left(\left(\left(2 - ux\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)} \]
13. Add Preprocessing

Alternative 10: 81.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 81.6

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

7. Applied rewrites 81.6%

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

Alternative 11: 81.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in uy around 0

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

   1. count-2-rev N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-+.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 81.6

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

9. Applied rewrites 81.6%

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

Alternative 12: 80.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. mul-1-neg N/A

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

   2. negate-sub N/A

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

   3. lower--.f32 96.9

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

9. Applied rewrites 96.9%

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

10. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lift-PI.f32 80.6

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

12. Applied rewrites 80.6%

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

Alternative 13: 75.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00015999999595806003:\\ \;\;\;\;\left(2 \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\right) \cdot \left(\pi \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00015999999595806003)
   (* (* 2.0 (* (sqrt (fma -2.0 maxCos 2.0)) (sqrt ux))) (* PI uy))
   (* (* PI (+ uy uy)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00015999999595806003f) {
		tmp = (2.0f * (sqrtf(fmaf(-2.0f, maxCos, 2.0f)) * sqrtf(ux))) * (((float) M_PI) * uy);
	} else {
		tmp = (((float) M_PI) * (uy + uy)) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00015999999595806003))
		tmp = Float32(Float32(Float32(2.0) * Float32(sqrt(fma(Float32(-2.0), maxCos, Float32(2.0))) * sqrt(ux))) * Float32(Float32(pi) * uy));
	else
		tmp = Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.59999996e-4

   1. Initial program 36.9%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 78.1

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

   7. Applied rewrites 78.1%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-fma.f32 N/A

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

      8. lower-sqrt.f32 78.1

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

   9. Applied rewrites 78.1%

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

   if 1.59999996e-4 < ux

   1. Initial program 89.2%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in maxCos around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift--.f32 72.2

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

   7. Applied rewrites 72.2%

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

Alternative 14: 66.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 66.0

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

7. Applied rewrites 66.0%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. lower-sqrt.f32 66.0

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

9. Applied rewrites 66.0%

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

Alternative 15: 66.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 66.0

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

7. Applied rewrites 66.0%

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

Alternative 16: 63.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 66.0

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

7. Applied rewrites 66.0%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(\sqrt{ux} \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(\sqrt{ux} \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \]

   6. lift-PI.f32 N/A

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

   7. lower-sqrt.f32 63.3

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

10. Applied rewrites 63.3%

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

Alternative 17: 63.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 66.0

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

7. Applied rewrites 66.0%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 63.3%

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

Alternative 18: 7.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025119 
(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)))))))