Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.0% → 99.1%

Time: 15.5s

Alternatives: 12

Speedup: 3.1×

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: 57.0% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

   1. sub-neg 58.1%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.00039999998989515007)
     (sqrt (- (log1p (- u1))))
     (*
      (cos t_0)
      (sqrt (* u1 (+ 1.0 (* u1 (- 0.5 (* u1 -0.3333333333333333))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.00039999998989515007f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (1.0f + (u1 * (0.5f - (u1 * -0.3333333333333333f))))));
	}
	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(0.00039999998989515007))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) - Float32(u1 * Float32(-0.3333333333333333))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.9999999e-4

   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 99.5%

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

   if 3.9999999e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.6%

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

   3. Taylor expanded in u1 around 0 91.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u1 \cdot \left(\cos \left(u2 \cdot \left(\pi \cdot -2\right)\right) \cdot \sqrt{0.5 + \frac{1}{u1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.00039999998989515007)
   (sqrt (- (log1p (- u1))))
   (* u1 (* (cos (* u2 (* PI -2.0))) (sqrt (+ 0.5 (/ 1.0 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00039999998989515007f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = u1 * (cosf((u2 * (((float) M_PI) * -2.0f))) * sqrtf((0.5f + (1.0f / u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00039999998989515007))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(u1 * Float32(cos(Float32(u2 * Float32(Float32(pi) * Float32(-2.0)))) * sqrt(Float32(Float32(0.5) + Float32(Float32(1.0) / u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.9999999e-4

   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 99.5%

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

   if 3.9999999e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.6%

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

   3. Taylor expanded in u1 around 0 88.4%

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

   4. Taylor expanded in u1 around inf 88.5%

      \[\leadsto \sqrt{-\color{blue}{-1 \cdot \left({u1}^{2} \cdot \left(0.5 + \frac{1}{u1}\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. mul-1-neg 88.5%

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

      2. *-commutative 88.5%

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

      3. distribute-rgt-neg-in 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in u2 around inf 88.1%

      \[\leadsto \color{blue}{\left(u1 \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \sqrt{0.5 + \frac{1}{u1}}} \]
   8. Step-by-step derivation

      1. associate-*l* 88.1%

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

      2. associate-*r* 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*r* 88.1%

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

      5. *-commutative 88.1%

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

      6. associate-*r* 88.1%

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

      7. metadata-eval 88.1%

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

      8. distribute-rgt-neg-in 88.1%

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

      9. associate-*r* 88.1%

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

      10. rem-cube-cbrt 88.1%

          \[\leadsto u1 \cdot \left(\cos \left(-u2 \cdot \left(\pi \cdot \color{blue}{{\left(\sqrt[3]{-2}\right)}^{3}}\right)\right) \cdot \sqrt{0.5 + \frac{1}{u1}}\right) \]

      11. cos-neg 88.1%

          \[\leadsto u1 \cdot \left(\color{blue}{\cos \left(u2 \cdot \left(\pi \cdot {\left(\sqrt[3]{-2}\right)}^{3}\right)\right)} \cdot \sqrt{0.5 + \frac{1}{u1}}\right) \]

      12. rem-cube-cbrt 88.1%

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

      13. +-commutative 88.1%

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

   9. Simplified 88.1%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u1 \cdot \left(\cos \left(u2 \cdot \left(\pi \cdot -2\right)\right) \cdot \sqrt{0.5 + \frac{1}{u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.00039999998989515007)
   (sqrt (- (log1p (- u1))))
   (* (sqrt (* u1 (- 1.0 (* u1 -0.5)))) (cos (* 2.0 (* PI u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00039999998989515007f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((u1 * (1.0f - (u1 * -0.5f)))) * cosf((2.0f * (((float) M_PI) * u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00039999998989515007))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) - Float32(u1 * Float32(-0.5))))) * cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.9999999e-4

   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 99.5%

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

   if 3.9999999e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.6%

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

   3. Taylor expanded in u1 around 0 88.4%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. pow1/2 88.4%

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

      2. distribute-rgt-neg-in 88.4%

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

      3. unpow-prod-down 88.0%

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

      4. pow1/2 88.0%

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

      5. *-commutative 88.0%

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

      6. fma-neg 88.0%

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

      7. metadata-eval 88.0%

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

   5. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot {\left(-\mathsf{fma}\left(u1, -0.5, -1\right)\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. unpow1/2 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in u2 around inf 88.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \left(\sqrt{u1} \cdot \left(1 + u1 \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.00039999998989515007)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (* (sqrt u1) (+ 1.0 (* u1 0.25)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.00039999998989515007f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * (sqrtf(u1) * (1.0f + (u1 * 0.25f)));
	}
	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(0.00039999998989515007))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(u1 * Float32(0.25)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.9999999e-4

   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 99.5%

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

   if 3.9999999e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.6%

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

   3. Taylor expanded in u1 around 0 88.4%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. pow1/2 88.4%

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

      2. distribute-rgt-neg-in 88.4%

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

      3. unpow-prod-down 88.0%

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

      4. pow1/2 88.0%

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

      5. *-commutative 88.0%

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

      6. fma-neg 88.0%

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

      7. metadata-eval 88.0%

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

   5. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot {\left(-\mathsf{fma}\left(u1, -0.5, -1\right)\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. unpow1/2 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in u1 around 0 88.6%

      \[\leadsto \left(\sqrt{u1} \cdot \color{blue}{\left(1 + 0.25 \cdot u1\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative 88.6%

         \[\leadsto \left(\sqrt{u1} \cdot \left(1 + \color{blue}{u1 \cdot 0.25}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. Simplified 88.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\sqrt{u1} \cdot \left(1 + u1 \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (cos (* (* 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 cosf(((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(cos(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 = cos(((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 58.1%

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

3. Taylor expanded in u1 around 0 94.1%

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

4. Final simplification 94.1%

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

Alternative 7: 90.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.027000000700354576:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.027000000700354576)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.027000000700354576f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	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(0.027000000700354576))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0270000007

   1. Initial program 59.2%

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

      1. sub-neg 59.2%

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

      2. log1p-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in u2 around 0 95.3%

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

   if 0.0270000007 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.4%

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

      1. pow1/2 55.4%

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

      2. pow-to-exp 55.4%

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

      3. add-sqr-sqrt 55.4%

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

      4. sqrt-unprod 55.4%

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

      5. sqr-neg 55.4%

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

      6. sqrt-unprod 1.5%

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

      7. add-sqr-sqrt 1.5%

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

      8. sub-neg 1.5%

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

      9. log1p-undefine -0.0%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 76.2%

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

      12. sqr-neg 76.2%

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

      13. sqrt-unprod 76.2%

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

      14. add-sqr-sqrt 76.2%

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

   4. Applied egg-rr 76.2%

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

   5. Taylor expanded in u1 around 0 79.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.027000000700354576:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

   1. sub-neg 58.1%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 78.3%

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

6. Final simplification 78.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
7. Add Preprocessing

Alternative 9: 77.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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 * (1.0e0 + (u1 * (0.5e0 + (u1 * (0.3333333333333333e0 - (u1 * (-0.25e0)))))))))
end function

Julia

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

Derivation

1. Initial program 58.1%

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

   1. sub-neg 58.1%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 78.3%

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

6. Taylor expanded in u1 around 0 75.3%

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

7. Final simplification 75.3%

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

Alternative 10: 75.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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 * (1.0e0 + (u1 * (0.5e0 - (u1 * (-0.3333333333333333e0)))))))
end function

Julia

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

Derivation

1. Initial program 58.1%

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

   1. sub-neg 58.1%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 78.3%

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

6. Taylor expanded in u1 around 0 74.2%

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

7. Final simplification 74.2%

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

Alternative 11: 73.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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 * (1.0e0 - (u1 * (-0.5e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in u1 around 0 89.1%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. pow1/2 89.1%

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

   2. distribute-rgt-neg-in 89.1%

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

   3. unpow-prod-down 88.9%

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

   4. pow1/2 88.9%

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

   5. *-commutative 88.9%

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

   6. fma-neg 88.9%

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

   7. metadata-eval 88.9%

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

5. Applied egg-rr 88.9%

   \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot {\left(-\mathsf{fma}\left(u1, -0.5, -1\right)\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. unpow1/2 88.9%

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

7. Simplified 88.9%

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

8. Taylor expanded in u2 around 0 72.0%

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

9. Final simplification 72.0%

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

Alternative 12: 65.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

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)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 58.1%

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

   1. pow1/2 58.1%

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

   2. pow-to-exp 58.2%

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

   3. add-sqr-sqrt 58.2%

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

   4. sqrt-unprod 58.2%

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

   5. sqr-neg 58.2%

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

   6. sqrt-unprod 1.3%

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

   7. add-sqr-sqrt 1.3%

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

   8. sub-neg 1.3%

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

   9. log1p-undefine -0.0%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 74.1%

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

   12. sqr-neg 74.1%

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

   13. sqrt-unprod 74.1%

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

   14. add-sqr-sqrt 74.1%

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

4. Applied egg-rr 74.1%

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

5. Taylor expanded in u1 around 0 77.2%

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

6. Taylor expanded in u2 around 0 63.4%

   \[\leadsto \color{blue}{\sqrt{u1}} \]

7. Final simplification 63.4%

   \[\leadsto \sqrt{u1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(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))))