UniformSampleCone, y

Percentage Accurate: 57.0% → 98.3%

Time: 14.0s

Alternatives: 9

Speedup: 2.0×

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: 57.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. associate-*l* 58.1%

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

   2. sub-neg 58.1%

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

   3. +-commutative 58.1%

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

   4. distribute-rgt-neg-in 58.1%

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

   5. fma-define 58.2%

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

3. Simplified 58.5%

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

5. Taylor expanded in uy around inf 58.2%

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

7. Simplified 58.3%

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

8. Taylor expanded in ux around -inf 98.3%

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

   1. sub-neg 98.3%

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

   2. +-commutative 98.3%

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

   3. mul-1-neg 98.3%

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

   4. unsub-neg 98.3%

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

   5. +-commutative 98.3%

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

   6. mul-1-neg 98.3%

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

   7. unsub-neg 98.3%

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

   8. sub-neg 98.3%

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

   9. metadata-eval 98.3%

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

   10. +-commutative 98.3%

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

   11. mul-1-neg 98.3%

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

   12. remove-double-neg 98.3%

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

   13. *-commutative 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in ux around 0 98.3%

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

    1. associate--l+ 98.4%

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

    2. associate-*r* 98.4%

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

    3. sub-neg 98.4%

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

    4. metadata-eval 98.4%

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

13. Simplified 98.4%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. associate-*l* 58.1%

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

   2. sub-neg 58.1%

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

   3. +-commutative 58.1%

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

   4. distribute-rgt-neg-in 58.1%

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

   5. fma-define 58.2%

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

3. Simplified 58.5%

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

5. Taylor expanded in uy around inf 58.2%

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

7. Simplified 58.3%

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

8. Taylor expanded in ux around -inf 98.3%

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

   1. sub-neg 98.3%

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

   2. +-commutative 98.3%

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

   3. mul-1-neg 98.3%

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

   4. unsub-neg 98.3%

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

   5. +-commutative 98.3%

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

   6. mul-1-neg 98.3%

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

   7. unsub-neg 98.3%

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

   8. sub-neg 98.3%

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

   9. metadata-eval 98.3%

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

   10. +-commutative 98.3%

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

   11. mul-1-neg 98.3%

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

   12. remove-double-neg 98.3%

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

   13. *-commutative 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in uy around inf 98.2%

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

    1. associate-*l* 98.0%

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

    2. +-commutative 98.0%

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

    3. associate--l+ 98.0%

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

    4. associate-*r/ 98.0%

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

    5. metadata-eval 98.0%

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

    6. associate-*r/ 98.0%

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

    7. div-sub 98.1%

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

    8. sub-neg 98.1%

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

    9. metadata-eval 98.1%

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

13. Simplified 98.1%

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

Alternative 3: 92.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. associate-*l* 58.1%

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

   2. sub-neg 58.1%

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

   3. +-commutative 58.1%

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

   4. distribute-rgt-neg-in 58.1%

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

   5. fma-define 58.2%

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

3. Simplified 58.5%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 93.6%

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

   1. *-commutative 93.6%

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

   2. mul-1-neg 93.6%

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

   3. unsub-neg 93.6%

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

8. Simplified 93.6%

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

Alternative 4: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. associate-*l* 58.1%

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

   2. sub-neg 58.1%

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

   3. +-commutative 58.1%

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

   4. distribute-rgt-neg-in 58.1%

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

   5. fma-define 58.2%

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

3. Simplified 58.5%

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

5. Taylor expanded in uy around inf 58.2%

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

7. Simplified 58.3%

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

8. Taylor expanded in ux around -inf 98.3%

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

   1. sub-neg 98.3%

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

   2. +-commutative 98.3%

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

   3. mul-1-neg 98.3%

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

   4. unsub-neg 98.3%

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

   5. +-commutative 98.3%

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

   6. mul-1-neg 98.3%

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

   7. unsub-neg 98.3%

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

   8. sub-neg 98.3%

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

   9. metadata-eval 98.3%

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

   10. +-commutative 98.3%

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

   11. mul-1-neg 98.3%

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

   12. remove-double-neg 98.3%

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

   13. *-commutative 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in uy around 0 83.0%

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

12. Final simplification 83.0%

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

Alternative 5: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. associate-*l* 58.1%

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

   2. sub-neg 58.1%

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

   3. +-commutative 58.1%

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

   4. distribute-rgt-neg-in 58.1%

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

   5. fma-define 58.2%

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

3. Simplified 58.5%

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

5. Taylor expanded in uy around inf 58.2%

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

7. Simplified 58.3%

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

8. Taylor expanded in ux around -inf 98.3%

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

   1. sub-neg 98.3%

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

   2. +-commutative 98.3%

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

   3. mul-1-neg 98.3%

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

   4. unsub-neg 98.3%

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

   5. +-commutative 98.3%

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

   6. mul-1-neg 98.3%

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

   7. unsub-neg 98.3%

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

   8. sub-neg 98.3%

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

   9. metadata-eval 98.3%

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

   10. +-commutative 98.3%

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

   11. mul-1-neg 98.3%

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

   12. remove-double-neg 98.3%

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

   13. *-commutative 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in uy around 0 83.0%

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

    1. associate-*l* 82.8%

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

    2. +-commutative 82.8%

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

    3. associate--l+ 82.8%

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

    4. associate-*r/ 82.8%

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

    5. metadata-eval 82.8%

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

    6. associate-*r/ 82.8%

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

    7. div-sub 82.9%

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

    8. sub-neg 82.9%

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

    9. metadata-eval 82.9%

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

13. Simplified 82.9%

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

Alternative 6: 77.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around inf 98.2%

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

4. Taylor expanded in maxCos around 0 93.5%

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

5. Taylor expanded in uy around 0 79.8%

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

   1. associate-*l* 79.7%

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

   2. sub-neg 79.7%

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

   3. metadata-eval 79.7%

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

   4. +-commutative 79.7%

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

   5. associate-*r/ 79.7%

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

   6. metadata-eval 79.7%

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

7. Simplified 79.7%

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

8. Taylor expanded in uy around 0 79.8%

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

   1. *-commutative 79.8%

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

   2. sub-neg 79.8%

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

   3. metadata-eval 79.8%

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

   4. +-commutative 79.8%

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

   5. associate-*r/ 79.8%

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

   6. metadata-eval 79.8%

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

   7. metadata-eval 79.8%

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

   8. associate-*r/ 79.8%

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

   9. +-commutative 79.8%

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

   10. associate-*r/ 79.8%

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

   11. metadata-eval 79.8%

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

10. Simplified 79.8%

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

11. Final simplification 79.8%

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

Alternative 7: 77.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around inf 98.2%

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

4. Taylor expanded in maxCos around 0 93.5%

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

5. Taylor expanded in uy around 0 79.8%

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

   1. associate-*l* 79.7%

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

   2. sub-neg 79.7%

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

   3. metadata-eval 79.7%

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

   4. +-commutative 79.7%

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

   5. associate-*r/ 79.7%

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

   6. metadata-eval 79.7%

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

7. Simplified 79.7%

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

8. Taylor expanded in uy around 0 79.7%

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

   1. sub-neg 79.7%

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

   2. metadata-eval 79.7%

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

   3. +-commutative 79.7%

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

   4. associate-*r/ 79.7%

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

   5. metadata-eval 79.7%

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

   6. associate-*l* 79.7%

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

   7. metadata-eval 79.7%

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

   8. associate-*r/ 79.7%

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

   9. +-commutative 79.7%

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

   10. associate-*r/ 79.7%

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

   11. metadata-eval 79.7%

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

10. Simplified 79.7%

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

11. Final simplification 79.7%

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

Alternative 8: -0.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around inf 98.2%

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

4. Taylor expanded in maxCos around 0 93.5%

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

5. Taylor expanded in uy around 0 79.8%

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

   1. associate-*l* 79.7%

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

   2. sub-neg 79.7%

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

   3. metadata-eval 79.7%

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

   4. +-commutative 79.7%

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

   5. associate-*r/ 79.7%

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

   6. metadata-eval 79.7%

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

7. Simplified 79.7%

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

8. Taylor expanded in ux around inf -0.0%

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

Alternative 9: 7.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{0}\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* 2.0 (* uy (* PI (sqrt 0.0)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. associate-*l* 58.1%

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

   2. sub-neg 58.1%

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

   3. +-commutative 58.1%

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

   4. distribute-rgt-neg-in 58.1%

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

   5. fma-define 58.2%

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

3. Simplified 58.5%

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

5. Taylor expanded in uy around 0 52.5%

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

7. Simplified 52.6%

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

8. Taylor expanded in ux around 0 7.1%

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

9. Final simplification 7.1%

   \[\leadsto 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{0}\right)\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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)))))))