Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 99.0%

Time: 14.3s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

Alternative 2: 96.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0014199999859556556:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(-0.25 \cdot u1 - 0.3333333333333333\right) - 0.5\right) - 1\right)} \cdot \cos t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0014199999859556556)
     (* (sqrt (- (log1p (- u1)))) 1.0)
     (*
      (sqrt
       (-
        (*
         u1
         (- (* u1 (- (* u1 (- (* -0.25 u1) 0.3333333333333333)) 0.5)) 1.0))))
      (cos t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0014199999859556556f) {
		tmp = sqrtf(-log1pf(-u1)) * 1.0f;
	} else {
		tmp = sqrtf(-(u1 * ((u1 * ((u1 * ((-0.25f * u1) - 0.3333333333333333f)) - 0.5f)) - 1.0f))) * cosf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0014199999859556556))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(-Float32(u1 * Float32(Float32(u1 * Float32(Float32(u1 * Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0))))) * cos(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00141999999

   1. Initial program 59.4%

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

      1. sub-neg 59.4%

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

      2. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

   if 0.00141999999 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 50.2%

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

   3. Taylor expanded in u1 around 0 93.1%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(-0.25 \cdot u1 - 0.3333333333333333\right) - 0.5\right) - 1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0014199999859556556:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)} \cdot \cos t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0014199999859556556)
     (* (sqrt (- (log1p (- u1)))) 1.0)
     (*
      (sqrt (- (* u1 (- (* u1 (- (* -0.3333333333333333 u1) 0.5)) 1.0))))
      (cos t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0014199999859556556f) {
		tmp = sqrtf(-log1pf(-u1)) * 1.0f;
	} else {
		tmp = sqrtf(-(u1 * ((u1 * ((-0.3333333333333333f * u1) - 0.5f)) - 1.0f))) * cosf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0014199999859556556))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(-Float32(u1 * Float32(Float32(u1 * Float32(Float32(Float32(-0.3333333333333333) * u1) - Float32(0.5))) - Float32(1.0))))) * cos(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00141999999

   1. Initial program 59.4%

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

      1. sub-neg 59.4%

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

      2. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

   if 0.00141999999 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 50.2%

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

   3. Taylor expanded in u1 around 0 91.8%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0014199999859556556:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-u1 \cdot \left(-0.5 \cdot u1 - 1\right)} \cdot \cos t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0014199999859556556)
     (* (sqrt (- (log1p (- u1)))) 1.0)
     (* (sqrt (- (* u1 (- (* -0.5 u1) 1.0)))) (cos t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0014199999859556556f) {
		tmp = sqrtf(-log1pf(-u1)) * 1.0f;
	} else {
		tmp = sqrtf(-(u1 * ((-0.5f * u1) - 1.0f))) * cosf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0014199999859556556))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(-Float32(u1 * Float32(Float32(Float32(-0.5) * u1) - Float32(1.0))))) * cos(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00141999999

   1. Initial program 59.4%

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

      1. sub-neg 59.4%

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

      2. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

   if 0.00141999999 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 50.2%

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

   3. Taylor expanded in u1 around 0 89.5%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0035000001080334187)
     (* (sqrt (- (log1p (- u1)))) 1.0)
     (* (sqrt u1) (cos t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0035000001080334187f) {
		tmp = sqrtf(-log1pf(-u1)) * 1.0f;
	} else {
		tmp = sqrtf(u1) * cosf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0035000001080334187))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(1.0));
	else
		tmp = Float32(sqrt(u1) * cos(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00350000011

   1. Initial program 58.7%

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

      1. sub-neg 58.7%

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

      2. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 98.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

   if 0.00350000011 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 50.1%

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

      1. sub-neg 50.1%

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

      2. log1p-define 97.9%

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

   3. Simplified 97.9%

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

      1. neg-mul-1 97.9%

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

      2. log1p-undefine 50.1%

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

      3. sub-neg 50.1%

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

      4. neg-mul-1 50.1%

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

      5. pow1/2 50.1%

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

      6. add-sqr-sqrt 50.1%

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

      7. sqrt-unprod 50.1%

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

      8. sqr-neg 50.1%

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

      9. sqrt-unprod 2.0%

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

      10. add-sqr-sqrt 2.0%

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

      11. sub-neg 2.0%

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

      12. log1p-undefine -0.0%

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

      13. add-sqr-sqrt -0.0%

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

      14. sqrt-unprod 79.3%

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

      15. sqr-neg 79.3%

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

      16. sqrt-unprod 79.4%

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

      17. add-sqr-sqrt 79.3%

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

   6. Applied egg-rr 79.3%

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

   7. Taylor expanded in u1 around 0 81.0%

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

Alternative 6: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * 1.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(1.0))
end

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
6. Add Preprocessing

Alternative 7: 76.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-(u1 * ((u1 * ((u1 * (((-0.25e0) * u1) - 0.3333333333333333e0)) - 0.5e0)) - 1.0e0))) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 76.0%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(-0.25 \cdot u1 - 0.3333333333333333\right) - 0.5\right) - 1\right)}} \cdot 1 \]
7. Add Preprocessing

Alternative 8: 74.9% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (* u1 (- (* u1 (- (* -0.3333333333333333 u1) 0.5)) 1.0)))) 1.0))

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-(u1 * ((u1 * (((-0.3333333333333333e0) * u1) - 0.5e0)) - 1.0e0))) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 74.6%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}} \cdot 1 \]
7. Add Preprocessing

Alternative 9: 72.5% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (+ (* u1 (* u1 -0.5)) (* u1 -1.0)))) 1.0))

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-((u1 * (u1 * (-0.5e0))) + (u1 * (-1.0e0)))) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 71.8%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot 1 \]
7. Step-by-step derivation

   1. add-sqr-sqrt 71.8%

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

   2. sqrt-unprod 71.8%

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

   3. sqr-neg 71.8%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt -0.0%

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

   6. sub-neg -0.0%

      \[\leadsto \sqrt{-\left(-u1\right) \cdot \color{blue}{\left(-0.5 \cdot u1 + \left(-1\right)\right)}} \cdot 1 \]

   7. distribute-lft-in -0.0%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-u1\right) \cdot \left(-0.5 \cdot u1\right) + \left(-u1\right) \cdot \left(-1\right)\right)}} \cdot 1 \]

   8. add-sqr-sqrt -0.0%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(\sqrt{-u1} \cdot \sqrt{-u1}\right)} \cdot \left(-0.5 \cdot u1\right) + \left(-u1\right) \cdot \left(-1\right)\right)} \cdot 1 \]

   9. sqrt-unprod -0.0%

      \[\leadsto \sqrt{-\left(\color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}} \cdot \left(-0.5 \cdot u1\right) + \left(-u1\right) \cdot \left(-1\right)\right)} \cdot 1 \]

   10. sqr-neg -0.0%

       \[\leadsto \sqrt{-\left(\sqrt{\color{blue}{u1 \cdot u1}} \cdot \left(-0.5 \cdot u1\right) + \left(-u1\right) \cdot \left(-1\right)\right)} \cdot 1 \]

   11. sqrt-unprod -0.0%

       \[\leadsto \sqrt{-\left(\color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1}\right)} \cdot \left(-0.5 \cdot u1\right) + \left(-u1\right) \cdot \left(-1\right)\right)} \cdot 1 \]

   12. add-sqr-sqrt -0.0%

       \[\leadsto \sqrt{-\left(\color{blue}{u1} \cdot \left(-0.5 \cdot u1\right) + \left(-u1\right) \cdot \left(-1\right)\right)} \cdot 1 \]

   13. *-commutative -0.0%

       \[\leadsto \sqrt{-\left(u1 \cdot \color{blue}{\left(u1 \cdot -0.5\right)} + \left(-u1\right) \cdot \left(-1\right)\right)} \cdot 1 \]

   14. add-sqr-sqrt -0.0%

       \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) + \color{blue}{\left(\sqrt{-u1} \cdot \sqrt{-u1}\right)} \cdot \left(-1\right)\right)} \cdot 1 \]

   15. sqrt-unprod 71.9%

       \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) + \color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}} \cdot \left(-1\right)\right)} \cdot 1 \]

   16. sqr-neg 71.9%

       \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) + \sqrt{\color{blue}{u1 \cdot u1}} \cdot \left(-1\right)\right)} \cdot 1 \]

   17. sqrt-unprod 71.8%

       \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) + \color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1}\right)} \cdot \left(-1\right)\right)} \cdot 1 \]

   18. add-sqr-sqrt 71.9%

       \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) + \color{blue}{u1} \cdot \left(-1\right)\right)} \cdot 1 \]

   19. metadata-eval 71.9%

       \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) + u1 \cdot \color{blue}{-1}\right)} \cdot 1 \]

8. Applied egg-rr 71.9%

   \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) + u1 \cdot -1\right)}} \cdot 1 \]
9. Add Preprocessing

Alternative 10: 72.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-(u1 * (((-0.5e0) * u1) - 1.0e0))) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 71.8%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot 1 \]
7. Add Preprocessing

Alternative 11: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{--1 \cdot u1} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (sqrt (- (* -1.0 u1))) 1.0))

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-((-1.0e0) * u1)) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(Float32(-1.0) * u1))) * Float32(1.0))
end

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 63.9%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot 1 \]
7. Add Preprocessing

Alternative 12: 18.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(u1 \cdot \sqrt{0.5}\right) \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (* u1 (sqrt 0.5)) 1.0))

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u1 * sqrt(0.5e0)) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u1 * sqrt(Float32(0.5))) * Float32(1.0))
end

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. sub-neg 55.8%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 71.8%

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

7. Taylor expanded in u1 around inf 18.9%

   \[\leadsto \color{blue}{\left(u1 \cdot \sqrt{0.5}\right)} \cdot 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024077 -o generate:simplify
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))