Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.5% → 98.4%

Time: 13.2s

Alternatives: 11

Speedup: 2.9×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}^{3} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt (* (pow (sin (* PI (* 2.0 u2))) 3.0) (pow (- (log1p (- u1))) 1.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf((powf(sinf((((float) M_PI) * (2.0f * u2))), 3.0f) * powf(-log1pf(-u1), 1.5f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return cbrt(Float32((sin(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) ^ Float32(3.0)) * (Float32(-log1p(Float32(-u1))) ^ Float32(1.5))))
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 57.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(e^{\log \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)} \]

   2. associate-*l* 57.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(e^{\log \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)}}\right) \]

4. Applied egg-rr 57.7%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(e^{\log \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)} \]
5. Step-by-step derivation

   1. *-commutative 57.7%

      \[\leadsto \color{blue}{\sin \left(e^{\log \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}} \]

   2. rem-exp-log 58.3%

      \[\leadsto \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \sqrt{-\log \left(1 - u1\right)} \]

   3. *-commutative 58.3%

      \[\leadsto \sin \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)} \]

   4. associate-*r* 58.3%

      \[\leadsto \sin \color{blue}{\left(\left(2 \cdot u2\right) \cdot \pi\right)} \cdot \sqrt{-\log \left(1 - u1\right)} \]

   5. *-commutative 58.3%

      \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \cdot \sqrt{-\log \left(1 - u1\right)} \]

   6. add-cbrt-cube 58.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}} \cdot \sqrt{-\log \left(1 - u1\right)} \]

   7. add-cbrt-cube 58.3%

      \[\leadsto \sqrt[3]{\left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}}} \]

   8. cbrt-unprod 58.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right) \cdot \left(\left(\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right)}} \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}^{3} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}}} \]
7. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* PI (* 2.0 u2))) (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((((float) M_PI) * (2.0f * u2))) * sqrtf(-log1pf(-u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(Float32(-log1p(Float32(-u1)))))
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 58.3%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.3%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. *-commutative 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right) \]

   4. associate-*l* 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)} \]
4. Add Preprocessing

5. Final simplification 98.3%

   \[\leadsto \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
6. Add Preprocessing

Alternative 3: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot \left(u1 \cdot -0.25 - 0.3333333333333333\right)\right)\right)} \cdot \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (* u1 (+ 1.0 (* u1 (- 0.5 (* u1 (- (* u1 -0.25) 0.3333333333333333)))))))
  (sin (* u2 (* PI 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * (0.5f - (u1 * ((u1 * -0.25f) - 0.3333333333333333f))))))) * sinf((u2 * (((float) M_PI) * 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) - Float32(u1 * Float32(Float32(u1 * Float32(-0.25)) - Float32(0.3333333333333333)))))))) * sin(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * (single(0.5) - (u1 * ((u1 * single(-0.25)) - single(0.3333333333333333)))))))) * sin((u2 * (single(pi) * single(2.0))));
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 91.7%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(-0.25 \cdot u1 - 0.3333333333333333\right) - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 91.7%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot \left(u1 \cdot -0.25 - 0.3333333333333333\right)\right)\right)} \cdot \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* u2 (* PI 2.0)))
  (sqrt (* u1 (+ 1.0 (* u1 (- 0.5 (* u1 -0.3333333333333333))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (((float) M_PI) * 2.0f))) * sqrtf((u1 * (1.0f + (u1 * (0.5f - (u1 * -0.3333333333333333f))))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) - Float32(u1 * Float32(-0.3333333333333333))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(pi) * single(2.0)))) * sqrt((u1 * (single(1.0) + (u1 * (single(0.5) - (u1 * single(-0.3333333333333333)))))));
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 89.8%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 89.8%

   \[\leadsto \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* u2 (* PI 2.0))) (sqrt (* u1 (- 1.0 (* u1 -0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (((float) M_PI) * 2.0f))) * sqrtf((u1 * (1.0f - (u1 * -0.5f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))) * sqrt(Float32(u1 * Float32(Float32(1.0) - Float32(u1 * Float32(-0.5))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(pi) * single(2.0)))) * sqrt((u1 * (single(1.0) - (u1 * single(-0.5)))));
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 86.3%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 86.3%

   \[\leadsto \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(\frac{1}{\frac{1}{\pi \cdot \left(2 \cdot u2\right)}}\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (/ 1.0 (/ 1.0 (* PI (* 2.0 u2))))) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((1.0f / (1.0f / (((float) M_PI) * (2.0f * u2))))) * sqrtf(u1);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(Float32(pi) * Float32(Float32(2.0) * u2))))) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(1.0) / (single(1.0) / (single(pi) * (single(2.0) * u2))))) * sqrt(u1);
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 57.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(e^{\log \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)} \]

   2. associate-*l* 57.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(e^{\log \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)}}\right) \]

4. Applied egg-rr 57.7%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(e^{\log \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)} \]
5. Step-by-step derivation

   1. rem-exp-log 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

   2. *-commutative 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right) \]

   3. pow1 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left({\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{1}\right)} \]

   4. metadata-eval 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{\color{blue}{\left(2 - 1\right)}}\right) \]

   5. pow-div 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\frac{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2}}{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{1}}\right)} \]

   6. pow1 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2}}{\color{blue}{2 \cdot \left(u2 \cdot \pi\right)}}\right) \]

   7. clear-num 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\frac{1}{\frac{2 \cdot \left(u2 \cdot \pi\right)}{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2}}}\right)} \]

   8. pow1 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{1}{\frac{\color{blue}{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{1}}}{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2}}}\right) \]

   9. pow-div 58.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{1}{\color{blue}{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{\left(1 - 2\right)}}}\right) \]

   10. metadata-eval 58.3%

       \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{1}{{\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{\color{blue}{-1}}}\right) \]

   11. inv-pow 58.3%

       \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{1}{\color{blue}{\frac{1}{2 \cdot \left(u2 \cdot \pi\right)}}}\right) \]

   12. associate-*r* 58.3%

       \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{1}{\frac{1}{\color{blue}{\left(2 \cdot u2\right) \cdot \pi}}}\right) \]

   13. *-commutative 58.3%

       \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{1}{\frac{1}{\color{blue}{\pi \cdot \left(2 \cdot u2\right)}}}\right) \]

6. Applied egg-rr 58.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\frac{1}{\frac{1}{\pi \cdot \left(2 \cdot u2\right)}}\right)} \]

7. Taylor expanded in u1 around 0 75.4%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot \sin \left(\frac{1}{\frac{1}{\pi \cdot \left(2 \cdot u2\right)}}\right) \]
8. Step-by-step derivation

   1. mul-1-neg 75.4%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\frac{1}{\frac{1}{\pi \cdot \left(2 \cdot u2\right)}}\right) \]

9. Simplified 75.4%

   \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\frac{1}{\frac{1}{\pi \cdot \left(2 \cdot u2\right)}}\right) \]

10. Final simplification 75.4%

    \[\leadsto \sin \left(\frac{1}{\frac{1}{\pi \cdot \left(2 \cdot u2\right)}}\right) \cdot \sqrt{u1} \]
11. Add Preprocessing

Alternative 7: 76.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* u2 (* PI 2.0))) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (((float) M_PI) * 2.0f))) * sqrtf(u1);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(pi) * single(2.0)))) * sqrt(u1);
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 58.3%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.3%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. *-commutative 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right) \]

   4. associate-*l* 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 98.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

   2. pow1/3 95.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

6. Applied egg-rr 71.8%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]
7. Step-by-step derivation

   1. unpow1/3 73.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

8. Simplified 73.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

9. Taylor expanded in u1 around 0 75.4%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 75.4%

       \[\leadsto \sqrt{u1} \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right) \]

    2. associate-*r* 75.4%

       \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

11. Simplified 75.4%

    \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

12. Final simplification 75.4%

    \[\leadsto \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1} \]
13. Add Preprocessing

Alternative 8: 66.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 2.0 (* PI (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 2.0f * (((float) M_PI) * (u2 * sqrtf(u1)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(Float32(pi) * Float32(u2 * sqrt(u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(2.0) * (single(pi) * (u2 * sqrt(u1)));
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 58.3%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.3%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. *-commutative 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right) \]

   4. associate-*l* 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 98.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

   2. pow1/3 95.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

6. Applied egg-rr 71.8%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]
7. Step-by-step derivation

   1. unpow1/3 73.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

8. Simplified 73.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

9. Taylor expanded in u1 around 0 75.3%

   \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{u1}^{3}}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

10. Taylor expanded in u2 around 0 62.8%

    \[\leadsto \color{blue}{2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]
11. Step-by-step derivation

    1. associate-*r* 62.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot \pi\right)} \]

    2. *-commutative 62.8%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \cdot \pi\right) \]

12. Simplified 62.8%

    \[\leadsto \color{blue}{2 \cdot \left(\left(u2 \cdot \sqrt{u1}\right) \cdot \pi\right)} \]

13. Final simplification 62.8%

    \[\leadsto 2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right) \]
14. Add Preprocessing

Alternative 9: 66.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\sqrt{u1} \cdot \left(\pi \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 2.0 (* (sqrt u1) (* PI u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 2.0f * (sqrtf(u1) * (((float) M_PI) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(sqrt(u1) * Float32(Float32(pi) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(2.0) * (sqrt(u1) * (single(pi) * u2));
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 58.3%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.3%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. *-commutative 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right) \]

   4. associate-*l* 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 98.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

   2. pow1/3 95.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

6. Applied egg-rr 71.8%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]
7. Step-by-step derivation

   1. unpow1/3 73.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

8. Simplified 73.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

9. Taylor expanded in u1 around 0 75.3%

   \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{u1}^{3}}}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

10. Taylor expanded in u2 around 0 62.8%

    \[\leadsto \color{blue}{2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]

11. Final simplification 62.8%

    \[\leadsto 2 \cdot \left(\sqrt{u1} \cdot \left(\pi \cdot u2\right)\right) \]
12. Add Preprocessing

Alternative 10: 4.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* PI (* u2 (sqrt u1))) -2.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return (((float) M_PI) * (u2 * sqrtf(u1))) * -2.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(pi) * Float32(u2 * sqrt(u1))) * Float32(-2.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(pi) * (u2 * sqrt(u1))) * single(-2.0);
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative -0.0%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} \]

   2. unpow2 -0.0%

      \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \]

   3. rem-square-sqrt 4.0%

      \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{-1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \]

   4. *-commutative 4.0%

      \[\leadsto \sqrt{u1} \cdot \left(-1 \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right)\right) \]

   5. neg-mul-1 4.0%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(-\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)} \]

   6. *-commutative 4.0%

      \[\leadsto \sqrt{u1} \cdot \left(-\sin \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right)\right) \]

5. Simplified 4.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(-\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} \]

6. Taylor expanded in u2 around 0 4.6%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]
7. Step-by-step derivation

   1. add0 4.6%

      \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right) + 0\right)} \]

   2. *-commutative 4.6%

      \[\leadsto -2 \cdot \left(\color{blue}{\left(u2 \cdot \pi\right) \cdot \sqrt{u1}} + 0\right) \]

8. Applied egg-rr 4.6%

   \[\leadsto -2 \cdot \color{blue}{\left(\left(u2 \cdot \pi\right) \cdot \sqrt{u1} + 0\right)} \]
9. Step-by-step derivation

   1. add0 4.6%

      \[\leadsto -2 \cdot \color{blue}{\left(\left(u2 \cdot \pi\right) \cdot \sqrt{u1}\right)} \]

   2. associate-*l* 4.6%

      \[\leadsto -2 \cdot \color{blue}{\left(u2 \cdot \left(\pi \cdot \sqrt{u1}\right)\right)} \]

   3. *-commutative 4.6%

      \[\leadsto -2 \cdot \left(u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \pi\right)}\right) \]

   4. associate-*l* 4.6%

      \[\leadsto -2 \cdot \color{blue}{\left(\left(u2 \cdot \sqrt{u1}\right) \cdot \pi\right)} \]

10. Simplified 4.6%

    \[\leadsto -2 \cdot \color{blue}{\left(\left(u2 \cdot \sqrt{u1}\right) \cdot \pi\right)} \]

11. Final simplification 4.6%

    \[\leadsto \left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right) \cdot -2 \]
12. Add Preprocessing

Alternative 11: 4.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{u1} \cdot \left(\pi \cdot u2\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* (sqrt u1) (* PI u2)) -2.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return (sqrtf(u1) * (((float) M_PI) * u2)) * -2.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(sqrt(u1) * Float32(Float32(pi) * u2)) * Float32(-2.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (sqrt(u1) * (single(pi) * u2)) * single(-2.0);
end

Derivation

1. Initial program 58.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative -0.0%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} \]

   2. unpow2 -0.0%

      \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \]

   3. rem-square-sqrt 4.0%

      \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{-1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \]

   4. *-commutative 4.0%

      \[\leadsto \sqrt{u1} \cdot \left(-1 \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right)\right) \]

   5. neg-mul-1 4.0%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(-\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)} \]

   6. *-commutative 4.0%

      \[\leadsto \sqrt{u1} \cdot \left(-\sin \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right)\right) \]

5. Simplified 4.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(-\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} \]

6. Taylor expanded in u2 around 0 4.6%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]

7. Final simplification 4.6%

   \[\leadsto \left(\sqrt{u1} \cdot \left(\pi \cdot u2\right)\right) \cdot -2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))