UniformSampleCone, y

Percentage Accurate: 57.8% → 98.3%

Time: 20.2s

Alternatives: 12

Speedup: N/A×

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: 57.8% 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, N/A× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((fmaf(-ux, powf((maxCos - 1.0f), 2.0f), 2.0f) - (maxCos * 2.0f)) * 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(maxCos - Float32(1.0)) ^ Float32(2.0)), Float32(2.0)) - Float32(maxCos * Float32(2.0))) * ux)))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

Alternative 2: 98.3% accurate, N/A× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((fmaf(-2.0f, maxCos, 2.0f) / ux) - powf(fmaf(-1.0f, maxCos, 1.0f), 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(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux) - (fma(Float32(-1.0), maxCos, Float32(1.0)) ^ Float32(2.0))) * Float32(ux * ux))))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.4%

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

Alternative 3: 98.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(maxCos - 1\right)}^{2}\\ t_1 := \left(-ux\right) \cdot t\_0\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{t\_1}^{3} + 8}{\mathsf{fma}\left(t\_1, t\_1, 4 + \left(ux \cdot t\_0\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (pow (- maxCos 1.0) 2.0)) (t_1 (* (- ux) t_0)))
   (*
    (sin (* (* uy 2.0) PI))
    (sqrt
     (*
      (-
       (/ (+ (pow t_1 3.0) 8.0) (fma t_1 t_1 (+ 4.0 (* (* ux t_0) 2.0))))
       (* maxCos 2.0))
      ux)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = powf((maxCos - 1.0f), 2.0f);
	float t_1 = -ux * t_0;
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((((powf(t_1, 3.0f) + 8.0f) / fmaf(t_1, t_1, (4.0f + ((ux * t_0) * 2.0f)))) - (maxCos * 2.0f)) * ux));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(maxCos - Float32(1.0)) ^ Float32(2.0)
	t_1 = Float32(Float32(-ux) * t_0)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(Float32((t_1 ^ Float32(3.0)) + Float32(8.0)) / fma(t_1, t_1, Float32(Float32(4.0) + Float32(Float32(ux * t_0) * Float32(2.0))))) - Float32(maxCos * Float32(2.0))) * ux)))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

   1. lift-neg.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. flip3-+ N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   6. lower-/.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   7. lower-+.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   8. lower-pow.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   9. lower-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   10. lift-neg.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   11. lift-pow.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   12. lift--.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   13. metadata-eval N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

7. Applied rewrites 98.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\mathsf{fma}\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}, \left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}, 4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

8. Final simplification 98.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\mathsf{fma}\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}, \left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}, 4 + \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]
9. Add Preprocessing

Alternative 4: 98.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 8 - {ux}^{3}\\ t_1 := 4 - -2 \cdot ux\\ t_2 := ux \cdot t\_1\\ t_3 := -4 \cdot t\_1\\ t_4 := \mathsf{fma}\left(-8, t\_2, -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, t\_3\right)\\ t_5 := {ux}^{6} + {t\_1}^{3}\\ t_6 := \frac{{ux}^{3} \cdot t\_4}{t\_5}\\ t_7 := {t\_5}^{2}\\ t_8 := \mathsf{fma}\left(4, t\_2, 16 \cdot \left(ux \cdot ux\right)\right)\\ t_9 := \mathsf{fma}\left(4, t\_2, \mathsf{fma}\left(16, ux \cdot ux, 28 \cdot {ux}^{4}\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(2, ux, \mathsf{fma}\left(6, t\_1, 16 \cdot ux\right)\right)\\ t_10 := {t\_1}^{2}\\ t_11 := \left(ux \cdot ux\right) \cdot t\_1\\ t_12 := \left({ux}^{4} + t\_10\right) - t\_11\\ t_13 := \frac{{ux}^{3} \cdot t\_12}{t\_5}\\ t_14 := \mathsf{fma}\left(6, t\_13, \frac{t\_0 \cdot t\_4}{t\_5}\right)\\ t_15 := ux \cdot t\_10\\ t_16 := \mathsf{fma}\left(2, t\_15, \mathsf{fma}\left(32, t\_11, \mathsf{fma}\left(66, {ux}^{6}, t\_8 \cdot t\_1\right)\right)\right)\\ t_17 := \mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, t\_15, -4 \cdot t\_15\right)\right)\\ t_18 := \frac{t\_0 \cdot \left(t\_17 \cdot t\_12\right)}{t\_7}\\ t_19 := t\_14 - t\_18\\ t_20 := \mathsf{fma}\left(-15, t\_13, \mathsf{fma}\left(6, t\_6, \frac{t\_0 \cdot t\_9}{t\_5}\right)\right) - \left(\frac{t\_0 \cdot \left(t\_16 \cdot t\_12\right)}{t\_7} + \frac{t\_17 \cdot t\_19}{t\_5}\right)\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, \mathsf{fma}\left(maxCos, \mathsf{fma}\left(maxCos, ux \cdot \left(\mathsf{fma}\left(-15, t\_6, \mathsf{fma}\left(6, \frac{{ux}^{3} \cdot t\_9}{t\_5}, \mathsf{fma}\left(20, t\_13, \frac{t\_0 \cdot \left(\mathsf{fma}\left(-56, {ux}^{4}, -16 \cdot \left(ux \cdot ux\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-24, ux, \mathsf{fma}\left(-8, ux, t\_3\right)\right)\right)}{t\_5}\right)\right)\right) - \left(\frac{t\_0 \cdot \left(\mathsf{fma}\left(-220, {ux}^{6}, \mathsf{fma}\left(-32, t\_11, -4 \cdot \left(ux \cdot t\_8\right)\right)\right) \cdot t\_12\right)}{t\_7} + \frac{\mathsf{fma}\left(t\_17, t\_20, t\_16 \cdot t\_19\right)}{t\_5}\right)\right), ux \cdot t\_20\right), ux \cdot \left(t\_14 - \left(2 + t\_18\right)\right)\right), \frac{ux \cdot \left(t\_0 \cdot t\_12\right)}{t\_5}\right)} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 8.0 (pow ux 3.0)))
        (t_1 (- 4.0 (* -2.0 ux)))
        (t_2 (* ux t_1))
        (t_3 (* -4.0 t_1))
        (t_4
         (-
          (fma -8.0 t_2 (* -8.0 (pow ux 4.0)))
          (* (* ux ux) (fma -4.0 ux t_3))))
        (t_5 (+ (pow ux 6.0) (pow t_1 3.0)))
        (t_6 (/ (* (pow ux 3.0) t_4) t_5))
        (t_7 (pow t_5 2.0))
        (t_8 (fma 4.0 t_2 (* 16.0 (* ux ux))))
        (t_9
         (-
          (fma 4.0 t_2 (fma 16.0 (* ux ux) (* 28.0 (pow ux 4.0))))
          (* (* ux ux) (fma 2.0 ux (fma 6.0 t_1 (* 16.0 ux))))))
        (t_10 (pow t_1 2.0))
        (t_11 (* (* ux ux) t_1))
        (t_12 (- (+ (pow ux 4.0) t_10) t_11))
        (t_13 (/ (* (pow ux 3.0) t_12) t_5))
        (t_14 (fma 6.0 t_13 (/ (* t_0 t_4) t_5)))
        (t_15 (* ux t_10))
        (t_16
         (fma 2.0 t_15 (fma 32.0 t_11 (fma 66.0 (pow ux 6.0) (* t_8 t_1)))))
        (t_17 (fma -12.0 (pow ux 6.0) (fma -8.0 t_15 (* -4.0 t_15))))
        (t_18 (/ (* t_0 (* t_17 t_12)) t_7))
        (t_19 (- t_14 t_18))
        (t_20
         (-
          (fma -15.0 t_13 (fma 6.0 t_6 (/ (* t_0 t_9) t_5)))
          (+ (/ (* t_0 (* t_16 t_12)) t_7) (/ (* t_17 t_19) t_5)))))
   (*
    (sin (* (* uy 2.0) PI))
    (sqrt
     (fma
      maxCos
      (fma
       maxCos
       (fma
        maxCos
        (*
         ux
         (-
          (fma
           -15.0
           t_6
           (fma
            6.0
            (/ (* (pow ux 3.0) t_9) t_5)
            (fma
             20.0
             t_13
             (/
              (*
               t_0
               (-
                (fma -56.0 (pow ux 4.0) (* -16.0 (* ux ux)))
                (* (* ux ux) (fma -24.0 ux (fma -8.0 ux t_3)))))
              t_5))))
          (+
           (/
            (*
             t_0
             (*
              (fma -220.0 (pow ux 6.0) (fma -32.0 t_11 (* -4.0 (* ux t_8))))
              t_12))
            t_7)
           (/ (fma t_17 t_20 (* t_16 t_19)) t_5))))
        (* ux t_20))
       (* ux (- t_14 (+ 2.0 t_18))))
      (/ (* ux (* t_0 t_12)) t_5))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 8.0f - powf(ux, 3.0f);
	float t_1 = 4.0f - (-2.0f * ux);
	float t_2 = ux * t_1;
	float t_3 = -4.0f * t_1;
	float t_4 = fmaf(-8.0f, t_2, (-8.0f * powf(ux, 4.0f))) - ((ux * ux) * fmaf(-4.0f, ux, t_3));
	float t_5 = powf(ux, 6.0f) + powf(t_1, 3.0f);
	float t_6 = (powf(ux, 3.0f) * t_4) / t_5;
	float t_7 = powf(t_5, 2.0f);
	float t_8 = fmaf(4.0f, t_2, (16.0f * (ux * ux)));
	float t_9 = fmaf(4.0f, t_2, fmaf(16.0f, (ux * ux), (28.0f * powf(ux, 4.0f)))) - ((ux * ux) * fmaf(2.0f, ux, fmaf(6.0f, t_1, (16.0f * ux))));
	float t_10 = powf(t_1, 2.0f);
	float t_11 = (ux * ux) * t_1;
	float t_12 = (powf(ux, 4.0f) + t_10) - t_11;
	float t_13 = (powf(ux, 3.0f) * t_12) / t_5;
	float t_14 = fmaf(6.0f, t_13, ((t_0 * t_4) / t_5));
	float t_15 = ux * t_10;
	float t_16 = fmaf(2.0f, t_15, fmaf(32.0f, t_11, fmaf(66.0f, powf(ux, 6.0f), (t_8 * t_1))));
	float t_17 = fmaf(-12.0f, powf(ux, 6.0f), fmaf(-8.0f, t_15, (-4.0f * t_15)));
	float t_18 = (t_0 * (t_17 * t_12)) / t_7;
	float t_19 = t_14 - t_18;
	float t_20 = fmaf(-15.0f, t_13, fmaf(6.0f, t_6, ((t_0 * t_9) / t_5))) - (((t_0 * (t_16 * t_12)) / t_7) + ((t_17 * t_19) / t_5));
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(fmaf(maxCos, fmaf(maxCos, fmaf(maxCos, (ux * (fmaf(-15.0f, t_6, fmaf(6.0f, ((powf(ux, 3.0f) * t_9) / t_5), fmaf(20.0f, t_13, ((t_0 * (fmaf(-56.0f, powf(ux, 4.0f), (-16.0f * (ux * ux))) - ((ux * ux) * fmaf(-24.0f, ux, fmaf(-8.0f, ux, t_3))))) / t_5)))) - (((t_0 * (fmaf(-220.0f, powf(ux, 6.0f), fmaf(-32.0f, t_11, (-4.0f * (ux * t_8)))) * t_12)) / t_7) + (fmaf(t_17, t_20, (t_16 * t_19)) / t_5)))), (ux * t_20)), (ux * (t_14 - (2.0f + t_18)))), ((ux * (t_0 * t_12)) / t_5)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(8.0) - (ux ^ Float32(3.0)))
	t_1 = Float32(Float32(4.0) - Float32(Float32(-2.0) * ux))
	t_2 = Float32(ux * t_1)
	t_3 = Float32(Float32(-4.0) * t_1)
	t_4 = Float32(fma(Float32(-8.0), t_2, Float32(Float32(-8.0) * (ux ^ Float32(4.0)))) - Float32(Float32(ux * ux) * fma(Float32(-4.0), ux, t_3)))
	t_5 = Float32((ux ^ Float32(6.0)) + (t_1 ^ Float32(3.0)))
	t_6 = Float32(Float32((ux ^ Float32(3.0)) * t_4) / t_5)
	t_7 = t_5 ^ Float32(2.0)
	t_8 = fma(Float32(4.0), t_2, Float32(Float32(16.0) * Float32(ux * ux)))
	t_9 = Float32(fma(Float32(4.0), t_2, fma(Float32(16.0), Float32(ux * ux), Float32(Float32(28.0) * (ux ^ Float32(4.0))))) - Float32(Float32(ux * ux) * fma(Float32(2.0), ux, fma(Float32(6.0), t_1, Float32(Float32(16.0) * ux)))))
	t_10 = t_1 ^ Float32(2.0)
	t_11 = Float32(Float32(ux * ux) * t_1)
	t_12 = Float32(Float32((ux ^ Float32(4.0)) + t_10) - t_11)
	t_13 = Float32(Float32((ux ^ Float32(3.0)) * t_12) / t_5)
	t_14 = fma(Float32(6.0), t_13, Float32(Float32(t_0 * t_4) / t_5))
	t_15 = Float32(ux * t_10)
	t_16 = fma(Float32(2.0), t_15, fma(Float32(32.0), t_11, fma(Float32(66.0), (ux ^ Float32(6.0)), Float32(t_8 * t_1))))
	t_17 = fma(Float32(-12.0), (ux ^ Float32(6.0)), fma(Float32(-8.0), t_15, Float32(Float32(-4.0) * t_15)))
	t_18 = Float32(Float32(t_0 * Float32(t_17 * t_12)) / t_7)
	t_19 = Float32(t_14 - t_18)
	t_20 = Float32(fma(Float32(-15.0), t_13, fma(Float32(6.0), t_6, Float32(Float32(t_0 * t_9) / t_5))) - Float32(Float32(Float32(t_0 * Float32(t_16 * t_12)) / t_7) + Float32(Float32(t_17 * t_19) / t_5)))
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(fma(maxCos, fma(maxCos, fma(maxCos, Float32(ux * Float32(fma(Float32(-15.0), t_6, fma(Float32(6.0), Float32(Float32((ux ^ Float32(3.0)) * t_9) / t_5), fma(Float32(20.0), t_13, Float32(Float32(t_0 * Float32(fma(Float32(-56.0), (ux ^ Float32(4.0)), Float32(Float32(-16.0) * Float32(ux * ux))) - Float32(Float32(ux * ux) * fma(Float32(-24.0), ux, fma(Float32(-8.0), ux, t_3))))) / t_5)))) - Float32(Float32(Float32(t_0 * Float32(fma(Float32(-220.0), (ux ^ Float32(6.0)), fma(Float32(-32.0), t_11, Float32(Float32(-4.0) * Float32(ux * t_8)))) * t_12)) / t_7) + Float32(fma(t_17, t_20, Float32(t_16 * t_19)) / t_5)))), Float32(ux * t_20)), Float32(ux * Float32(t_14 - Float32(Float32(2.0) + t_18)))), Float32(Float32(ux * Float32(t_0 * t_12)) / t_5))))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

   1. lift-neg.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. flip3-+ N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   6. lower-/.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   7. lower-+.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   8. lower-pow.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   9. lower-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   10. lift-neg.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   11. lift-pow.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   12. lift--.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + {2}^{3}}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   13. metadata-eval N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 \cdot 2 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

7. Applied rewrites 98.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\mathsf{fma}\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}, \left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}, 4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]
8. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   2. lift-neg.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   3. lift-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   4. lift--.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   5. lift-pow.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   6. lift-neg.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   7. lift-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   8. lift--.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   9. lift-pow.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   10. lift--.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   11. lift-*.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   12. lift-neg.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   13. lift-*.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   14. lift--.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

   15. lift-pow.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(4 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)} - maxCos \cdot 2\right) \cdot ux} \]

9. Applied rewrites 98.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{{\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{3} + 8}{\frac{{\left({\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{2}\right)}^{3} + {\left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)}^{3}}{\mathsf{fma}\left({\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{2}, {\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{2}, \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right) \cdot \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right) - {\left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right)}^{2} \cdot \left(4 - \left(\left(-ux\right) \cdot {\left(maxCos - 1\right)}^{2}\right) \cdot 2\right)\right)}} - maxCos \cdot 2\right) \cdot ux} \]

10. Taylor expanded in maxCos around 0

    \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{maxCos \cdot \left(maxCos \cdot \left(maxCos \cdot \left(ux \cdot \left(\left(-15 \cdot \frac{{ux}^{3} \cdot \left(\left(-8 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + -8 \cdot {ux}^{4}\right) - {ux}^{2} \cdot \left(-4 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \left(6 \cdot \frac{{ux}^{3} \cdot \left(\left(4 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + \left(16 \cdot {ux}^{2} + 28 \cdot {ux}^{4}\right)\right) - {ux}^{2} \cdot \left(2 \cdot ux + \left(6 \cdot \left(4 - -2 \cdot ux\right) + 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \left(20 \cdot \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-56 \cdot {ux}^{4} + -16 \cdot {ux}^{2}\right) - {ux}^{2} \cdot \left(-24 \cdot ux + \left(-8 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)\right) - \left(\frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-220 \cdot {ux}^{6} + \left(-32 \cdot \left({ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right) + -4 \cdot \left(ux \cdot \left(4 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + 16 \cdot {ux}^{2}\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \left(\frac{\left(-12 \cdot {ux}^{6} + \left(-8 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left(-15 \cdot \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \left(6 \cdot \frac{{ux}^{3} \cdot \left(\left(-8 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + -8 \cdot {ux}^{4}\right) - {ux}^{2} \cdot \left(-4 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(4 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + \left(16 \cdot {ux}^{2} + 28 \cdot {ux}^{4}\right)\right) - {ux}^{2} \cdot \left(2 \cdot ux + \left(6 \cdot \left(4 - -2 \cdot ux\right) + 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right) - \left(\frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(2 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + \left(32 \cdot \left({ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right) + \left(66 \cdot {ux}^{6} + \left(4 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + 16 \cdot {ux}^{2}\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\left(-12 \cdot {ux}^{6} + \left(-8 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left(6 \cdot \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-8 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + -8 \cdot {ux}^{4}\right) - {ux}^{2} \cdot \left(-4 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-12 \cdot {ux}^{6} + \left(-8 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(2 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + \left(32 \cdot \left({ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right) + \left(66 \cdot {ux}^{6} + \left(4 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + 16 \cdot {ux}^{2}\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\left(6 \cdot \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-8 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + -8 \cdot {ux}^{4}\right) - {ux}^{2} \cdot \left(-4 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-12 \cdot {ux}^{6} + \left(-8 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)\right)\right) + ux \cdot \left(\left(-15 \cdot \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \left(6 \cdot \frac{{ux}^{3} \cdot \left(\left(-8 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + -8 \cdot {ux}^{4}\right) - {ux}^{2} \cdot \left(-4 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(4 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + \left(16 \cdot {ux}^{2} + 28 \cdot {ux}^{4}\right)\right) - {ux}^{2} \cdot \left(2 \cdot ux + \left(6 \cdot \left(4 - -2 \cdot ux\right) + 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right) - \left(\frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(2 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + \left(32 \cdot \left({ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right) + \left(66 \cdot {ux}^{6} + \left(4 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + 16 \cdot {ux}^{2}\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\left(-12 \cdot {ux}^{6} + \left(-8 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left(6 \cdot \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-8 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + -8 \cdot {ux}^{4}\right) - {ux}^{2} \cdot \left(-4 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-12 \cdot {ux}^{6} + \left(-8 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)\right) + ux \cdot \left(\left(6 \cdot \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}} + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-8 \cdot \left(ux \cdot \left(4 - -2 \cdot ux\right)\right) + -8 \cdot {ux}^{4}\right) - {ux}^{2} \cdot \left(-4 \cdot ux + -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \left(2 + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left(-12 \cdot {ux}^{6} + \left(-8 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right) + -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)\right)\right) + \color{blue}{\frac{ux \cdot \left(\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - {ux}^{2} \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}}} \]

12. Applied rewrites 98.4%

    \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, \color{blue}{\mathsf{fma}\left(maxCos, \mathsf{fma}\left(maxCos, ux \cdot \left(\mathsf{fma}\left(-15, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(16, ux \cdot ux, 28 \cdot {ux}^{4}\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(2, ux, \mathsf{fma}\left(6, 4 - -2 \cdot ux, 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(20, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-56, {ux}^{4}, -16 \cdot \left(ux \cdot ux\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-24, ux, \mathsf{fma}\left(-8, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)\right) - \left(\frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-220, {ux}^{6}, \mathsf{fma}\left(-32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), -4 \cdot \left(ux \cdot \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right), \mathsf{fma}\left(-15, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(16, ux \cdot ux, 28 \cdot {ux}^{4}\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(2, ux, \mathsf{fma}\left(6, 4 - -2 \cdot ux, 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right) - \left(\frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(2, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, \mathsf{fma}\left(32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(66, {ux}^{6}, \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right), \mathsf{fma}\left(2, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, \mathsf{fma}\left(32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(66, {ux}^{6}, \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right), ux \cdot \left(\mathsf{fma}\left(-15, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(16, ux \cdot ux, 28 \cdot {ux}^{4}\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(2, ux, \mathsf{fma}\left(6, 4 - -2 \cdot ux, 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right) - \left(\frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(2, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, \mathsf{fma}\left(32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(66, {ux}^{6}, \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)\right), ux \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \left(2 + \frac{\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)\right)\right)}, \frac{ux \cdot \left(\left(8 + -1 \cdot {ux}^{3}\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)} \]

13. Final simplification 98.4%

    \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, \mathsf{fma}\left(maxCos, \mathsf{fma}\left(maxCos, ux \cdot \left(\mathsf{fma}\left(-15, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(16, ux \cdot ux, 28 \cdot {ux}^{4}\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(2, ux, \mathsf{fma}\left(6, 4 - -2 \cdot ux, 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(20, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-56, {ux}^{4}, -16 \cdot \left(ux \cdot ux\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-24, ux, \mathsf{fma}\left(-8, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)\right) - \left(\frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-220, {ux}^{6}, \mathsf{fma}\left(-32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), -4 \cdot \left(ux \cdot \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right), \mathsf{fma}\left(-15, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(16, ux \cdot ux, 28 \cdot {ux}^{4}\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(2, ux, \mathsf{fma}\left(6, 4 - -2 \cdot ux, 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right) - \left(\frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(2, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, \mathsf{fma}\left(32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(66, {ux}^{6}, \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right), \mathsf{fma}\left(2, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, \mathsf{fma}\left(32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(66, {ux}^{6}, \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right), ux \cdot \left(\mathsf{fma}\left(-15, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(16, ux \cdot ux, 28 \cdot {ux}^{4}\right)\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(2, ux, \mathsf{fma}\left(6, 4 - -2 \cdot ux, 16 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right) - \left(\frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(2, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, \mathsf{fma}\left(32, \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right), \mathsf{fma}\left(66, {ux}^{6}, \mathsf{fma}\left(4, ux \cdot \left(4 - -2 \cdot ux\right), 16 \cdot \left(ux \cdot ux\right)\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}} + \frac{\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)\right)\right), ux \cdot \left(\mathsf{fma}\left(6, \frac{{ux}^{3} \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}, \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-8, ux \cdot \left(4 - -2 \cdot ux\right), -8 \cdot {ux}^{4}\right) - \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-4, ux, -4 \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right) - \left(2 + \frac{\left(8 - {ux}^{3}\right) \cdot \left(\mathsf{fma}\left(-12, {ux}^{6}, \mathsf{fma}\left(-8, ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}, -4 \cdot \left(ux \cdot {\left(4 - -2 \cdot ux\right)}^{2}\right)\right)\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{\left({ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}\right)}^{2}}\right)\right)\right), \frac{ux \cdot \left(\left(8 - {ux}^{3}\right) \cdot \left(\left({ux}^{4} + {\left(4 - -2 \cdot ux\right)}^{2}\right) - \left(ux \cdot ux\right) \cdot \left(4 - -2 \cdot ux\right)\right)\right)}{{ux}^{6} + {\left(4 - -2 \cdot ux\right)}^{3}}\right)} \]
14. Add Preprocessing

Alternative 5: 95.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - 2 \cdot maxCos\\ t_1 := ux \cdot t\_0\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\sqrt{ux}, \sqrt{t\_0}, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.5, \sqrt{\frac{1}{t\_1}} \cdot {\left(maxCos - 1\right)}^{2}, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{1}{{t\_1}^{3}}} \cdot {\left(maxCos - 1\right)}^{4}, -0.0625 \cdot \left(\sqrt{\frac{1}{ux \cdot {t\_0}^{5}}} \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 2.0 (* 2.0 maxCos))) (t_1 (* ux t_0)))
   (*
    (sin (* (* uy 2.0) PI))
    (fma
     (sqrt ux)
     (sqrt t_0)
     (*
      (* ux ux)
      (fma
       -0.5
       (* (sqrt (/ 1.0 t_1)) (pow (- maxCos 1.0) 2.0))
       (*
        (* ux ux)
        (fma
         -0.125
         (* (sqrt (/ 1.0 (pow t_1 3.0))) (pow (- maxCos 1.0) 4.0))
         (*
          -0.0625
          (*
           (sqrt (/ 1.0 (* ux (pow t_0 5.0))))
           (pow (- maxCos 1.0) 6.0)))))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f - (2.0f * maxCos);
	float t_1 = ux * t_0;
	return sinf(((uy * 2.0f) * ((float) M_PI))) * fmaf(sqrtf(ux), sqrtf(t_0), ((ux * ux) * fmaf(-0.5f, (sqrtf((1.0f / t_1)) * powf((maxCos - 1.0f), 2.0f)), ((ux * ux) * fmaf(-0.125f, (sqrtf((1.0f / powf(t_1, 3.0f))) * powf((maxCos - 1.0f), 4.0f)), (-0.0625f * (sqrtf((1.0f / (ux * powf(t_0, 5.0f)))) * powf((maxCos - 1.0f), 6.0f))))))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))
	t_1 = Float32(ux * t_0)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * fma(sqrt(ux), sqrt(t_0), Float32(Float32(ux * ux) * fma(Float32(-0.5), Float32(sqrt(Float32(Float32(1.0) / t_1)) * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))), Float32(Float32(ux * ux) * fma(Float32(-0.125), Float32(sqrt(Float32(Float32(1.0) / (t_1 ^ Float32(3.0)))) * (Float32(maxCos - Float32(1.0)) ^ Float32(4.0))), Float32(Float32(-0.0625) * Float32(sqrt(Float32(Float32(1.0) / Float32(ux * (t_0 ^ Float32(5.0))))) * (Float32(maxCos - Float32(1.0)) ^ Float32(6.0))))))))))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in ux around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} + {ux}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \cdot {\left(maxCos - 1\right)}^{2}\right) + {ux}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\sqrt{\frac{1}{{ux}^{3} \cdot {\left(2 - 2 \cdot maxCos\right)}^{3}}} \cdot {\left(maxCos - 1\right)}^{4}\right) + \frac{-1}{16} \cdot \left(\sqrt{\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}} \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)} \]

8. Applied rewrites 95.6%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt{ux}, \sqrt{2 - 2 \cdot maxCos}, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.5, \sqrt{\frac{1}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \cdot {\left(maxCos - 1\right)}^{2}, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}} \cdot {\left(maxCos - 1\right)}^{4}, -0.0625 \cdot \left(\sqrt{\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}} \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 6: 95.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ t_1 := \mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux\\ \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \sqrt{\frac{\frac{1}{ux}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, {\left(maxCos - 1\right)}^{2} \cdot t\_0, \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{6} \cdot t\_0\right) \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(\mathsf{fma}\left(-2, maxCos, 2\right)\right)}^{5}}}, -0.0625, \left(\sqrt{\frac{1}{{t\_1}^{3}}} \cdot \left({\left(maxCos - 1\right)}^{4} \cdot t\_0\right)\right) \cdot -0.125\right) \cdot \left(ux \cdot ux\right)\right), ux \cdot ux, t\_0 \cdot \sqrt{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* PI (* 2.0 uy)))) (t_1 (* (fma -2.0 maxCos 2.0) ux)))
   (fma
    (fma
     (* -0.5 (sqrt (/ (/ 1.0 ux) (fma -2.0 maxCos 2.0))))
     (* (pow (- maxCos 1.0) 2.0) t_0)
     (*
      (fma
       (*
        (* (pow (- maxCos 1.0) 6.0) t_0)
        (sqrt (/ (/ 1.0 ux) (pow (fma -2.0 maxCos 2.0) 5.0))))
       -0.0625
       (*
        (* (sqrt (/ 1.0 (pow t_1 3.0))) (* (pow (- maxCos 1.0) 4.0) t_0))
        -0.125))
      (* ux ux)))
    (* ux ux)
    (* t_0 (sqrt t_1)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((((float) M_PI) * (2.0f * uy)));
	float t_1 = fmaf(-2.0f, maxCos, 2.0f) * ux;
	return fmaf(fmaf((-0.5f * sqrtf(((1.0f / ux) / fmaf(-2.0f, maxCos, 2.0f)))), (powf((maxCos - 1.0f), 2.0f) * t_0), (fmaf(((powf((maxCos - 1.0f), 6.0f) * t_0) * sqrtf(((1.0f / ux) / powf(fmaf(-2.0f, maxCos, 2.0f), 5.0f)))), -0.0625f, ((sqrtf((1.0f / powf(t_1, 3.0f))) * (powf((maxCos - 1.0f), 4.0f) * t_0)) * -0.125f)) * (ux * ux))), (ux * ux), (t_0 * sqrtf(t_1)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	t_1 = Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)
	return fma(fma(Float32(Float32(-0.5) * sqrt(Float32(Float32(Float32(1.0) / ux) / fma(Float32(-2.0), maxCos, Float32(2.0))))), Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * t_0), Float32(fma(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(6.0)) * t_0) * sqrt(Float32(Float32(Float32(1.0) / ux) / (fma(Float32(-2.0), maxCos, Float32(2.0)) ^ Float32(5.0))))), Float32(-0.0625), Float32(Float32(sqrt(Float32(Float32(1.0) / (t_1 ^ Float32(3.0)))) * Float32((Float32(maxCos - Float32(1.0)) ^ Float32(4.0)) * t_0)) * Float32(-0.125))) * Float32(ux * ux))), Float32(ux * ux), Float32(t_0 * sqrt(t_1)))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + {ux}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + {ux}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\sqrt{\frac{1}{{ux}^{3} \cdot {\left(2 - 2 \cdot maxCos\right)}^{3}}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right)\right) + \frac{-1}{16} \cdot \left(\sqrt{\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)} \]

5. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \sqrt{\frac{\frac{1}{ux}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, {\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right), \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{6} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(\mathsf{fma}\left(-2, maxCos, 2\right)\right)}^{5}}}, -0.0625, \left(\sqrt{\frac{1}{{\left(\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux\right)}^{3}}} \cdot \left({\left(maxCos - 1\right)}^{4} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\right) \cdot -0.125\right) \cdot \left(ux \cdot ux\right)\right), ux \cdot ux, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]
6. Add Preprocessing

Alternative 7: 89.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot t\_0\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, t\_0 \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* PI (* 2.0 uy)))))
   (fma
    (*
     (* (pow (- maxCos 1.0) 2.0) t_0)
     (sqrt (/ (pow ux 3.0) (fma -2.0 maxCos 2.0))))
    -0.5
    (* t_0 (* (sqrt (fma -2.0 maxCos 2.0)) (sqrt ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((((float) M_PI) * (2.0f * uy)));
	return fmaf(((powf((maxCos - 1.0f), 2.0f) * t_0) * sqrtf((powf(ux, 3.0f) / fmaf(-2.0f, maxCos, 2.0f)))), -0.5f, (t_0 * (sqrtf(fmaf(-2.0f, maxCos, 2.0f)) * sqrtf(ux))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	return fma(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * t_0) * sqrt(Float32((ux ^ Float32(3.0)) / fma(Float32(-2.0), maxCos, Float32(2.0))))), Float32(-0.5), Float32(t_0 * Float32(sqrt(fma(Float32(-2.0), maxCos, Float32(2.0))) * sqrt(ux))))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]
6. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. lower-sqrt.f32 90.2

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\right) \]

7. Applied rewrites 90.2%

   \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\right) \]
8. Add Preprocessing

Alternative 8: 89.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot t\_0\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* PI (* 2.0 uy)))))
   (fma
    (*
     (* (pow (- maxCos 1.0) 2.0) t_0)
     (sqrt (/ (pow ux 3.0) (fma -2.0 maxCos 2.0))))
    -0.5
    (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((((float) M_PI) * (2.0f * uy)));
	return fmaf(((powf((maxCos - 1.0f), 2.0f) * t_0) * sqrtf((powf(ux, 3.0f) / fmaf(-2.0f, maxCos, 2.0f)))), -0.5f, (t_0 * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	return fma(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * t_0) * sqrt(Float32((ux ^ Float32(3.0)) / fma(Float32(-2.0), maxCos, Float32(2.0))))), Float32(-0.5), Float32(t_0 * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux))))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]
6. Add Preprocessing

Alternative 9: 89.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot t\_0\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, t\_0 \cdot \sqrt{\left(maxCos \cdot \left(2 \cdot \frac{1}{maxCos} - 2\right)\right) \cdot ux}\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* PI (* 2.0 uy)))))
   (fma
    (*
     (* (pow (- maxCos 1.0) 2.0) t_0)
     (sqrt (/ (pow ux 3.0) (fma -2.0 maxCos 2.0))))
    -0.5
    (* t_0 (sqrt (* (* maxCos (- (* 2.0 (/ 1.0 maxCos)) 2.0)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((((float) M_PI) * (2.0f * uy)));
	return fmaf(((powf((maxCos - 1.0f), 2.0f) * t_0) * sqrtf((powf(ux, 3.0f) / fmaf(-2.0f, maxCos, 2.0f)))), -0.5f, (t_0 * sqrtf(((maxCos * ((2.0f * (1.0f / maxCos)) - 2.0f)) * ux))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	return fma(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * t_0) * sqrt(Float32((ux ^ Float32(3.0)) / fma(Float32(-2.0), maxCos, Float32(2.0))))), Float32(-0.5), Float32(t_0 * sqrt(Float32(Float32(maxCos * Float32(Float32(Float32(2.0) * Float32(Float32(1.0) / maxCos)) - Float32(2.0))) * ux))))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

6. Taylor expanded in maxCos around inf

   \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\left(maxCos \cdot \left(2 \cdot \frac{1}{maxCos} - 2\right)\right) \cdot ux}\right) \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 90.1

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\left(maxCos \cdot \left(2 \cdot \frac{1}{maxCos} - 2\right)\right) \cdot ux}\right) \]

8. Applied rewrites 90.1%

   \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\left(maxCos \cdot \left(2 \cdot \frac{1}{maxCos} - 2\right)\right) \cdot ux}\right) \]
9. Add Preprocessing

Alternative 10: 89.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ t_1 := \left(\left({\left(maxCos - 1\right)}^{2} \cdot t\_0\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}\right) \cdot -0.5\\ t_2 := t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \frac{t\_1 \cdot t\_1 - t\_2 \cdot t\_2}{t\_1 - t\_2} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* PI (* 2.0 uy))))
        (t_1
         (*
          (*
           (* (pow (- maxCos 1.0) 2.0) t_0)
           (sqrt (/ (pow ux 3.0) (fma -2.0 maxCos 2.0))))
          -0.5))
        (t_2 (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux)))))
   (/ (- (* t_1 t_1) (* t_2 t_2)) (- t_1 t_2))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((((float) M_PI) * (2.0f * uy)));
	float t_1 = ((powf((maxCos - 1.0f), 2.0f) * t_0) * sqrtf((powf(ux, 3.0f) / fmaf(-2.0f, maxCos, 2.0f)))) * -0.5f;
	float t_2 = t_0 * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	return ((t_1 * t_1) - (t_2 * t_2)) / (t_1 - t_2);
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	t_1 = Float32(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * t_0) * sqrt(Float32((ux ^ Float32(3.0)) / fma(Float32(-2.0), maxCos, Float32(2.0))))) * Float32(-0.5))
	t_2 = Float32(t_0 * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)))
	return Float32(Float32(Float32(t_1 * t_1) - Float32(t_2 * t_2)) / Float32(t_1 - t_2))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

7. Applied rewrites 90.1%

   \[\leadsto \frac{\left(\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}\right) \cdot -0.5\right) \cdot \left(\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}\right) \cdot -0.5\right) - \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)}{\color{blue}{\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}\right) \cdot -0.5 - \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}}} \]
8. Add Preprocessing

Alternative 11: 89.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ t_1 := 2 + -2 \cdot maxCos\\ \left(ux \cdot ux\right) \cdot \frac{\mathsf{fma}\left(-0.5, \sqrt{\frac{{ux}^{3}}{t\_1}} \cdot \left(t\_0 \cdot {\left(maxCos - 1\right)}^{2}\right), \sqrt{ux \cdot t\_1} \cdot t\_0\right)}{ux \cdot ux} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* uy PI)))) (t_1 (+ 2.0 (* -2.0 maxCos))))
   (*
    (* ux ux)
    (/
     (fma
      -0.5
      (* (sqrt (/ (pow ux 3.0) t_1)) (* t_0 (pow (- maxCos 1.0) 2.0)))
      (* (sqrt (* ux t_1)) t_0))
     (* ux ux)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((2.0f * (uy * ((float) M_PI))));
	float t_1 = 2.0f + (-2.0f * maxCos);
	return (ux * ux) * (fmaf(-0.5f, (sqrtf((powf(ux, 3.0f) / t_1)) * (t_0 * powf((maxCos - 1.0f), 2.0f))), (sqrtf((ux * t_1)) * t_0)) / (ux * ux));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))
	t_1 = Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos))
	return Float32(Float32(ux * ux) * Float32(fma(Float32(-0.5), Float32(sqrt(Float32((ux ^ Float32(3.0)) / t_1)) * Float32(t_0 * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0)))), Float32(sqrt(Float32(ux * t_1)) * t_0)) / Float32(ux * ux)))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

6. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

8. Applied rewrites 89.9%

   \[\leadsto \left(ux \cdot ux\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{\frac{1}{ux \cdot \left(2 + -2 \cdot maxCos\right)}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right), \sqrt{\frac{2 + -2 \cdot maxCos}{{ux}^{3}}} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]

9. Taylor expanded in ux around 0

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

    1. lower-/.f32 N/A

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

11. Applied rewrites 90.0%

    \[\leadsto \left(ux \cdot ux\right) \cdot \frac{\mathsf{fma}\left(-0.5, \sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right), \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}{ux \cdot \color{blue}{ux}} \]
12. Add Preprocessing

Alternative 12: 85.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + -2 \cdot maxCos\\ t_1 := \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ e^{\log ux \cdot 2} \cdot \mathsf{fma}\left(-0.5, \sqrt{\frac{1}{ux \cdot t\_0}} \cdot \left(t\_1 \cdot {\left(maxCos - 1\right)}^{2}\right), \sqrt{\frac{t\_0}{{ux}^{3}}} \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ 2.0 (* -2.0 maxCos))) (t_1 (sin (* 2.0 (* uy PI)))))
   (*
    (exp (* (log ux) 2.0))
    (fma
     -0.5
     (* (sqrt (/ 1.0 (* ux t_0))) (* t_1 (pow (- maxCos 1.0) 2.0)))
     (* (sqrt (/ t_0 (pow ux 3.0))) t_1)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f + (-2.0f * maxCos);
	float t_1 = sinf((2.0f * (uy * ((float) M_PI))));
	return expf((logf(ux) * 2.0f)) * fmaf(-0.5f, (sqrtf((1.0f / (ux * t_0))) * (t_1 * powf((maxCos - 1.0f), 2.0f))), (sqrtf((t_0 / powf(ux, 3.0f))) * t_1));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos))
	t_1 = sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))
	return Float32(exp(Float32(log(ux) * Float32(2.0))) * fma(Float32(-0.5), Float32(sqrt(Float32(Float32(1.0) / Float32(ux * t_0))) * Float32(t_1 * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0)))), Float32(sqrt(Float32(t_0 / (ux ^ Float32(3.0)))) * t_1)))
end

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

6. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

8. Applied rewrites 89.9%

   \[\leadsto \left(ux \cdot ux\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{\frac{1}{ux \cdot \left(2 + -2 \cdot maxCos\right)}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right), \sqrt{\frac{2 + -2 \cdot maxCos}{{ux}^{3}}} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 85.1

      \[\leadsto e^{\log ux \cdot 2} \cdot \mathsf{fma}\left(-0.5, \sqrt{\frac{\color{blue}{1}}{ux \cdot \left(2 + -2 \cdot maxCos\right)}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right), \sqrt{\frac{2 + -2 \cdot maxCos}{{ux}^{3}}} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

10. Applied rewrites 85.1%

    \[\leadsto e^{\log ux \cdot 2} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\sqrt{\frac{1}{ux \cdot \left(2 + -2 \cdot maxCos\right)}}} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right), \sqrt{\frac{2 + -2 \cdot maxCos}{{ux}^{3}}} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))))