UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 7.3s

Alternatives: 14

Speedup: 4.8×

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: 57.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 57.5%

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 57.5%

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

3. Applied rewrites 57.5%

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

4. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. lower-*.f32 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 98.3%

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

9. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 57.5%

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 57.5%

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

3. Applied rewrites 57.5%

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

4. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. lower-*.f32 98.3%

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

6. Applied rewrites 98.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 98.3%

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

8. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 57.5%

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 57.5%

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

3. Applied rewrites 57.5%

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

4. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. lower-*.f32 98.3%

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.3%

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

Alternative 4: 97.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 57.5%

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 57.5%

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

3. Applied rewrites 57.5%

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

4. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. lower-*.f32 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in ux around 0

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

9. Applied rewrites 97.2%

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

Alternative 5: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathbf{if}\;maxCos \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(0 - \left(maxCos \cdot ux - ux\right)\right) \cdot \left(2 - \left(ux - maxCos \cdot ux\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* PI (+ uy uy))))
  (if (<= maxCos 6.000000212225132e-7)
    (* (sqrt (* (- 2.0 ux) ux)) (sin t_0))
    (*
     t_0
     (sqrt
      (*
       (- 0.0 (- (* maxCos ux) ux))
       (- 2.0 (- ux (* maxCos ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float tmp;
	if (maxCos <= 6.000000212225132e-7f) {
		tmp = sqrtf(((2.0f - ux) * ux)) * sinf(t_0);
	} else {
		tmp = t_0 * sqrtf(((0.0f - ((maxCos * ux) - ux)) * (2.0f - (ux - (maxCos * ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	tmp = Float32(0.0)
	if (maxCos <= Float32(6.000000212225132e-7))
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * sin(t_0));
	else
		tmp = Float32(t_0 * sqrt(Float32(Float32(Float32(0.0) - Float32(Float32(maxCos * ux) - ux)) * Float32(Float32(2.0) - Float32(ux - Float32(maxCos * ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(pi) * (uy + uy);
	tmp = single(0.0);
	if (maxCos <= single(6.000000212225132e-7))
		tmp = sqrt(((single(2.0) - ux) * ux)) * sin(t_0);
	else
		tmp = t_0 * sqrt(((single(0.0) - ((maxCos * ux) - ux)) * (single(2.0) - (ux - (maxCos * ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 6.00000021e-7

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f32 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 57.5%

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 57.5%

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

   3. Applied rewrites 57.5%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 92.3%

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

   6. Applied rewrites 92.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 92.3%

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

   8. Applied rewrites 92.3%

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

   if 6.00000021e-7 < maxCos

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.2%

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

   4. Applied rewrites 50.2%

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

   6. Applied rewrites 81.7%

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{\sqrt{\left(0 - \left(maxCos \cdot ux - ux\right)\right) \cdot \left(2 - \left(ux - maxCos \cdot ux\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \mathbf{if}\;uy \leq 0.002899999963119626:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{-t\_0 \cdot \left(-2 + t\_0\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (- ux (* maxCos ux))))
  (if (<= uy 0.002899999963119626)
    (* 2.0 (* uy (* PI (sqrt (- (* t_0 (+ -2.0 t_0)))))))
    (* (sin (* (* uy 2.0) PI)) (sqrt (* ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	float tmp;
	if (uy <= 0.002899999963119626f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(-(t_0 * (-2.0f + t_0)))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	tmp = Float32(0.0)
	if (uy <= Float32(0.002899999963119626))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(-Float32(t_0 * Float32(Float32(-2.0) + t_0)))))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux - (maxCos * ux);
	tmp = single(0.0);
	if (uy <= single(0.002899999963119626))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(-(t_0 * (single(-2.0) + t_0)))));
	else
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00289999996

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift--.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. associate--l+ N/A

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

      6. sub-flip N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. sub-flip N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. associate-+l- N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. lift--.f32 N/A

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

      16. sub-square-pow N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift--.f32 N/A

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

      21. pow2 N/A

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

   6. Applied rewrites 81.7%

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

      1. lift--.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate--l- N/A

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

      4. sub0-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. sqr-neg-rev N/A

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

      9. lift--.f32 N/A

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

      10. sub-negate-rev N/A

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

      11. lift--.f32 N/A

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

      12. lift--.f32 N/A

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

      13. sub-negate-rev N/A

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

      14. lift--.f32 N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f32 N/A

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

      17. lower-+.f32 81.7%

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

   8. Applied rewrites 81.7%

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

   if 0.00289999996 < uy

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f32 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 57.5%

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 57.5%

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

   3. Applied rewrites 57.5%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in ux around 0

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

   9. Applied rewrites 72.8%

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

Alternative 7: 81.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.2%

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

4. Applied rewrites 50.2%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. associate--l+ N/A

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

   6. sub-flip N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. sub-flip N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-+l- N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift--.f32 N/A

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

   16. sub-square-pow N/A

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

   17. metadata-eval N/A

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. lift--.f32 N/A

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

   21. pow2 N/A

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

6. Applied rewrites 81.7%

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

   1. lift--.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate--l- N/A

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

   4. sub0-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. sqr-neg-rev N/A

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

   9. lift--.f32 N/A

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

   10. sub-negate-rev N/A

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

   11. lift--.f32 N/A

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

   12. lift--.f32 N/A

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

   13. sub-negate-rev N/A

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

   14. lift--.f32 N/A

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

   15. distribute-rgt-out N/A

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

   16. lower-*.f32 N/A

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

   17. lower-+.f32 81.7%

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

8. Applied rewrites 81.7%

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

Alternative 8: 81.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.2%

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

4. Applied rewrites 50.2%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. associate--l+ N/A

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

   6. sub-flip N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. sub-flip N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-+l- N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift--.f32 N/A

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

   16. sub-square-pow N/A

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

   17. metadata-eval N/A

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. lift--.f32 N/A

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

   21. pow2 N/A

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

6. Applied rewrites 81.7%

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

8. Applied rewrites 81.7%

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

Alternative 9: 81.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.2%

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

4. Applied rewrites 50.2%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. associate--l+ N/A

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

   6. sub-flip N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. sub-flip N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-+l- N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift--.f32 N/A

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

   16. sub-square-pow N/A

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

   17. metadata-eval N/A

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. lift--.f32 N/A

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

   21. pow2 N/A

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

6. Applied rewrites 81.7%

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

8. Applied rewrites 81.7%

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

Alternative 10: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995800256729126:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \cdot \pi\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (* t_0 t_0) 0.9995800256729126)
    (* 2.0 (* uy (* PI (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))
    (* (* (+ uy uy) (sqrt (* (fma -2.0 maxCos 2.0) ux))) PI))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9995800256729126f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))))));
	} else {
		tmp = ((uy + uy) * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux))) * ((float) M_PI);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9995800256729126))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))));
	else
		tmp = Float32(Float32(Float32(uy + uy) * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux))) * Float32(pi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999580026

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-pow.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift-fma.f32 N/A

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

      9. pow2 N/A

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

      10. lower-*.f32 50.3%

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

      11. lift-fma.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift--.f32 N/A

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

      14. associate-+r- N/A

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

      15. +-commutative N/A

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

      16. lift-+.f32 N/A

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

      17. lift--.f32 50.3%

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

      18. lift-+.f32 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f32 N/A

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

      21. lower-fma.f32 50.3%

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

      22. lift-fma.f32 N/A

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

      23. lift-*.f32 N/A

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

      24. lift--.f32 N/A

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

      25. associate-+r- N/A

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

      26. +-commutative N/A

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

      27. lift-+.f32 N/A

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

      28. lift--.f32 50.2%

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

   6. Applied rewrites 50.2%

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

   7. Taylor expanded in ux around 0

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in ux around 0

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

   12. Applied rewrites 48.9%

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

   if 0.999580026 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.2%

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in ux around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 66.1%

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

   7. Applied rewrites 66.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

   9. Applied rewrites 66.1%

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

Alternative 11: 66.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.2%

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

4. Applied rewrites 50.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 66.1%

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

7. Applied rewrites 66.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

9. Applied rewrites 66.1%

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

Alternative 12: 66.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.2%

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

4. Applied rewrites 50.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 66.1%

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

7. Applied rewrites 66.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f32 N/A

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

   9. lower-*.f32 66.1%

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

9. Applied rewrites 66.1%

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

Alternative 13: 63.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.2%

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

4. Applied rewrites 50.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 66.1%

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

7. Applied rewrites 66.1%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 63.4%

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

10. Applied rewrites 63.4%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lift-*.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f32 N/A

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

12. Applied rewrites 63.4%

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

Alternative 14: 63.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.2%

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

4. Applied rewrites 50.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 66.1%

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

7. Applied rewrites 66.1%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 63.4%

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

10. Applied rewrites 63.4%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lift-*.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. associate-*r* N/A

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

    8. lift-*.f32 N/A

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

    9. lower-*.f32 63.4%

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

12. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2025325 
(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)))))))