Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.4% → 98.3%

Time: 13.3s

Alternatives: 6

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: 57.4% 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: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.4%

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

3. Simplified 98.4%

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

   1. add-cbrt-cube 98.4%

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

   2. add-cbrt-cube 98.4%

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

   3. cbrt-unprod 98.5%

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

   4. pow3 98.5%

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

   5. pow3 98.5%

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

6. Applied egg-rr 98.5%

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

Alternative 2: 98.3% 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 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.4%

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

3. Simplified 98.4%

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

Alternative 3: 90.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (* (* 2.0 PI) u2) 0.006000000052154064)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (((float) M_PI) * u2);
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.006000000052154064f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(Float32(pi) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.006000000052154064))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00600000005

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. log1p-define 98.5%

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

   3. Simplified 98.5%

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

      1. add-cbrt-cube 98.5%

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

      2. add-cbrt-cube 98.5%

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

      3. cbrt-unprod 98.5%

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

      4. pow3 98.5%

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

      5. pow3 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in u2 around 0 96.9%

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

   if 0.00600000005 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.8%

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

      1. sub-neg 55.8%

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

      2. log1p-define 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in u1 around 0 -0.0%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. 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 3.6%

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

      4. *-commutative 3.6%

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

      5. associate-*l* 3.6%

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

      6. *-commutative 3.6%

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

      7. mul-1-neg 3.6%

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

   7. Simplified 3.6%

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

      1. expm1-log1p-u 3.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)\right)\right)} \]

      2. expm1-undefine 3.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)\right)} - 1} \]

      3. *-commutative 3.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\sqrt{u1}\right) \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)} - 1 \]

      4. add-sqr-sqrt -0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-\sqrt{u1}} \cdot \sqrt{-\sqrt{u1}}\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

      5. sqrt-unprod 58.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-\sqrt{u1}\right) \cdot \left(-\sqrt{u1}\right)}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

      6. sqr-neg 58.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

      7. add-sqr-sqrt 58.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

      8. *-commutative 58.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right)\right)} - 1 \]

      9. associate-*r* 58.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)} - 1 \]

      10. *-commutative 58.3%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right)\right)} - 1 \]

      11. associate-*l* 58.3%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)}\right)} - 1 \]

   9. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 58.3%

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

       2. rem-exp-log 58.3%

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

       3. +-commutative 58.3%

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

       4. associate--l+ 76.5%

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

       5. *-commutative 76.5%

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

       6. *-commutative 76.5%

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

       7. associate-*r* 76.5%

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

       8. metadata-eval 76.5%

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

       9. mul0-lft 76.5%

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

       10. distribute-rgt-in 76.5%

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

       11. +-rgt-identity 76.5%

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

       12. associate-*r* 76.5%

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

       13. *-commutative 76.5%

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

       14. *-commutative 76.5%

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

       15. associate-*r* 76.5%

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

   11. Simplified 76.5%

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
6. 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%

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

7. Simplified 4.0%

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

   1. expm1-log1p-u 4.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)\right)\right)} \]

   2. expm1-undefine 5.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)\right)} - 1} \]

   3. *-commutative 5.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\sqrt{u1}\right) \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)} - 1 \]

   4. add-sqr-sqrt -0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-\sqrt{u1}} \cdot \sqrt{-\sqrt{u1}}\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

   5. sqrt-unprod 33.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-\sqrt{u1}\right) \cdot \left(-\sqrt{u1}\right)}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

   6. sqr-neg 33.5%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

   7. add-sqr-sqrt 33.5%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} - 1 \]

   8. *-commutative 33.5%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right)\right)} - 1 \]

   9. associate-*r* 33.5%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)} - 1 \]

   10. *-commutative 33.5%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right)\right)} - 1 \]

   11. associate-*l* 33.5%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)}\right)} - 1 \]

9. Applied egg-rr 33.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right)} - 1} \]
10. Step-by-step derivation

    1. log1p-undefine 33.5%

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

    2. rem-exp-log 33.5%

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

    3. +-commutative 33.5%

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

    4. associate--l+ 76.3%

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

    5. *-commutative 76.3%

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

    6. *-commutative 76.3%

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

    7. associate-*r* 76.3%

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

    8. metadata-eval 76.3%

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

    9. mul0-lft 76.3%

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

    10. distribute-rgt-in 76.3%

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

    11. +-rgt-identity 76.3%

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

    12. associate-*r* 76.3%

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

    13. *-commutative 76.3%

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

    14. *-commutative 76.3%

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

    15. associate-*r* 76.3%

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

11. Simplified 76.3%

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

12. Final simplification 76.3%

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

Alternative 5: 65.9% accurate, 2.9× speedup

Math

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

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(Float32(pi) * 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 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
6. 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%

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

7. Simplified 4.0%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 76.3%

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

   3. sqr-neg 76.3%

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

   4. add-sqr-sqrt 76.3%

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

   5. *-commutative 76.3%

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

   6. add-exp-log 73.2%

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

   7. *-commutative 73.2%

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

   8. associate-*r* 73.2%

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

   9. *-commutative 73.2%

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

   10. associate-*l* 73.2%

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

9. Applied egg-rr 73.2%

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

10. Taylor expanded in u2 around 0 65.0%

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

11. Final simplification 65.0%

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

Alternative 6: 4.7% accurate, 2.9× speedup

Math

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

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(Float32(pi) * 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 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
6. 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%

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

7. Simplified 4.0%

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

8. Taylor expanded in u2 around 0 4.4%

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

9. Final simplification 4.4%

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

Reproduce

herbie shell --seed 2024083 
(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))))