UniformSampleCone, y

Percentage Accurate: 57.4% → 98.4%

Time: 4.1s

Alternatives: 11

Speedup: 5.0×

Specification

\[\left(\left(\frac{2328306437}{10000000000000000000} \leq ux \land ux \leq 1\right) \land \left(\frac{2328306437}{10000000000000000000} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1 ux) (* ux maxCos))))
  (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* 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: 57.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1 ux) (* ux maxCos))))
  (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* 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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (sqrt
  (*
   (- (- (* (* (- 1 maxCos) (- maxCos 1)) ux) -2) (+ maxCos maxCos))
   ux))
 (sin (* PI (+ uy uy)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.4%

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

4. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.4%

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

6. Applied rewrites 98.4%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (sqrt (* (- 1 maxCos) (* ux (- (- (* maxCos ux) -2) ux))))
 (sin (* PI (+ uy uy)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   3. lower-*.f32 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

5. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. sub-flip N/A

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

   7. lift--.f32 N/A

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

   8. lower-*.f32 98.4%

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

7. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-+l- N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. add-flip-rev N/A

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

   12. +-commutative N/A

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

   13. lift-+.f32 N/A

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

   14. lower-*.f32 98.4%

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. add-flip-rev N/A

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

   18. lower--.f32 N/A

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

   19. metadata-eval 98.4%

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

9. Applied rewrites 98.4%

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

Alternative 3: 97.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (sqrt (* (- (+ 1 1) ux) (* (- 1 maxCos) ux)))
 (sin (* PI (+ uy uy)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   3. lower-*.f32 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

5. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. sub-flip N/A

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

   7. lift--.f32 N/A

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

   8. lower-*.f32 98.4%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 97.1%

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

Alternative 4: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq \frac{2233383}{4294967296}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(ux \cdot \left(maxCos + 2 \cdot \frac{1}{ux}\right) - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 2233383/4294967296)
  (*
   2
   (*
    uy
    (*
     PI
     (sqrt
      (*
       (- ux (* maxCos ux))
       (- (* ux (+ maxCos (* 2 (/ 1 ux)))) ux))))))
  (* (sqrt (* ux (- 2 ux))) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0005200000014156103f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((ux - (maxCos * ux)) * ((ux * (maxCos + (2.0f * (1.0f / ux)))) - ux)))));
	} else {
		tmp = sqrtf((ux * (2.0f - ux))) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0005200000014156103))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux - Float32(maxCos * ux)) * Float32(Float32(ux * Float32(maxCos + Float32(Float32(2.0) * Float32(Float32(1.0) / ux)))) - ux))))));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0005200000014156103))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((ux - (maxCos * ux)) * ((ux * (maxCos + (single(2.0) * (single(1.0) / ux)))) - ux)))));
	else
		tmp = sqrt((ux * (single(2.0) - ux))) * sin((single(pi) * (uy + uy)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 5.20000001e-4

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. metadata-eval N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. sub-negate N/A

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

      11. associate-+l- N/A

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

      12. associate-+l- N/A

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

      13. lift--.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 81.5%

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

   9. Applied rewrites 81.5%

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

   if 5.20000001e-4 < uy

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. metadata-eval N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. sub-negate N/A

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

      11. associate-+l- N/A

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

      12. associate-+l- N/A

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

      13. lift--.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

      3. lower-*.f32 98.3%

         \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   5. Applied rewrites 98.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. sub-flip N/A

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

      7. lift--.f32 N/A

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

      8. lower-*.f32 98.4%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 92.6%

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

   10. Applied rewrites 92.6%

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

Alternative 5: 89.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq \frac{15375983}{17179869184}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(ux \cdot \left(maxCos + 2 \cdot \frac{1}{ux}\right) - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 15375983/17179869184)
  (*
   2
   (*
    uy
    (*
     PI
     (sqrt
      (*
       (- ux (* maxCos ux))
       (- (* ux (+ maxCos (* 2 (/ 1 ux)))) ux))))))
  (* (sin (* (* uy 2) PI)) (sqrt (* 2 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0008950000046752393f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((ux - (maxCos * ux)) * ((ux * (maxCos + (2.0f * (1.0f / ux)))) - ux)))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0008950000046752393))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux - Float32(maxCos * ux)) * Float32(Float32(ux * Float32(maxCos + Float32(Float32(2.0) * Float32(Float32(1.0) / ux)))) - ux))))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0008950000046752393))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((ux - (maxCos * ux)) * ((ux * (maxCos + (single(2.0) * (single(1.0) / ux)))) - ux)))));
	else
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 8.95000005e-4

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. metadata-eval N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. sub-negate N/A

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

      11. associate-+l- N/A

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

      12. associate-+l- N/A

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

      13. lift--.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 81.5%

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

   9. Applied rewrites 81.5%

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

   if 8.95000005e-4 < uy

   1. Initial program 57.4%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.5%

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 73.1%

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

   7. Applied rewrites 73.1%

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

Alternative 6: 81.5% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 2
 (*
  uy
  (*
   PI
   (sqrt
    (*
     (- ux (* maxCos ux))
     (- (* ux (+ maxCos (* 2 (/ 1 ux)))) ux)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 81.5%

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

9. Applied rewrites 81.5%

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

Alternative 7: 81.5% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (* PI (+ uy uy))
 (sqrt (* (- (- (* maxCos ux) -2) ux) (- ux (* maxCos ux))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f32 N/A

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

   9. lower-*.f32 81.5%

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. count-2 N/A

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

   15. lift-+.f32 N/A

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

   16. lift-*.f32 81.5%

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

8. Applied rewrites 81.5%

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

Alternative 8: 80.7% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (* 2 (* uy (* PI (sqrt (* (- ux (* maxCos ux)) (- 2 ux)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in ux around 0

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

9. Applied rewrites 80.7%

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

Alternative 9: 77.4% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (* (* (+ PI PI) uy) (sqrt (* (- 2 ux) ux))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 77.4%

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

9. Applied rewrites 77.4%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-*r* N/A

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

    5. associate-*r* N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. lift-*.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. lower-*.f32 77.4%

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

11. Applied rewrites 77.4%

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

Alternative 10: 77.4% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (* (* (+ uy uy) (sqrt (* (- 2 ux) ux))) PI))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 77.4%

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

9. Applied rewrites 77.4%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. count-2 N/A

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

    5. lift-+.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f32 N/A

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

11. Applied rewrites 77.4%

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

Alternative 11: 63.4% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (* 2 (* uy (* PI (sqrt (* 2 ux))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\left(maxCos \cdot ux - -1\right) - \left(ux - 1\right)\right)}} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 77.4%

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

9. Applied rewrites 77.4%

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

10. Taylor expanded in ux around 0

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

    1. lower-*.f32 63.4%

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

12. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2328306437/10000000000000000000 ux) (<= ux 1)) (and (<= 2328306437/10000000000000000000 uy) (<= uy 1))) (and (<= 0 maxCos) (<= maxCos 1)))
  (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* (+ (- 1 ux) (* ux maxCos)) (+ (- 1 ux) (* ux maxCos)))))))