Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 99.1%

Time: 13.1s

Alternatives: 13

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: 57.3% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

   1. cos-neg 58.2%

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

   2. distribute-rgt-neg-out 58.2%

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

   3. sub-neg 58.2%

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

   4. log1p-define 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-out 99.1%

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

   9. distribute-rgt-neg-in 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9980000257492065:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot u2\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* PI 2.0)))))
   (if (<= t_0 0.9980000257492065)
     (*
      t_0
      (sqrt
       (*
        u1
        (+ 1.0 (* u1 (+ 0.5 (* u1 (- 0.3333333333333333 (* u1 -0.25)))))))))
     (* (+ 1.0 (* -2.0 (* u2 (* PI (* PI u2))))) (sqrt (- (log1p (- u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (((float) M_PI) * 2.0f)));
	float tmp;
	if (t_0 <= 0.9980000257492065f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f - (u1 * -0.25f))))))));
	} else {
		tmp = (1.0f + (-2.0f * (u2 * (((float) M_PI) * (((float) M_PI) * u2))))) * sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9980000257492065))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(Float32(0.3333333333333333) - Float32(u1 * Float32(-0.25))))))))));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(u2 * Float32(Float32(pi) * Float32(Float32(pi) * u2))))) * sqrt(Float32(-log1p(Float32(-u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.998000026

   1. Initial program 55.1%

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

   3. Taylor expanded in u1 around 0 93.2%

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

   if 0.998000026 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   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. cos-neg 59.4%

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

      2. distribute-rgt-neg-out 59.4%

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

      3. sub-neg 59.4%

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

      4. log1p-define 99.3%

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

      5. associate-*l* 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*l* 99.3%

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

      8. distribute-lft-neg-out 99.3%

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

      9. distribute-rgt-neg-in 99.3%

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

      10. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in u2 around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

      6. *-commutative 99.3%

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

   9. Simplified 99.3%

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

       1. unpow2 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*r* 99.3%

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

       4. *-commutative 99.3%

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

   11. Applied egg-rr 99.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -2 \cdot \color{blue}{\left(\left(\left(\pi \cdot u2\right) \cdot \pi\right) \cdot u2\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \leq 0.9980000257492065:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot u2\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9980000257492065:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot u2\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* PI 2.0)))))
   (if (<= t_0 0.9980000257492065)
     (* t_0 (sqrt (* u1 (- 1.0 (* u1 -0.5)))))
     (* (+ 1.0 (* -2.0 (* u2 (* PI (* PI u2))))) (sqrt (- (log1p (- u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (((float) M_PI) * 2.0f)));
	float tmp;
	if (t_0 <= 0.9980000257492065f) {
		tmp = t_0 * sqrtf((u1 * (1.0f - (u1 * -0.5f))));
	} else {
		tmp = (1.0f + (-2.0f * (u2 * (((float) M_PI) * (((float) M_PI) * u2))))) * sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9980000257492065))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) - Float32(u1 * Float32(-0.5))))));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(u2 * Float32(Float32(pi) * Float32(Float32(pi) * u2))))) * sqrt(Float32(-log1p(Float32(-u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.998000026

   1. Initial program 55.1%

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

   3. Taylor expanded in u1 around 0 89.2%

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

   if 0.998000026 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   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. cos-neg 59.4%

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

      2. distribute-rgt-neg-out 59.4%

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

      3. sub-neg 59.4%

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

      4. log1p-define 99.3%

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

      5. associate-*l* 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*l* 99.3%

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

      8. distribute-lft-neg-out 99.3%

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

      9. distribute-rgt-neg-in 99.3%

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

      10. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in u2 around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

      6. *-commutative 99.3%

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

   9. Simplified 99.3%

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

       1. unpow2 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*r* 99.3%

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

       4. *-commutative 99.3%

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

   11. Applied egg-rr 99.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -2 \cdot \color{blue}{\left(\left(\left(\pi \cdot u2\right) \cdot \pi\right) \cdot u2\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \leq 0.9980000257492065:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot u2\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.05999999865889549:\\ \;\;\;\;\left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot u2\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.05999999865889549)
     (* (+ 1.0 (* -2.0 (* u2 (* PI (* PI u2))))) (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 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.05999999865889549f) {
		tmp = (1.0f + (-2.0f * (u2 * (((float) M_PI) * (((float) M_PI) * u2))))) * 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(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05999999865889549))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(u2 * Float32(Float32(pi) * Float32(Float32(pi) * u2))))) * 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) < 0.0599999987

   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. cos-neg 59.4%

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

      2. distribute-rgt-neg-out 59.4%

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

      3. sub-neg 59.4%

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

      4. log1p-define 99.3%

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

      5. associate-*l* 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*l* 99.3%

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

      8. distribute-lft-neg-out 99.3%

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

      9. distribute-rgt-neg-in 99.3%

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

      10. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in u2 around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

      6. *-commutative 99.3%

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

   9. Simplified 99.3%

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

       1. unpow2 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*r* 99.3%

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

       4. *-commutative 99.3%

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

   11. Applied egg-rr 99.3%

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

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.1%

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

   3. Taylor expanded in u1 around 0 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. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.05999999865889549:\\ \;\;\;\;\left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot u2\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0006699999794363976)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt (* u1 (- 1.0 (* u1 -0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.0006699999794363976f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (1.0f - (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0006699999794363976))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) - Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 6.6999998e-4

   1. Initial program 61.3%

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

      1. cos-neg 61.3%

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

      2. distribute-rgt-neg-out 61.3%

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

      3. sub-neg 61.3%

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

      4. log1p-define 99.5%

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

      5. associate-*l* 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*l* 99.5%

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

      8. distribute-lft-neg-out 99.5%

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

      9. distribute-rgt-neg-in 99.5%

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

      10. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in u2 around 0 99.2%

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

   if 6.6999998e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 54.6%

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

   3. Taylor expanded in u1 around 0 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. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u1 \cdot \left(\sqrt{0.5 + \frac{1}{u1}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.0006699999794363976)
   (sqrt (- (log1p (- u1))))
   (* u1 (* (sqrt (+ 0.5 (/ 1.0 u1))) (cos (* 2.0 (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.0006699999794363976f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = u1 * (sqrtf((0.5f + (1.0f / u1))) * cosf((2.0f * (((float) M_PI) * u2))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.0006699999794363976))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(u1 * Float32(sqrt(Float32(Float32(0.5) + Float32(Float32(1.0) / u1))) * 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) < 6.6999998e-4

   1. Initial program 61.3%

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

      1. cos-neg 61.3%

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

      2. distribute-rgt-neg-out 61.3%

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

      3. sub-neg 61.3%

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

      4. log1p-define 99.5%

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

      5. associate-*l* 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*l* 99.5%

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

      8. distribute-lft-neg-out 99.5%

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

      9. distribute-rgt-neg-in 99.5%

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

      10. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in u2 around 0 99.2%

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

   if 6.6999998e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 54.6%

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

   3. Taylor expanded in u1 around 0 89.5%

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

   4. Taylor expanded in u1 around inf 89.4%

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

      1. mul-1-neg 89.4%

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

      2. *-commutative 89.4%

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

      3. distribute-rgt-neg-in 89.4%

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

      4. +-commutative 89.4%

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

   6. Simplified 89.4%

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

   7. Taylor expanded in u2 around inf 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*r* 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*r* 89.1%

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

   9. Simplified 89.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u1 \cdot \left(\sqrt{0.5 + \frac{1}{u1}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.007499999832361937:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.007499999832361937)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.007499999832361937f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.007499999832361937))
		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) < 0.00749999983

   1. Initial program 59.7%

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

      1. cos-neg 59.7%

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

      2. distribute-rgt-neg-out 59.7%

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

      3. sub-neg 59.7%

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

      4. log1p-define 99.4%

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

      5. associate-*l* 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*l* 99.4%

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

      8. distribute-lft-neg-out 99.4%

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

      9. distribute-rgt-neg-in 99.4%

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

      10. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in u2 around 0 96.9%

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

   if 0.00749999983 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.6%

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

      1. add-cube-cbrt 55.5%

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

      2. pow3 55.5%

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

   4. Applied egg-rr 75.5%

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

   5. Taylor expanded in u1 around 0 77.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.007499999832361937:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% 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 58.2%

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

   1. cos-neg 58.2%

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

   2. distribute-rgt-neg-out 58.2%

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

   3. sub-neg 58.2%

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

   4. log1p-define 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-out 99.1%

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

   9. distribute-rgt-neg-in 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 75.9%

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

6. Final simplification 75.9%

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

Alternative 9: 77.0% 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 58.2%

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

   1. cos-neg 58.2%

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

   2. distribute-rgt-neg-out 58.2%

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

   3. sub-neg 58.2%

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

   4. log1p-define 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-out 99.1%

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

   9. distribute-rgt-neg-in 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 75.9%

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

6. Taylor expanded in u1 around 0 71.2%

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

7. Final simplification 71.2%

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

Alternative 10: 75.7% 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 58.2%

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

   1. cos-neg 58.2%

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

   2. distribute-rgt-neg-out 58.2%

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

   3. sub-neg 58.2%

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

   4. log1p-define 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-out 99.1%

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

   9. distribute-rgt-neg-in 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 75.9%

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

6. Taylor expanded in u1 around 0 70.1%

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

7. Final simplification 70.1%

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

Alternative 11: 73.1% 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 58.2%

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

   1. cos-neg 58.2%

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

   2. distribute-rgt-neg-out 58.2%

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

   3. sub-neg 58.2%

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

   4. log1p-define 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-out 99.1%

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

   9. distribute-rgt-neg-in 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 75.9%

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

6. Taylor expanded in u1 around 0 67.7%

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

7. Final simplification 67.7%

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

Alternative 12: 73.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \sqrt{0.5 + \frac{1}{u1}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * sqrtf((0.5f + (1.0f / 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 = u1 * sqrt((0.5e0 + (1.0e0 / u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in u1 around 0 86.9%

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

4. Taylor expanded in u1 around inf 86.9%

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

   1. mul-1-neg 86.9%

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

   2. *-commutative 86.9%

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

   3. distribute-rgt-neg-in 86.9%

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

   4. +-commutative 86.9%

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

6. Simplified 86.9%

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

7. Taylor expanded in u2 around 0 67.6%

   \[\leadsto \color{blue}{u1 \cdot \sqrt{0.5 + \frac{1}{u1}}} \]
8. Add Preprocessing

Alternative 13: 65.1% 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 58.2%

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

   1. cos-neg 58.2%

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

   2. distribute-rgt-neg-out 58.2%

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

   3. sub-neg 58.2%

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

   4. log1p-define 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-out 99.1%

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

   9. distribute-rgt-neg-in 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 75.9%

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

6. Taylor expanded in u1 around 0 60.6%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot 1 \]
7. Step-by-step derivation

   1. mul-1-neg 60.6%

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

8. Simplified 60.6%

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

9. Final simplification 60.6%

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

Reproduce

herbie shell --seed 2024107 
(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))))