UniformSampleCone, x

Percentage Accurate: 57.4% → 99.0%

Time: 15.4s

Alternatives: 9

Speedup: 1.0×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

   1. associate-*l* 56.6%

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

   2. sub-neg 56.6%

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

   3. +-commutative 56.6%

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

   4. distribute-rgt-neg-in 56.6%

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

   5. fma-define 56.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in ux around 0 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

   1. associate-*l* 56.6%

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

   2. sub-neg 56.6%

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

   3. +-commutative 56.6%

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

   4. distribute-rgt-neg-in 56.6%

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

   5. fma-define 56.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. +-commutative 99.0%

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

   5. distribute-lft-in 99.0%

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

   6. metadata-eval 99.0%

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

   7. neg-mul-1 99.0%

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

   8. sub-neg 99.0%

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

   9. sub-neg 99.0%

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

   10. metadata-eval 99.0%

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

   11. +-commutative 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

   1. associate-*l* 56.6%

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

   2. sub-neg 56.6%

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

   3. +-commutative 56.6%

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

   4. distribute-rgt-neg-in 56.6%

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

   5. fma-define 56.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in ux around 0 99.0%

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

7. Taylor expanded in uy around inf 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in uy around inf 98.9%

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

11. Final simplification 98.9%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

   1. associate-*l* 56.6%

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

   2. sub-neg 56.6%

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

   3. +-commutative 56.6%

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

   4. distribute-rgt-neg-in 56.6%

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

   5. fma-define 56.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in ux around 0 99.0%

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

7. Taylor expanded in maxCos around 0 98.1%

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

8. Final simplification 98.1%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

   1. associate-*l* 56.6%

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

   2. sub-neg 56.6%

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

   3. +-commutative 56.6%

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

   4. distribute-rgt-neg-in 56.6%

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

   5. fma-define 56.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in uy around inf 98.4%

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

   1. associate-*l* 98.4%

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

   2. associate-*r* 98.4%

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

   3. associate--l+ 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

   7. associate-/l* 98.4%

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

   8. distribute-lft-in 98.4%

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

   9. metadata-eval 98.4%

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

   10. neg-mul-1 98.4%

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

   11. sub-neg 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in maxCos around 0 97.6%

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

    1. associate--l+ 97.6%

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

    2. associate-*r/ 97.6%

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

    3. metadata-eval 97.6%

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

    4. associate-*r/ 97.6%

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

    5. metadata-eval 97.6%

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

11. Simplified 97.6%

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

12. Final simplification 97.6%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0 98.1%

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

7. Final simplification 98.1%

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

Alternative 7: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0 46.5%

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

4. Taylor expanded in ux around inf 77.9%

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

   1. associate-*r* 77.9%

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

   2. expm1-log1p-u 78.0%

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(2 \cdot uy\right) \cdot \pi\right)\right)\right)} \]

   3. expm1-undefine 78.0%

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(2 \cdot uy\right) \cdot \pi\right)\right)} - 1\right)} \]

   4. log1p-undefine 77.9%

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

   5. rem-exp-log 77.9%

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

   6. associate-*r* 77.9%

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

6. Applied egg-rr 77.9%

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

   1. associate--l+ 78.0%

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

8. Simplified 78.0%

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

9. Final simplification 78.0%

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

Alternative 8: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0 46.5%

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

4. Taylor expanded in ux around inf 77.9%

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

5. Final simplification 77.9%

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

Alternative 9: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0 46.5%

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

4. Taylor expanded in maxCos around 0 77.9%

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

   1. distribute-lft-out-- 77.9%

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

6. Simplified 77.9%

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

7. Final simplification 77.9%

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

Reproduce

herbie shell --seed 2024179 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))