Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.3% → 98.4%

Time: 11.8s

Alternatives: 8

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: 57.3% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

   1. sub-neg 57.8%

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

   2. log1p-define 98.2%

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

3. Simplified 98.2%

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

Alternative 2: 93.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in u1 around 0 92.9%

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

4. Final simplification 92.9%

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

Alternative 3: 91.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in u1 around 0 91.1%

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

4. Final simplification 91.1%

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

Alternative 4: 87.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in u1 around 0 87.5%

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

4. Final simplification 87.5%

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

Alternative 5: 76.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

   1. sub-neg 57.8%

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

   2. log1p-define 98.2%

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

3. Simplified 98.2%

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

   1. pow1/2 98.2%

      \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-undefine 57.8%

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

   3. sub-neg 57.8%

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

   4. pow-to-exp 57.8%

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

   5. add-sqr-sqrt 57.9%

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

   6. sqrt-unprod 57.8%

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

   7. sqr-neg 57.8%

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

   8. sqrt-unprod 1.5%

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

   9. add-sqr-sqrt 1.5%

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

   10. sub-neg 1.5%

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

   11. log1p-undefine -0.0%

       \[\leadsto e^{\log \color{blue}{\left(\mathsf{log1p}\left(-u1\right)\right)} \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. add-sqr-sqrt -0.0%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{\sqrt{-u1} \cdot \sqrt{-u1}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. sqrt-unprod 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. sqr-neg 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\sqrt{\color{blue}{u1 \cdot u1}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. sqrt-unprod 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. add-sqr-sqrt 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{u1}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Applied egg-rr 73.9%

   \[\leadsto \color{blue}{e^{\log \left(\mathsf{log1p}\left(u1\right)\right) \cdot 0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Taylor expanded in u1 around 0 77.0%

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

8. Final simplification 77.0%

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

Alternative 6: 66.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 4.0%

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

   4. *-commutative 4.0%

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

   5. associate-*l* 4.0%

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

   6. *-commutative 4.0%

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

   7. mul-1-neg 4.0%

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

5. Simplified 4.0%

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

   1. add-sqr-sqrt 0.1%

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

   2. sqrt-unprod 74.2%

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

   3. *-commutative 74.2%

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

   4. *-commutative 74.2%

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

   5. *-commutative 74.2%

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

   6. *-commutative 74.2%

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

   7. swap-sqr 74.0%

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

   8. sqr-neg 74.0%

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

   9. add-sqr-sqrt 74.2%

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

   10. pow2 74.2%

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

7. Applied egg-rr 74.2%

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

8. Taylor expanded in u2 around 0 66.0%

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

   1. *-commutative 66.0%

      \[\leadsto \sqrt{\color{blue}{\left(4 \cdot \left({u2}^{2} \cdot {\pi}^{2}\right)\right) \cdot u1}} \]

   2. sqrt-prod 66.0%

      \[\leadsto \color{blue}{\sqrt{4 \cdot \left({u2}^{2} \cdot {\pi}^{2}\right)} \cdot \sqrt{u1}} \]

   3. *-commutative 66.0%

      \[\leadsto \sqrt{\color{blue}{\left({u2}^{2} \cdot {\pi}^{2}\right) \cdot 4}} \cdot \sqrt{u1} \]

   4. sqrt-prod 66.0%

      \[\leadsto \color{blue}{\left(\sqrt{{u2}^{2} \cdot {\pi}^{2}} \cdot \sqrt{4}\right)} \cdot \sqrt{u1} \]

   5. pow-prod-down 66.1%

      \[\leadsto \left(\sqrt{\color{blue}{{\left(u2 \cdot \pi\right)}^{2}}} \cdot \sqrt{4}\right) \cdot \sqrt{u1} \]

   6. *-commutative 66.1%

      \[\leadsto \left(\sqrt{{\color{blue}{\left(\pi \cdot u2\right)}}^{2}} \cdot \sqrt{4}\right) \cdot \sqrt{u1} \]

   7. sqrt-pow1 66.1%

      \[\leadsto \left(\color{blue}{{\left(\pi \cdot u2\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{4}\right) \cdot \sqrt{u1} \]

   8. metadata-eval 66.1%

      \[\leadsto \left({\left(\pi \cdot u2\right)}^{\color{blue}{1}} \cdot \sqrt{4}\right) \cdot \sqrt{u1} \]

   9. pow1 66.1%

      \[\leadsto \left(\color{blue}{\left(\pi \cdot u2\right)} \cdot \sqrt{4}\right) \cdot \sqrt{u1} \]

   10. metadata-eval 66.1%

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

   11. associate-*l* 66.1%

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

10. Applied egg-rr 66.1%

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

11. Final simplification 66.1%

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

Alternative 7: 66.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

   1. sub-neg 57.8%

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

   2. log1p-define 98.2%

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

3. Simplified 98.2%

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

   1. pow1/2 98.2%

      \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-undefine 57.8%

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

   3. sub-neg 57.8%

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

   4. pow-to-exp 57.8%

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

   5. add-sqr-sqrt 57.9%

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

   6. sqrt-unprod 57.8%

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

   7. sqr-neg 57.8%

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

   8. sqrt-unprod 1.5%

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

   9. add-sqr-sqrt 1.5%

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

   10. sub-neg 1.5%

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

   11. log1p-undefine -0.0%

       \[\leadsto e^{\log \color{blue}{\left(\mathsf{log1p}\left(-u1\right)\right)} \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. add-sqr-sqrt -0.0%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{\sqrt{-u1} \cdot \sqrt{-u1}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. sqrt-unprod 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. sqr-neg 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\sqrt{\color{blue}{u1 \cdot u1}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. sqrt-unprod 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. add-sqr-sqrt 73.9%

       \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{u1}\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Applied egg-rr 73.9%

   \[\leadsto \color{blue}{e^{\log \left(\mathsf{log1p}\left(u1\right)\right) \cdot 0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Taylor expanded in u2 around 0 38.0%

   \[\leadsto \color{blue}{u2 \cdot \left(-1.3333333333333333 \cdot \left(\left({u2}^{2} \cdot {\pi}^{3}\right) \cdot \sqrt{\log \left(1 + u1\right)}\right) + 2 \cdot \left(\pi \cdot \sqrt{\log \left(1 + u1\right)}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 38.0%

      \[\leadsto u2 \cdot \left(\color{blue}{\left(-1.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{3}\right)\right) \cdot \sqrt{\log \left(1 + u1\right)}} + 2 \cdot \left(\pi \cdot \sqrt{\log \left(1 + u1\right)}\right)\right) \]

   2. associate-*r* 38.0%

      \[\leadsto u2 \cdot \left(\left(-1.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{3}\right)\right) \cdot \sqrt{\log \left(1 + u1\right)} + \color{blue}{\left(2 \cdot \pi\right) \cdot \sqrt{\log \left(1 + u1\right)}}\right) \]

   3. distribute-rgt-out 38.0%

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\log \left(1 + u1\right)} \cdot \left(-1.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right)} \]

   4. log1p-define 68.6%

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \cdot \left(-1.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]

   5. +-commutative 68.6%

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \color{blue}{\left(2 \cdot \pi + -1.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{3}\right)\right)}\right) \]

   6. *-commutative 68.6%

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \left(\color{blue}{\pi \cdot 2} + -1.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{3}\right)\right)\right) \]

   7. fma-define 68.6%

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\pi, 2, -1.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{3}\right)\right)}\right) \]

   8. *-commutative 68.6%

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \mathsf{fma}\left(\pi, 2, \color{blue}{\left({u2}^{2} \cdot {\pi}^{3}\right) \cdot -1.3333333333333333}\right)\right) \]

9. Simplified 68.6%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \mathsf{fma}\left(\pi, 2, \left({u2}^{2} \cdot {\pi}^{3}\right) \cdot -1.3333333333333333\right)\right)} \]

10. Taylor expanded in u2 around 0 37.2%

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

    1. associate-*r* 37.2%

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

    2. log1p-define 64.5%

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

12. Simplified 64.5%

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

13. Taylor expanded in u1 around 0 66.0%

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

Alternative 8: 4.6% 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 57.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 4.0%

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

   4. *-commutative 4.0%

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

   5. associate-*l* 4.0%

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

   6. *-commutative 4.0%

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

   7. mul-1-neg 4.0%

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

5. Simplified 4.0%

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

6. Taylor expanded in u2 around 0 4.8%

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

7. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2024096 
(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))))