Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 98.9%

Time: 12.2s

Alternatives: 12

Speedup: 21.0×

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: 57.6% 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: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(Float32(pi) * log(Float32(exp(1))))) * u2)))
end

Derivation

1. Initial program 55.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-PI.f32 N/A

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

   2. add-log-exp N/A

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

   3. *-un-lft-identity N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \log \left(e^{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}\right)\right) \cdot u2\right) \]

   4. lift-PI.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \log \left(e^{1 \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)\right) \cdot u2\right) \]

   5. exp-prod N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \log \color{blue}{\left({\left(e^{1}\right)}^{\mathsf{PI}\left(\right)}\right)}\right) \cdot u2\right) \]

   6. log-pow N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(e^{1}\right)\right)}\right) \cdot u2\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(e^{1}\right)\right)}\right) \cdot u2\right) \]

   8. lower-log.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\log \left(e^{1}\right)}\right)\right) \cdot u2\right) \]

   9. exp-1-e N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \log \color{blue}{\mathsf{E}\left(\right)}\right)\right) \cdot u2\right) \]

   10. lower-E.f32 99.0

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

6. Applied rewrites 99.0%

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

Alternative 2: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - u1 \cdot \left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot -0.5, -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p u1) (* u1 (* u1 (fma u1 (* u1 -0.5) -1.0))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf((log1pf(u1) - (u1 * (u1 * fmaf(u1, (u1 * -0.5f), -1.0f)))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(log1p(u1) - Float32(u1 * Float32(u1 * fma(u1, Float32(u1 * Float32(-0.5)), Float32(-1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999949992

   1. Initial program 50.0%

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 80.6

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

   7. Applied rewrites 80.6%

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

   if 0.999949992 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.4%

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 - {u1}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

      6. unpow2 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - {u1}^{2} \cdot \left(\frac{-1}{2} \cdot {u1}^{2} - 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.8%

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - u1 \cdot \left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot -0.5, -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - u1 \cdot \left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot -0.5, -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), 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)) < 0.999949992

   1. Initial program 50.0%

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 80.6

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

   7. Applied rewrites 80.6%

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

   if 0.999949992 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.4%

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 - {u1}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

      6. unpow2 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.2%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 5: 94.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (cos (* u2 (* 2.0 PI)))
  (sqrt
   (-
    (fma (* u1 u1) (fma u1 (fma u1 -0.25 0.3333333333333333) -0.5) u1)
    (*
     (* u1 u1)
     (fma
      u1
      (* u1 (fma (* u1 u1) (fma u1 (* u1 -0.25) -0.3333333333333333) -0.5))
      -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((u2 * (2.0f * ((float) M_PI)))) * sqrtf((fmaf((u1 * u1), fmaf(u1, fmaf(u1, -0.25f, 0.3333333333333333f), -0.5f), u1) - ((u1 * u1) * fmaf(u1, (u1 * fmaf((u1 * u1), fmaf(u1, (u1 * -0.25f), -0.3333333333333333f), -0.5f)), -1.0f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(-0.25), Float32(0.3333333333333333)), Float32(-0.5)), u1) - Float32(Float32(u1 * u1) * fma(u1, Float32(u1 * fma(Float32(u1 * u1), fma(u1, Float32(u1 * Float32(-0.25)), Float32(-0.3333333333333333)), Float32(-0.5))), Float32(-1.0))))))
end

Derivation

1. Initial program 55.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\left(\color{blue}{{u1}^{2} \cdot \left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\color{blue}{{u1}^{2} \cdot \left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot \left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot \left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \color{blue}{\left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \left(\color{blue}{\left(u1 \cdot u1\right)} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \left(\color{blue}{u1 \cdot \left(u1 \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot \left(u1 \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right)\right) + \color{blue}{-1}\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right), -1\right)} - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

7. Applied rewrites 97.3%

   \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \color{blue}{u1 \cdot \left(1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - u1 \cdot \color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}\right) + 1\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \color{blue}{\left(u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}\right)\right) + u1 \cdot 1\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \left(\color{blue}{\left(u1 \cdot u1\right) \cdot \left(u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}\right)} + u1 \cdot 1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \left(\color{blue}{{u1}^{2}} \cdot \left(u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}\right) + u1 \cdot 1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. *-rgt-identity N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \left({u1}^{2} \cdot \left(u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}\right) + \color{blue}{u1}\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \color{blue}{\mathsf{fma}\left({u1}^{2}, u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}, u1\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \mathsf{fma}\left(\color{blue}{u1 \cdot u1}, u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}, u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \mathsf{fma}\left(\color{blue}{u1 \cdot u1}, u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) - \frac{1}{2}, u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \mathsf{fma}\left(u1 \cdot u1, \color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \mathsf{fma}\left(u1 \cdot u1, u1 \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot u1\right) + \color{blue}{\frac{-1}{2}}, u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \mathsf{fma}\left(u1 \cdot u1, \color{blue}{\mathsf{fma}\left(u1, \frac{1}{3} + \frac{-1}{4} \cdot u1, \frac{-1}{2}\right)}, u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. +-commutative N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{4} \cdot u1 + \frac{1}{3}}, \frac{-1}{2}\right), u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   13. *-commutative N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) - \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{4}} + \frac{1}{3}, \frac{-1}{2}\right), u1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   14. lower-fma.f32 95.2

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

10. Applied rewrites 95.2%

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

11. Final simplification 95.2%

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

Alternative 6: 91.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0006699999794363976))
		tmp = sqrt(Float32(log1p(u1) - Float32(u1 * Float32(u1 * fma(u1, Float32(u1 * Float32(-0.5)), Float32(-1.0))))));
	else
		tmp = Float32(cos(t_0) * sqrt(fma(u1, Float32(u1 * Float32(0.5)), u1)));
	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 57.4%

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 - {u1}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

      6. unpow2 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - {u1}^{2} \cdot \left(\frac{-1}{2} \cdot {u1}^{2} - 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - u1 \cdot \left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot -0.5, -1\right)\right)} \]

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

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 93.9%

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

Alternative 7: 93.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (cos (* u2 (* 2.0 PI)))
  (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((u2 * (2.0f * ((float) M_PI)))) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)))
end

Derivation

1. Initial program 55.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) + u1 \cdot 1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot u1\right) \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)} + u1 \cdot 1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\color{blue}{{u1}^{2}} \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + u1 \cdot 1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. *-rgt-identity N/A

      \[\leadsto \sqrt{{u1}^{2} \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + \color{blue}{u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{\mathsf{fma}\left(u1, \frac{1}{3} + \frac{1}{4} \cdot u1, \frac{1}{2}\right)}, u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. +-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, \frac{1}{2}\right), u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   13. lower-fma.f32 95.1

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

5. Applied rewrites 95.1%

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

6. Final simplification 95.1%

   \[\leadsto \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \]
7. Add Preprocessing

Alternative 8: 91.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (cos (* u2 (* 2.0 PI)))
  (sqrt (fma (* u1 u1) (fma u1 0.3333333333333333 0.5) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((u2 * (2.0f * ((float) M_PI)))) * sqrtf(fmaf((u1 * u1), fmaf(u1, 0.3333333333333333f, 0.5f), u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(fma(Float32(u1 * u1), fma(u1, Float32(0.3333333333333333), Float32(0.5)), u1)))
end

Derivation

1. Initial program 55.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) + 1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) + u1 \cdot 1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot u1\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)} + u1 \cdot 1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{2} + \frac{1}{3} \cdot u1, u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + \frac{1}{3} \cdot u1, u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + \frac{1}{3} \cdot u1, u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{\frac{1}{3} \cdot u1 + \frac{1}{2}}, u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{u1 \cdot \frac{1}{3}} + \frac{1}{2}, u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-fma.f32 93.4

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{\mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right)}, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Applied rewrites 93.4%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Final simplification 93.4%

   \[\leadsto \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)} \]
7. Add Preprocessing

Alternative 9: 76.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 - {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-log1p.f32 N/A

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

   4. sub-neg N/A

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

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. lower-neg.f32 82.8

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

7. Applied rewrites 82.8%

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

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 80.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \]
11. Add Preprocessing

Alternative 10: 75.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 - {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-log1p.f32 N/A

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

   4. sub-neg N/A

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

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. lower-neg.f32 82.8

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

7. Applied rewrites 82.8%

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

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 78.8%

    \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)} \]
11. Add Preprocessing

Alternative 11: 72.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 - {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-log1p.f32 N/A

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

   4. sub-neg N/A

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

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. lower-neg.f32 82.8

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

7. Applied rewrites 82.8%

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

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 76.0%

    \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)} \]
11. Add Preprocessing

Alternative 12: 64.6% accurate, 21.0× 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 55.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 - {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-log1p.f32 N/A

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

   4. sub-neg N/A

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

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. lower-neg.f32 82.8

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

7. Applied rewrites 82.8%

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

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
9. Step-by-step derivation

10. Applied rewrites 67.9%

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

Reproduce

herbie shell --seed 2024219 
(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))))