Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.5% → 98.3%

Time: 12.4s

Alternatives: 8

Speedup: 2.9×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

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

   1. sub-neg 58.3%

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

   2. log1p-define 98.3%

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

3. Simplified 98.3%

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

Alternative 2: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0006699999794363976)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (*
      (sin t_0)
      (sqrt
       (*
        u1
        (+ 1.0 (* u1 (+ 0.5 (* u1 (- 0.3333333333333333 (* 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 <= 0.0006699999794363976f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f - (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(0.0006699999794363976))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(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))))))))));
	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 61.5%

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

      1. sub-neg 61.5%

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

      2. log1p-define 98.8%

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

   3. Simplified 98.8%

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

      1. expm1-log1p-u 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

   6. Applied egg-rr 98.8%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 41.8%

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

      2. log1p-undefine 41.8%

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

      3. rem-exp-log 41.8%

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

      4. *-commutative 41.8%

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

      5. associate-*r* 41.8%

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

      6. +-commutative 41.8%

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

      7. associate-*r* 41.8%

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

      8. *-commutative 41.8%

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

      9. *-commutative 41.8%

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

   8. Applied egg-rr 41.8%

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

   9. Taylor expanded in u2 around 0 98.8%

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

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

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0 93.3%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0006699999794363976)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (*
      (sin 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 <= 0.0006699999794363976f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(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(0.0006699999794363976))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(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) < 6.6999998e-4

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. log1p-define 98.8%

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

   3. Simplified 98.8%

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

      1. expm1-log1p-u 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

   6. Applied egg-rr 98.8%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 41.8%

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

      2. log1p-undefine 41.8%

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

      3. rem-exp-log 41.8%

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

      4. *-commutative 41.8%

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

      5. associate-*r* 41.8%

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

      6. +-commutative 41.8%

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

      7. associate-*r* 41.8%

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

      8. *-commutative 41.8%

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

      9. *-commutative 41.8%

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

   8. Applied egg-rr 41.8%

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

   9. Taylor expanded in u2 around 0 98.8%

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

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

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0 91.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0006699999794363976f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * (1.0f - (u1 * -0.5f))));
	}
	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(0.0006699999794363976))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(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 61.5%

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

      1. sub-neg 61.5%

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

      2. log1p-define 98.8%

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

   3. Simplified 98.8%

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

      1. expm1-log1p-u 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

   6. Applied egg-rr 98.8%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 41.8%

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

      2. log1p-undefine 41.8%

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

      3. rem-exp-log 41.8%

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

      4. *-commutative 41.8%

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

      5. associate-*r* 41.8%

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

      6. +-commutative 41.8%

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

      7. associate-*r* 41.8%

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

      8. *-commutative 41.8%

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

      9. *-commutative 41.8%

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

   8. Applied egg-rr 41.8%

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

   9. Taylor expanded in u2 around 0 98.8%

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

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

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0 88.9%

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

4. Final simplification 94.3%

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

Alternative 5: 90.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.00860000029206276:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.00860000029206276)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin 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 <= 0.00860000029206276f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(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(0.00860000029206276))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(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) < 0.00860000029

   1. Initial program 60.0%

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

      1. sub-neg 60.0%

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

      2. log1p-define 98.7%

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

   3. Simplified 98.7%

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

      1. expm1-log1p-u 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*r* 98.7%

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

   6. Applied egg-rr 98.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 47.9%

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

      2. log1p-undefine 47.9%

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

      3. rem-exp-log 47.9%

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

      4. *-commutative 47.9%

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

      5. associate-*r* 47.9%

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

      6. +-commutative 47.9%

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

      7. associate-*r* 47.9%

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

      8. *-commutative 47.9%

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

      9. *-commutative 47.9%

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

   8. Applied egg-rr 47.9%

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

   9. Taylor expanded in u2 around 0 96.9%

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

   if 0.00860000029 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.2%

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

      1. sub-neg 55.2%

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

      2. log1p-define 97.6%

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

   3. Simplified 97.6%

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

      1. pow1/2 97.6%

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

      2. log1p-undefine 55.2%

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

      3. sub-neg 55.2%

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

      4. pow-to-exp 55.1%

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

      5. add-sqr-sqrt 55.2%

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

      6. sqrt-unprod 55.1%

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

      7. sqr-neg 55.1%

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

      8. sqrt-unprod 1.7%

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

      9. add-sqr-sqrt 1.7%

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

      10. sub-neg 1.7%

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

      11. log1p-undefine -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 74.2%

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

      14. sqr-neg 74.2%

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

      15. sqrt-unprod 74.2%

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

      16. add-sqr-sqrt 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in u1 around 0 77.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00860000029206276:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

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

   1. sub-neg 58.3%

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

   2. log1p-define 98.3%

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

3. Simplified 98.3%

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

   1. pow1/2 98.3%

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

   2. log1p-undefine 58.3%

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

   3. sub-neg 58.3%

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

   4. pow-to-exp 58.3%

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

   5. add-sqr-sqrt 58.3%

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

   6. sqrt-unprod 58.3%

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

   7. sqr-neg 58.3%

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

   8. sqrt-unprod 1.6%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. log1p-undefine -0.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 72.2%

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

   14. sqr-neg 72.2%

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

   15. sqrt-unprod 72.2%

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

   16. add-sqr-sqrt 72.2%

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

6. Applied egg-rr 72.2%

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

7. Taylor expanded in u1 around 0 75.4%

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

8. Final simplification 75.4%

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

Alternative 7: 66.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* PI (* u2 (sqrt (* u1 4.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return ((float) M_PI) * (u2 * sqrtf((u1 * 4.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(pi) * Float32(u2 * sqrt(Float32(u1 * Float32(4.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(pi) * (u2 * sqrt((u1 * single(4.0))));
end

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in u1 around 0 -0.0%

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

   1. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 4.0%

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

   4. *-commutative 4.0%

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

   5. associate-*l* 4.0%

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

   6. *-commutative 4.0%

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

   7. mul-1-neg 4.0%

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

5. Simplified 4.0%

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

6. Taylor expanded in u2 around 0 4.6%

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

   1. associate-*r* 4.6%

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

   2. *-commutative 4.6%

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

   3. *-commutative 4.6%

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

   4. *-commutative 4.6%

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

   5. *-commutative 4.6%

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

8. Simplified 4.6%

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

   1. pow1 4.6%

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

   2. *-commutative 4.6%

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

   3. associate-*l* 4.6%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 62.8%

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

   6. swap-sqr 62.8%

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

   7. add-sqr-sqrt 62.8%

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

   8. metadata-eval 62.8%

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

10. Applied egg-rr 62.8%

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

12. Simplified 62.8%

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

Alternative 8: 66.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in u1 around 0 -0.0%

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

   1. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 4.0%

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

   4. *-commutative 4.0%

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

   5. associate-*l* 4.0%

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

   6. *-commutative 4.0%

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

   7. mul-1-neg 4.0%

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

5. Simplified 4.0%

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

6. Taylor expanded in u2 around 0 4.6%

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

   1. associate-*r* 4.6%

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

   2. *-commutative 4.6%

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

   3. *-commutative 4.6%

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

   4. *-commutative 4.6%

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

   5. *-commutative 4.6%

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

8. Simplified 4.6%

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

   1. pow1 4.6%

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

   2. *-commutative 4.6%

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

   3. associate-*l* 4.6%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 62.8%

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

   6. swap-sqr 62.8%

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

   7. add-sqr-sqrt 62.8%

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

   8. metadata-eval 62.8%

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

10. Applied egg-rr 62.8%

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

12. Simplified 62.8%

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

13. Taylor expanded in u2 around 0 62.8%

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

14. Final simplification 62.8%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))