UniformSampleCone, y

Percentage Accurate: 57.9% → 98.2%

Time: 7.5s

Alternatives: 14

Speedup: 3.3×

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: 57.9% 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: 98.2% accurate, 0.2× 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 56.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 2: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower--.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-pow.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lower-*.f32 98.4

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

8. Applied rewrites 98.4%

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

9. Final simplification 98.4%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 98.4

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

8. Applied rewrites 98.4%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 56.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 98.4

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

7. Applied rewrites 98.4%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 97.6

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

8. Applied rewrites 97.6%

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

Alternative 6: 96.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt (* (- (fma (- ux) 1.0 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, 1.0f, 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(1.0), Float32(2.0)) - Float32(maxCos * Float32(2.0))) * ux)))
end

Derivation

1. Initial program 56.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 maxCos around 0

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

8. Applied rewrites 97.0%

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

Alternative 7: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0035000001080334187:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, 1, 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0035000001080334187)
   (* (* 2.0 (* uy PI)) (sqrt (* (- (fma (- ux) 1.0 2.0) (* maxCos 2.0)) ux)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (fma -2.0 maxCos 2.0) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0035000001080334187f) {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf(((fmaf(-ux, 1.0f, 2.0f) - (maxCos * 2.0f)) * ux));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(1.0), Float32(2.0)) - Float32(maxCos * Float32(2.0))) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00350000011

   1. Initial program 56.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 maxCos around 0

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lift-PI.f32 93.3

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

   11. Applied rewrites 93.3%

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

   if 0.00350000011 < uy

   1. Initial program 53.9%

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

   3. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 78.5

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

   5. Applied rewrites 78.5%

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

Alternative 8: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.1500000255182385 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 - ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.1500000255182385e-5)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (- 2.0 ux) ux)))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt
     (*
      (* maxCos maxCos)
      (fma
       -1.0
       (* ux ux)
       (+
        (/ (* ux (- 2.0 ux)) (* maxCos maxCos))
        (/ (* ux (- (* 2.0 ux) 2.0)) maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.1500000255182385e-5f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((2.0f - ux) * ux));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf(((maxCos * maxCos) * fmaf(-1.0f, (ux * ux), (((ux * (2.0f - ux)) / (maxCos * maxCos)) + ((ux * ((2.0f * ux) - 2.0f)) / maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.1500000255182385e-5))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(maxCos * maxCos) * fma(Float32(-1.0), Float32(ux * ux), Float32(Float32(Float32(ux * Float32(Float32(2.0) - ux)) / Float32(maxCos * maxCos)) + Float32(Float32(ux * Float32(Float32(Float32(2.0) * ux) - Float32(2.0))) / maxCos))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.15e-5

   1. Initial program 56.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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 maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 98.0

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

   8. Applied rewrites 98.0%

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

   if 1.15e-5 < maxCos

   1. Initial program 56.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. pow2 N/A

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

       5. lift-*.f32 N/A

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

       6. lift-pow.f32 N/A

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

       7. lift-PI.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lift-PI.f32 88.0

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 + -1 \cdot ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)} \]

   11. Applied rewrites 88.0%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\pi}^{3}, 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 + -1 \cdot ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)} \]
   12. Step-by-step derivation

       1. lift-PI.f32 N/A

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

       2. lift-pow.f32 N/A

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

       3. unpow3 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lift-PI.f32 N/A

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

       7. lift-PI.f32 N/A

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

       8. lift-PI.f32 88.0

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

   13. Applied rewrites 88.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.1500000255182385 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 - ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.3999999555717316 \cdot 10^{-21}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, 1, 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 - ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.3999999555717316e-21)
   (* (* 2.0 (* uy PI)) (sqrt (* (- (fma (- ux) 1.0 2.0) (* maxCos 2.0)) ux)))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt
     (*
      (* maxCos maxCos)
      (fma
       -1.0
       (* ux ux)
       (+
        (/ (* ux (- 2.0 ux)) (* maxCos maxCos))
        (/ (* ux (- (* 2.0 ux) 2.0)) maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.3999999555717316e-21f) {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf(((fmaf(-ux, 1.0f, 2.0f) - (maxCos * 2.0f)) * ux));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf(((maxCos * maxCos) * fmaf(-1.0f, (ux * ux), (((ux * (2.0f - ux)) / (maxCos * maxCos)) + ((ux * ((2.0f * ux) - 2.0f)) / maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.3999999555717316e-21))
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(1.0), Float32(2.0)) - Float32(maxCos * Float32(2.0))) * ux)));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(maxCos * maxCos) * fma(Float32(-1.0), Float32(ux * ux), Float32(Float32(Float32(ux * Float32(Float32(2.0) - ux)) / Float32(maxCos * maxCos)) + Float32(Float32(ux * Float32(Float32(Float32(2.0) * ux) - Float32(2.0))) / maxCos))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.4e-21

   1. Initial program 60.5%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lift-PI.f32 83.4

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

   11. Applied rewrites 83.4%

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

   if 1.4e-21 < maxCos

   1. Initial program 52.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. pow2 N/A

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

       5. lift-*.f32 N/A

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

       6. lift-pow.f32 N/A

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

       7. lift-PI.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lift-PI.f32 87.4

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 + -1 \cdot ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)} \]

   11. Applied rewrites 87.4%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\pi}^{3}, 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 + -1 \cdot ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)} \]
   12. Step-by-step derivation

       1. lift-PI.f32 N/A

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

       2. lift-pow.f32 N/A

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

       3. unpow3 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lift-PI.f32 N/A

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

       7. lift-PI.f32 N/A

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

       8. lift-PI.f32 87.4

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

   13. Applied rewrites 87.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.3999999555717316 \cdot 10^{-21}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, 1, 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos \cdot maxCos\right) \cdot \mathsf{fma}\left(-1, ux \cdot ux, \frac{ux \cdot \left(2 - ux\right)}{maxCos \cdot maxCos} + \frac{ux \cdot \left(2 \cdot ux - 2\right)}{maxCos}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 maxCos around 0

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

8. Applied rewrites 97.0%

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

9. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lift-PI.f32 79.8

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

11. Applied rewrites 79.8%

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

Alternative 11: 49.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in maxCos around 0

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in maxCos around 0

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

   1. lift--.f32 54.1

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

8. Applied rewrites 54.1%

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

9. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lift-PI.f32 47.7

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

11. Applied rewrites 47.7%

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

Alternative 12: 41.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 40.9

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

8. Applied rewrites 40.9%

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

Alternative 13: 40.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 40.9

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

8. Applied rewrites 40.9%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 40.0%

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

Alternative 14: 7.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - 1} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025082 
(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)))))))