UniformSampleCone, y

Percentage Accurate: 58.1% → 98.3%

Time: 4.5s

Alternatives: 14

Speedup: 4.6×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-+.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 55.1%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 97.5

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

10. Applied rewrites 97.5%

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

Alternative 3: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f32 97.5

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in maxCos around 0

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

   1. lower--.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 97.5

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

10. Applied rewrites 97.5%

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

Alternative 4: 96.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f32 97.5

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 96.8%

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

Alternative 5: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 3.4999999343199306 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 3.4999999343199306e-7)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (- 2.0 ux) ux)))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (+ PI PI)))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 3.4999999343199306e-7f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((2.0f - ux) * ux));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(3.4999999343199306e-7))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(pi) + Float32(pi)))) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 3.49999993e-7

   1. Initial program 58.1%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f32 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 92.4

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

   10. Applied rewrites 92.4%

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

   if 3.49999993e-7 < maxCos

   1. Initial program 58.1%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow3 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 88.7

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

   7. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \pi + \pi\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f32 97.5

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in maxCos around 0

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

   1. lower--.f32 92.4

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

10. Applied rewrites 92.4%

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

Alternative 7: 88.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f32 97.5

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow3 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 88.0

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

10. Applied rewrites 88.0%

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

Alternative 8: 80.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 80.9

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

7. Applied rewrites 80.9%

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

Alternative 9: 80.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f32 97.5

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 80.3

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

10. Applied rewrites 80.3%

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

Alternative 10: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.017000000923871994:\\ \;\;\;\;\left(uy + uy\right) \cdot \left(\pi \cdot \left(\sqrt{ux} \cdot \sqrt{2 - \left(maxCos + maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy + uy\right) \cdot \left(\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.017000000923871994)
     (* (+ uy uy) (* PI (* (sqrt ux) (sqrt (- 2.0 (+ maxCos maxCos))))))
     (* (+ uy uy) (* (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (sqrtf((1.0f - (t_0 * t_0))) <= 0.017000000923871994f) {
		tmp = (uy + uy) * (((float) M_PI) * (sqrtf(ux) * sqrtf((2.0f - (maxCos + maxCos)))));
	} else {
		tmp = (uy + uy) * (sqrtf((1.0f - ((1.0f - ux) * (1.0f - 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 (sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) <= Float32(0.017000000923871994))
		tmp = Float32(Float32(uy + uy) * Float32(Float32(pi) * Float32(sqrt(ux) * sqrt(Float32(Float32(2.0) - Float32(maxCos + maxCos))))));
	else
		tmp = Float32(Float32(uy + uy) * Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))) * Float32(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = single(0.0);
	if (sqrt((single(1.0) - (t_0 * t_0))) <= single(0.017000000923871994))
		tmp = (uy + uy) * (single(pi) * (sqrt(ux) * sqrt((single(2.0) - (maxCos + maxCos)))));
	else
		tmp = (uy + uy) * (sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux)))) * single(pi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0170000009

   1. Initial program 58.1%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f32 65.1

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

   7. Applied rewrites 65.1%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sqrt.f32 65.1

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

   9. Applied rewrites 65.1%

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

   if 0.0170000009 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 58.1%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in maxCos around 0

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

      1. lift--.f32 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in maxCos around 0

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

      1. lift--.f32 49.2

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

   10. Applied rewrites 49.2%

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

Alternative 11: 65.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. count-2-rev N/A

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

   7. lift-+.f32 65.1

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

7. Applied rewrites 65.1%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. sqrt-prod N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-sqrt.f32 65.1

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

9. Applied rewrites 65.1%

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

Alternative 12: 65.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. count-2-rev N/A

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

   7. lift-+.f32 65.1

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

7. Applied rewrites 65.1%

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

Alternative 13: 62.5% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. count-2-rev N/A

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

   7. lift-+.f32 65.1

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

7. Applied rewrites 65.1%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 62.5

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

10. Applied rewrites 62.5%

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

Alternative 14: 7.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025142 
(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)))))))