UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 9.0s

Alternatives: 12

Speedup: 5.0×

Specification

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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1 ux) (* ux maxCos))))
  (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* t_0 t_0))))))

C

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

Julia

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

MATLAB

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1 ux) (* ux maxCos))))
  (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* t_0 t_0))))))

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := maxCos \cdot ux - ux\\ t_1 := ux - maxCos \cdot ux\\ \sqrt{\frac{0 \cdot 0 - t\_0 \cdot t\_0}{t\_1} \cdot \left(t\_1 - 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (- (* maxCos ux) ux)) (t_1 (- ux (* maxCos ux))))
  (*
   (sqrt (* (/ (- (* 0 0) (* t_0 t_0)) t_1) (- t_1 2)))
   (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (maxCos * ux) - ux;
	float t_1 = ux - (maxCos * ux);
	return sqrtf(((((0.0f * 0.0f) - (t_0 * t_0)) / t_1) * (t_1 - 2.0f))) * sinf((((float) M_PI) * (uy + uy)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos * ux) - ux)
	t_1 = Float32(ux - Float32(maxCos * ux))
	return Float32(sqrt(Float32(Float32(Float32(Float32(Float32(0.0) * Float32(0.0)) - Float32(t_0 * t_0)) / t_1) * Float32(t_1 - Float32(2.0)))) * sin(Float32(Float32(pi) * Float32(uy + uy))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (maxCos * ux) - ux;
	t_1 = ux - (maxCos * ux);
	tmp = sqrt(((((single(0.0) * single(0.0)) - (t_0 * t_0)) / t_1) * (t_1 - single(2.0)))) * sin((single(pi) * (uy + uy)));
end

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-negate N/A

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

   4. flip-- N/A

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

   5. lower-unsound-+.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. +-commutative N/A

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

   8. +-rgt-identity N/A

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

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

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

5. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. --rgt-identity N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. sub-negate-rev N/A

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

   9. flip-- N/A

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

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

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

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

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

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

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 N/A

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

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

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

   20. lower-unsound-+.f32 98.2%

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f32 98.2%

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

5. Applied rewrites 98.2%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. --rgt-identity N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. sub-negate-rev N/A

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

   9. lower--.f32 98.3%

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 98.3%

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

5. Applied rewrites 98.3%

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

Alternative 4: 97.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (sqrt (* (- (- (- ux (* maxCos ux)) 0)) (- ux 2)))
 (sin (* PI (+ uy uy)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((-((ux - (maxCos * ux)) - 0.0f) * (ux - 2.0f))) * sinf((((float) M_PI) * (uy + uy)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(-Float32(Float32(ux - Float32(maxCos * ux)) - Float32(0.0))) * Float32(ux - Float32(2.0)))) * sin(Float32(Float32(pi) * Float32(uy + uy))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((-((ux - (maxCos * ux)) - single(0.0)) * (ux - single(2.0)))) * sin((single(pi) * (uy + uy)));
end

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in maxCos around 0

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

   1. lower--.f32 97.1%

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

6. Applied rewrites 97.1%

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

Alternative 5: 92.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 92.4%

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

6. Applied rewrites 92.4%

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

Alternative 6: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq \frac{13743895}{8589934592}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \left(ux \cdot maxCos - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 13743895/8589934592)
  (*
   (* (+ uy uy) PI)
   (sqrt (* (- (- ux (* ux maxCos)) 2) (- (* ux maxCos) ux))))
  (* (sqrt (* (- 2 (+ maxCos maxCos)) ux)) (sin (* (+ PI PI) uy)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0015999999595806003f) {
		tmp = ((uy + uy) * ((float) M_PI)) * sqrtf((((ux - (ux * maxCos)) - 2.0f) * ((ux * maxCos) - ux)));
	} else {
		tmp = sqrtf(((2.0f - (maxCos + maxCos)) * ux)) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0015999999595806003))
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * sqrt(Float32(Float32(Float32(ux - Float32(ux * maxCos)) - Float32(2.0)) * Float32(Float32(ux * maxCos) - ux))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - Float32(maxCos + maxCos)) * ux)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0015999999595806003))
		tmp = ((uy + uy) * single(pi)) * sqrt((((ux - (ux * maxCos)) - single(2.0)) * ((ux * maxCos) - ux)));
	else
		tmp = sqrt(((single(2.0) - (maxCos + maxCos)) * ux)) * sin(((single(pi) + single(pi)) * uy));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00159999996

   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

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.4%

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 81.5%

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

   6. Applied rewrites 81.5%

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

   8. Applied rewrites 81.4%

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

   if 0.00159999996 < uy

   1. Initial program 57.4%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.4%

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

   4. Applied rewrites 76.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 76.4%

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 76.4%

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

      7. lift-*.f32 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f32 76.4%

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

      10. lower-+.f32 N/A

          \[\leadsto \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux} \cdot \sin \mathsf{Rewrite=>}\left(lift-*.f32, \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \]

      11. lower-+.f32 N/A

          \[\leadsto \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(uy \cdot 2\right)\right) \cdot \pi\right) \]

      12. lower-+.f32 N/A

          \[\leadsto \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux} \cdot \sin \mathsf{Rewrite=>}\left(associate-*l*, \left(uy \cdot \left(2 \cdot \pi\right)\right)\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux} \cdot \sin \mathsf{Rewrite=>}\left(*-commutative, \left(\left(2 \cdot \pi\right) \cdot uy\right)\right) \]

      14. lower-+.f32 N/A

          \[\leadsto \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux} \cdot \sin \mathsf{Rewrite=>}\left(lower-*.f32, \left(\left(2 \cdot \pi\right) \cdot uy\right)\right) \]

   6. Applied rewrites 76.4%

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

Alternative 7: 89.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq \frac{8589935}{2147483648}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \left(ux \cdot maxCos - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 8589935/2147483648)
  (*
   (* (+ uy uy) PI)
   (sqrt (* (- (- ux (* ux maxCos)) 2) (- (* ux maxCos) ux))))
  (* (sin (* (* uy 2) PI)) (sqrt (* ux 2)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.004000000189989805f) {
		tmp = ((uy + uy) * ((float) M_PI)) * sqrtf((((ux - (ux * maxCos)) - 2.0f) * ((ux * maxCos) - ux)));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.004000000189989805))
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * sqrt(Float32(Float32(Float32(ux - Float32(ux * maxCos)) - Float32(2.0)) * Float32(Float32(ux * maxCos) - ux))));
	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)
	tmp = single(0.0);
	if (uy <= single(0.004000000189989805))
		tmp = ((uy + uy) * single(pi)) * sqrt((((ux - (ux * maxCos)) - single(2.0)) * ((ux * maxCos) - ux)));
	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 < 0.00400000019

   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

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.4%

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 81.5%

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

   6. Applied rewrites 81.5%

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

   8. Applied rewrites 81.4%

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

   if 0.00400000019 < uy

   1. Initial program 57.4%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.4%

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in maxCos around 0

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

   7. Applied rewrites 72.9%

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

Alternative 8: 81.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

8. Applied rewrites 81.4%

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

Alternative 9: 80.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in ux around 0

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

9. Applied rewrites 80.7%

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

Alternative 10: 77.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(2.0) * (uy * (single(pi) * sqrt((single(-1.0) * (ux * ((ux - single(1.0)) - single(1.0)))))));
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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 77.3%

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

9. Applied rewrites 77.3%

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

    1. lift--.f32 N/A

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

    2. metadata-eval N/A

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

    3. associate--r+ N/A

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

    4. lower--.f32 N/A

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

    5. lower--.f32 77.3%

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

11. Applied rewrites 77.3%

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

Alternative 11: 77.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 77.3%

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

9. Applied rewrites 77.3%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. count-2 N/A

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

    5. lift-+.f32 N/A

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

    6. lower-*.f32 77.3%

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 77.3%

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

11. Applied rewrites 77.3%

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

Alternative 12: 63.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.4%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 81.5%

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

6. Applied rewrites 81.5%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 77.3%

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

9. Applied rewrites 77.3%

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

10. Taylor expanded in ux around 0

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

    1. lower-*.f32 63.2%

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

12. Applied rewrites 63.2%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2328306437/10000000000000000000 ux) (<= ux 1)) (and (<= 2328306437/10000000000000000000 uy) (<= uy 1))) (and (<= 0 maxCos) (<= maxCos 1)))
  (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* (+ (- 1 ux) (* ux maxCos)) (+ (- 1 ux) (* ux maxCos)))))))