UniformSampleCone, y

Percentage Accurate: 57.7% → 98.3%

Time: 1.4min

Alternatives: 14

Speedup: 3.3×

• unsoundness detected in the egraph

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3%

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

6. Applied rewrites 98.3%

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

   1. lift-fma.f32 N/A

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

   2. add-flip N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. associate--r+ N/A

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

   6. *-commutative N/A

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

   7. lift--.f32 N/A

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

   8. distribute-rgt-out-- N/A

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

   9. *-lft-identity N/A

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

   10. lift-*.f32 N/A

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

   11. associate--r+ N/A

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

   12. lift-*.f32 N/A

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

   13. lift-fma.f32 N/A

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

   14. lift-fma.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. +-commutative N/A

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

   17. associate--r+ N/A

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

   18. associate--l- N/A

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

   19. lift-*.f32 N/A

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

   20. lift-fma.f32 N/A

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

   21. lower--.f32 N/A

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

   22. lower--.f32 98.3%

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

8. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3%

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

6. Applied rewrites 98.3%

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

Alternative 3: 97.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. lower--.f32 97.0%

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

9. Applied rewrites 97.0%

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

Alternative 4: 95.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.00019500000053085387:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 0.00019500000053085387)
  (*
   2.0
   (*
    uy
    (*
     PI
     (sqrt (* (* (- maxCos 1.0) ux) (- (* ux (- 1.0 maxCos)) 2.0))))))
  (* (sin (* (* uy 2.0) PI)) (sqrt (* -1.0 (* ux (- ux 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00019500000053085387f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((((maxCos - 1.0f) * ux) * ((ux * (1.0f - maxCos)) - 2.0f)))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((-1.0f * (ux * (ux - 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00019500000053085387))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(maxCos - Float32(1.0)) * ux) * Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) - Float32(2.0)))))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.00019500000053085387))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((((maxCos - single(1.0)) * ux) * ((ux * (single(1.0) - maxCos)) - single(2.0))))));
	else
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(-1.0) * (ux * (ux - single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.95000001e-4

   1. Initial program 57.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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. sub-flip N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-*.f32 N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

      9. difference-of-squares N/A

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

      10. distribute-neg-out N/A

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

      11. add-flip-rev N/A

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

      12. sub-negate-rev N/A

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

      13. sub-flip-reverse N/A

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

      14. distribute-neg-out N/A

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

      15. sub-flip-reverse N/A

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 81.3%

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

   6. Applied rewrites 81.3%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 81.3%

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

   8. Applied rewrites 81.3%

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

   if 1.95000001e-4 < uy

   1. Initial program 57.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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. sub-flip N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-*.f32 N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

      9. difference-of-squares N/A

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

      10. distribute-neg-out N/A

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

      11. add-flip-rev N/A

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

      12. sub-negate-rev N/A

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

      13. sub-flip-reverse N/A

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

      14. distribute-neg-out N/A

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

      15. sub-flip-reverse N/A

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 92.1%

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

   6. Applied rewrites 92.1%

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

Alternative 5: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.0017999999690800905:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 0.0017999999690800905)
  (*
   2.0
   (*
    uy
    (*
     PI
     (sqrt (* (* (- maxCos 1.0) ux) (- (* ux (- 1.0 maxCos)) 2.0))))))
  (* (sin (* (* uy 2.0) PI)) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0017999999690800905f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((((maxCos - 1.0f) * ux) * ((ux * (1.0f - maxCos)) - 2.0f)))));
	} 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(0.0017999999690800905))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(maxCos - Float32(1.0)) * ux) * Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) - Float32(2.0)))))));
	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(0.0017999999690800905))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((((maxCos - single(1.0)) * ux) * ((ux * (single(1.0) - maxCos)) - single(2.0))))));
	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 < 0.00179999997

   1. Initial program 57.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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. sub-flip N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-*.f32 N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

      9. difference-of-squares N/A

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

      10. distribute-neg-out N/A

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

      11. add-flip-rev N/A

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

      12. sub-negate-rev N/A

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

      13. sub-flip-reverse N/A

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

      14. distribute-neg-out N/A

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

      15. sub-flip-reverse N/A

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 81.3%

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

   6. Applied rewrites 81.3%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 81.3%

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

   8. Applied rewrites 81.3%

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

   if 0.00179999997 < uy

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   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 76.2%

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

   4. Applied rewrites 76.2%

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

   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 72.6%

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

   7. Applied rewrites 72.6%

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

Alternative 6: 81.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-lft-identity N/A

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

   4. distribute-rgt-out-- N/A

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

   5. lift--.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 81.3%

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

8. Applied rewrites 81.3%

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

Alternative 7: 81.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower--.f32 N/A

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

   4. metadata-eval N/A

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

   5. add-flip N/A

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

   6. lift-fma.f32 81.3%

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

8. Applied rewrites 81.3%

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

Alternative 8: 81.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower--.f32 N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lift--.f32 N/A

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

   7. distribute-rgt-out-- N/A

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

   8. *-lft-identity N/A

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

   9. lift-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. associate--l- N/A

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

   12. lower--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lower-fma.f32 81.3%

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

8. Applied rewrites 81.3%

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

Alternative 9: 80.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower--.f32 N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lift--.f32 N/A

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

   7. distribute-rgt-out-- N/A

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

   8. *-lft-identity N/A

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

   9. lift-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. associate--l- N/A

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

   12. lower--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lower-fma.f32 81.3%

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

8. Applied rewrites 81.3%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 80.4%

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

Alternative 10: 76.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 76.9%

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

9. Applied rewrites 76.9%

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

Alternative 11: 65.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

7. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 65.7%

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

9. Applied rewrites 65.7%

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

Alternative 12: 21.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

8. Applied rewrites 21.8%

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

Alternative 13: 21.4% accurate, 17.0× speedup

Math

\[2 \cdot uy \]

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 2.0e0 * uy
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

8. Applied rewrites 21.4%

   \[\leadsto 2 \cdot uy \]
9. Add Preprocessing

Alternative 14: 14.0% accurate, 68.0× speedup

Math

\[6.2831854820251465 \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  6.2831854820251465)

C

float code(float ux, float uy, float maxCos) {
	return 6.2831854820251465f;
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 6.2831854820251465e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(6.2831854820251465)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(6.2831854820251465);
end

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. sub-flip N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. sqr-neg-rev N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

   9. difference-of-squares N/A

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

   10. distribute-neg-out N/A

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

   11. add-flip-rev N/A

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

   12. sub-negate-rev N/A

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

   13. sub-flip-reverse N/A

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

   14. distribute-neg-out N/A

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

   15. sub-flip-reverse N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 81.3%

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

6. Applied rewrites 81.3%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. add-flip N/A

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

   4. flip-- N/A

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

8. Applied rewrites 14.0%

   \[\leadsto \pi + \color{blue}{\pi} \]

9. Evaluated real constant 14.0%

   \[\leadsto 6.2831854820251465 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))