Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.4% → 98.4%

Time: 8.9s

Alternatives: 29

Speedup: 6.2×

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

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :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))))

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: 57.4% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :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))))

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: 98.4% accurate, 0.8× speedup

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

Math

\[\sqrt{-\frac{1}{\frac{2}{2 \cdot \mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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 (- (/ 1.0 (/ 2.0 (* 2.0 (log1p (- u1)))))))
 (sin (* 6.2831854820251465 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(1.0f / (2.0f / (2.0f * log1pf(-u1))))) * sinf((6.2831854820251465f * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(Float32(1.0) / Float32(Float32(2.0) / Float32(Float32(2.0) * log1p(Float32(-u1))))))) * sin(Float32(Float32(6.2831854820251465) * u2)))
end

Derivation

1. Initial program 57.4%

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

   1. lift-log.f32 N/A

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

   2. pow1 N/A

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

   3. log-pow N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

   6. div-flip N/A

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

   7. remove-sound-/ N/A

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

   8. lower-/.f32 N/A

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

   9. remove-sound-/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-lft-identity N/A

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

   13. log-pow N/A

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

   14. pow1 N/A

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

   15. lift-log.f32 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 98.3%

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

6. Evaluated real constant 98.3%

   \[\leadsto \sqrt{-\frac{1}{\frac{2}{2 \cdot \mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
7. Add Preprocessing

Alternative 2: 98.3% accurate, 0.9× speedup

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

Math

\[\frac{\sin \left(u2 \cdot 6.2831854820251465\right)}{\frac{1}{\sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|}}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (/
 (sin (* u2 6.2831854820251465))
 (/ 1.0 (sqrt (fabs (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.2831854820251465f)) / (1.0f / sqrtf(fabsf(log1pf(-u1))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(6.2831854820251465))) / Float32(Float32(1.0) / sqrt(abs(log1p(Float32(-u1))))))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 57.4%

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

6. Evaluated real constant 57.4%

   \[\leadsto \frac{\sin \left(u2 \cdot 6.2831854820251465\right)}{\frac{1}{\sqrt{\left|\log \left(1 - u1\right)\right|}}} \]
7. Step-by-step derivation

8. Applied rewrites 98.2%

   \[\leadsto \frac{\sin \left(u2 \cdot 6.2831854820251465\right)}{\frac{1}{\sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|}}} \]
9. Add Preprocessing

Alternative 3: 98.2% accurate, 1.0× speedup

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

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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 (- (log1p (- u1)))) (sin (* 6.2831854820251465 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((6.2831854820251465f * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(6.2831854820251465) * u2)))
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Step-by-step derivation

4. Applied rewrites 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.002199999988079071:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \sin \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (- (log (- 1.0 u1)))))
  (if (<= t_0 0.002199999988079071)
    (*
     (fma (* (sqrt u1) u1) 0.25 (sqrt u1))
     (sin (* u2 6.2831854820251465)))
    (* (sqrt t_0) (sin (* 6.2831854820251465 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = -logf((1.0f - u1));
	float tmp;
	if (t_0 <= 0.002199999988079071f) {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1)) * sinf((u2 * 6.2831854820251465f));
	} else {
		tmp = sqrtf(t_0) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(-log(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.002199999988079071))
		tmp = Float32(fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1)) * sin(Float32(u2 * Float32(6.2831854820251465))));
	else
		tmp = Float32(sqrt(t_0) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.00219999999

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.2%

      \[\leadsto \left(\sqrt{u1} + 0.25 \cdot \frac{{u1}^{2}}{\sqrt{u1}}\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 88.2%

      \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \sin \left(u2 \cdot \left(\pi + \pi\right)\right) \]

   7. Evaluated real constant 88.2%

      \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \sin \left(u2 \cdot 6.2831854820251465\right) \]

   if 0.00219999999 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.0% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \sin \left(6.2831854820251465 \cdot u2\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (sin (* 6.2831854820251465 u2)))
       (t_1 (- (log (- 1.0 u1)))))
  (if (<= t_1 0.002199999988079071)
    (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_0)
    (* (sqrt t_1) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((6.2831854820251465f * u2));
	float t_1 = -logf((1.0f - u1));
	float tmp;
	if (t_1 <= 0.002199999988079071f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = sqrtf(t_1) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(6.2831854820251465) * u2))
	t_1 = Float32(-log(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.002199999988079071))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(sqrt(t_1) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.00219999999

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 88.0%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   if 0.00219999999 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.3% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.05000000074505806)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    u2
    (+ PI PI)
    (* (* u2 (* (* u2 u2) -1.3333333333333333)) 31.006277084350586)))
  (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.05000000074505806f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (((float) M_PI) + ((float) M_PI)), ((u2 * ((u2 * u2) * -1.3333333333333333f)) * 31.006277084350586f));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(u2 * Float32(Float32(u2 * u2) * Float32(-1.3333333333333333))) * Float32(31.006277084350586))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0500000007

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \]

   7. Evaluated real constant 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 89.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]

   if 0.0500000007 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 88.0%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.5% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.18000000715255737:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u1}{\sqrt{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.18000000715255737)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    u2
    (+ PI PI)
    (* (* u2 (* (* u2 u2) -1.3333333333333333)) 31.006277084350586)))
  (* (/ u1 (sqrt u1)) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.18000000715255737f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (((float) M_PI) + ((float) M_PI)), ((u2 * ((u2 * u2) * -1.3333333333333333f)) * 31.006277084350586f));
	} else {
		tmp = (u1 / sqrtf(u1)) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.18000000715255737))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(u2 * Float32(Float32(u2 * u2) * Float32(-1.3333333333333333))) * Float32(31.006277084350586))));
	else
		tmp = Float32(Float32(u1 / sqrt(u1)) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.180000007

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \]

   7. Evaluated real constant 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 89.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]

   if 0.180000007 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 76.8%

      \[\leadsto \sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f32 N/A

         \[\leadsto \frac{u1}{\sqrt{u1}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      2. pow1/2 N/A

         \[\leadsto \frac{u1}{{u1}^{\frac{1}{2}}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{u1}{{u1}^{\left(1 - \frac{1}{2}\right)}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      4. pow-div N/A

         \[\leadsto \frac{u1}{\frac{{u1}^{1}}{{u1}^{\frac{1}{2}}}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      5. pow1 N/A

         \[\leadsto \frac{u1}{\frac{u1}{{u1}^{\frac{1}{2}}}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      6. pow1/2 N/A

         \[\leadsto \frac{u1}{\frac{u1}{\sqrt{u1}}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      7. lift-sqrt.f32 N/A

         \[\leadsto \frac{u1}{\frac{u1}{\sqrt{u1}}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      8. lift-/.f32 76.8%

         \[\leadsto \frac{u1}{\frac{u1}{\sqrt{u1}}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   7. Applied rewrites 76.7%

      \[\leadsto \frac{u1}{\sqrt{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.5% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.18000000715255737:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.18000000715255737)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    u2
    (+ PI PI)
    (* (* u2 (* (* u2 u2) -1.3333333333333333)) 31.006277084350586)))
  (* (sqrt u1) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.18000000715255737f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (((float) M_PI) + ((float) M_PI)), ((u2 * ((u2 * u2) * -1.3333333333333333f)) * 31.006277084350586f));
	} else {
		tmp = sqrtf(u1) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.18000000715255737))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(u2 * Float32(Float32(u2 * u2) * Float32(-1.3333333333333333))) * Float32(31.006277084350586))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.180000007

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \]

   7. Evaluated real constant 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 89.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]

   if 0.180000007 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 76.8%

      \[\leadsto \sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.2% accurate, 1.5× speedup

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

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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 (- (log1p (- u1))))
 (fma
  u2
  (+ PI PI)
  (* (* u2 (* (* u2 u2) -1.3333333333333333)) 31.006277084350586))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * fmaf(u2, (((float) M_PI) + ((float) M_PI)), ((u2 * ((u2 * u2) * -1.3333333333333333f)) * 31.006277084350586f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(u2 * Float32(Float32(u2 * u2) * Float32(-1.3333333333333333))) * Float32(31.006277084350586))))
end

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
3. Step-by-step derivation

4. Applied rewrites 53.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 53.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \]

7. Evaluated real constant 53.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]
8. Step-by-step derivation

9. Applied rewrites 89.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]
10. Add Preprocessing

Alternative 10: 89.1% accurate, 1.7× speedup

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

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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 (- (log1p (- u1))))
 (* u2 (fma -41.34170150756836 (* u2 u2) (+ PI PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (u2 * fmaf(-41.34170150756836f, (u2 * u2), (((float) M_PI) + ((float) M_PI))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-41.34170150756836), Float32(u2 * u2), Float32(Float32(pi) + Float32(pi)))))
end

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
3. Step-by-step derivation

4. Applied rewrites 53.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 53.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), u2 \cdot u2, \pi + \pi\right)\right) \]

7. Evaluated real constant 53.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 89.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right) \]
10. Add Preprocessing

Alternative 11: 88.0% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9976000189781189:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi\right) + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (- 1.0 u1) 0.9976000189781189)
  (*
   (sqrt (- (log (- 1.0 u1))))
   (* u2 (+ (fma (* -41.34170150756836 u2) u2 PI) PI)))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (* u2 (fma (* -41.34170150756836 u2) u2 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9976000189781189f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (u2 * (fmaf((-41.34170150756836f * u2), u2, ((float) M_PI)) + ((float) M_PI)));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (u2 * fmaf((-41.34170150756836f * u2), u2, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9976000189781189))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(u2 * Float32(fma(Float32(Float32(-41.34170150756836) * u2), u2, Float32(pi)) + Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(u2 * fma(Float32(Float32(-41.34170150756836) * u2), u2, Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997600019

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), u2 \cdot u2, \pi + \pi\right)\right) \]

   7. Evaluated real constant 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi\right) + \pi\right)\right) \]

   if 0.997600019 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.0%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 80.2%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 80.2%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right) \cdot u2, u2, \pi + \pi\right)\right) \]

   10. Evaluated real constant 80.2%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.0% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9976000189781189:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, 6.2831854820251465\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (- 1.0 u1) 0.9976000189781189)
  (*
   (sqrt (- (log (- 1.0 u1))))
   (* u2 (fma -41.34170150756836 (* u2 u2) 6.2831854820251465)))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (* u2 (fma (* -41.34170150756836 u2) u2 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9976000189781189f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (u2 * fmaf(-41.34170150756836f, (u2 * u2), 6.2831854820251465f));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (u2 * fmaf((-41.34170150756836f * u2), u2, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9976000189781189))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(u2 * fma(Float32(-41.34170150756836), Float32(u2 * u2), Float32(6.2831854820251465))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(u2 * fma(Float32(Float32(-41.34170150756836) * u2), u2, Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997600019

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), u2 \cdot u2, \pi + \pi\right)\right) \]

   7. Evaluated real constant 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right) \]

   8. Evaluated real constant 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, 6.2831854820251465\right)\right) \]

   if 0.997600019 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.0%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 80.2%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 80.2%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right) \cdot u2, u2, \pi + \pi\right)\right) \]

   10. Evaluated real constant 80.2%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 84.6% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.007199999876320362:\\ \;\;\;\;\sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right) \cdot 9.869604110717773, \pi, \pi\right) + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.007199999876320362)
  (* (sqrt (fabs (log1p (- u1)))) (* u2 6.2831854820251465))
  (*
   (sqrt u1)
   (*
    u2
    (+
     (fma
      (* (* (* u2 u2) -1.3333333333333333) 9.869604110717773)
      PI
      PI)
     PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.007199999876320362f) {
		tmp = sqrtf(fabsf(log1pf(-u1))) * (u2 * 6.2831854820251465f);
	} else {
		tmp = sqrtf(u1) * (u2 * (fmaf((((u2 * u2) * -1.3333333333333333f) * 9.869604110717773f), ((float) M_PI), ((float) M_PI)) + ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.007199999876320362))
		tmp = Float32(sqrt(abs(log1p(Float32(-u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(fma(Float32(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)) * Float32(9.869604110717773)), Float32(pi), Float32(pi)) + Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00719999988

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]

   9. Evaluated real constant 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right) \]

   if 0.00719999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right) \cdot \left(\pi \cdot \pi\right), \pi, \pi\right) + \pi\right)\right) \]

   7. Evaluated real constant 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right) \cdot 9.869604110717773, \pi, \pi\right) + \pi\right)\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right) \cdot 9.869604110717773, \pi, \pi\right) + \pi\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right) \cdot 9.869604110717773, \pi, \pi\right) + \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 84.6% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.007199999876320362:\\ \;\;\;\;\sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, 9.869604110717773, 1\right), \pi, \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.007199999876320362)
  (* (sqrt (fabs (log1p (- u1)))) (* u2 6.2831854820251465))
  (*
   (sqrt u1)
   (*
    u2
    (fma
     (fma (* (* u2 u2) -1.3333333333333333) 9.869604110717773 1.0)
     PI
     PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.007199999876320362f) {
		tmp = sqrtf(fabsf(log1pf(-u1))) * (u2 * 6.2831854820251465f);
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf(fmaf(((u2 * u2) * -1.3333333333333333f), 9.869604110717773f, 1.0f), ((float) M_PI), ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.007199999876320362))
		tmp = Float32(sqrt(abs(log1p(Float32(-u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(9.869604110717773), Float32(1.0)), Float32(pi), Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00719999988

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]

   9. Evaluated real constant 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right) \]

   if 0.00719999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 1\right), \pi, \pi\right)\right) \]

   10. Evaluated real constant 70.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, 9.869604110717773, 1\right), \pi, \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 84.6% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.007199999876320362:\\ \;\;\;\;\sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(2, u2 \cdot \pi, \left(-41.34170150756836 \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.007199999876320362)
  (* (sqrt (fabs (log1p (- u1)))) (* u2 6.2831854820251465))
  (*
   (sqrt u1)
   (fma 2.0 (* u2 PI) (* (* -41.34170150756836 (* u2 u2)) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.007199999876320362f) {
		tmp = sqrtf(fabsf(log1pf(-u1))) * (u2 * 6.2831854820251465f);
	} else {
		tmp = sqrtf(u1) * fmaf(2.0f, (u2 * ((float) M_PI)), ((-41.34170150756836f * (u2 * u2)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.007199999876320362))
		tmp = Float32(sqrt(abs(log1p(Float32(-u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	else
		tmp = Float32(sqrt(u1) * fma(Float32(2.0), Float32(u2 * Float32(pi)), Float32(Float32(Float32(-41.34170150756836) * Float32(u2 * u2)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00719999988

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]

   9. Evaluated real constant 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right) \]

   if 0.00719999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]

   9. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(2, u2 \cdot \pi, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right) \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right) \]

   10. Evaluated real constant 70.8%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(2, u2 \cdot \pi, \left(-41.34170150756836 \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 84.6% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.007199999876320362:\\ \;\;\;\;\sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.007199999876320362)
  (* (sqrt (fabs (log1p (- u1)))) (* u2 6.2831854820251465))
  (* (sqrt u1) (* u2 (fma (* -41.34170150756836 u2) u2 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.007199999876320362f) {
		tmp = sqrtf(fabsf(log1pf(-u1))) * (u2 * 6.2831854820251465f);
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((-41.34170150756836f * u2), u2, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.007199999876320362))
		tmp = Float32(sqrt(abs(log1p(Float32(-u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(Float32(-41.34170150756836) * u2), u2, Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00719999988

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]

   9. Evaluated real constant 81.3%

      \[\leadsto \sqrt{\left|\mathsf{log1p}\left(-u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right) \]

   if 0.00719999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right) \cdot u2, u2, \pi + \pi\right)\right) \]

   10. Evaluated real constant 70.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 84.6% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.007199999876320362:\\ \;\;\;\;u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.007199999876320362)
  (* u2 (* 6.2831854820251465 (sqrt (- (log1p (- u1))))))
  (* (sqrt u1) (* u2 (fma (* -41.34170150756836 u2) u2 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.007199999876320362f) {
		tmp = u2 * (6.2831854820251465f * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((-41.34170150756836f * u2), u2, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.007199999876320362))
		tmp = Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(Float32(-41.34170150756836) * u2), u2, Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00719999988

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + {u2}^{2} \cdot \left(\frac{-762619864465648886625}{18446744073709551616} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 54.4%

      \[\leadsto u2 \cdot \mathsf{fma}\left(6.2831854820251465, \sqrt{-\log \left(1 - u1\right)}, {u2}^{2} \cdot \mathsf{fma}\left(-41.341705691712875, \sqrt{-\log \left(1 - u1\right)}, 81.6052606304404 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\right) \]

   6. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 81.3%

       \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \]

   if 0.00719999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right) \cdot u2, u2, \pi + \pi\right)\right) \]

   10. Evaluated real constant 70.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 84.6% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.007199999876320362:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (* (* 2.0 PI) u2) 0.007199999876320362)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (* (sqrt u1) (* u2 (fma (* -41.34170150756836 u2) u2 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.007199999876320362f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((-41.34170150756836f * u2), u2, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.007199999876320362))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(Float32(-41.34170150756836) * u2), u2, Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00719999988

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 50.4%

      \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 81.3%

      \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \]

   if 0.00719999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right) \cdot u2, u2, \pi + \pi\right)\right) \]

   10. Evaluated real constant 70.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 81.0% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;-t\_0 \leq 0.000375000003259629:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t\_0\right|} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (log (- 1.0 u1))))
  (if (<= (- t_0) 0.000375000003259629)
    (* (sqrt u1) (* u2 (fma (* -41.34170150756836 u2) u2 (+ PI PI))))
    (* (sqrt (fabs t_0)) (* u2 6.2831854820251465)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (-t_0 <= 0.000375000003259629f) {
		tmp = sqrtf(u1) * (u2 * fmaf((-41.34170150756836f * u2), u2, (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = sqrtf(fabsf(t_0)) * (u2 * 6.2831854820251465f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (Float32(-t_0) <= Float32(0.000375000003259629))
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(Float32(-41.34170150756836) * u2), u2, Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = Float32(sqrt(abs(t_0)) * Float32(u2 * Float32(6.2831854820251465)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 3.75000003e-4

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right) \cdot u2, u2, \pi + \pi\right)\right) \]

   10. Evaluated real constant 70.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right) \]

   if 3.75000003e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]

   7. Evaluated real constant 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 80.5% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;-t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t\_0\right|} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (log (- 1.0 u1))))
  (if (<= (- t_0) 0.0024999999441206455)
    (* u2 (* 6.2831854820251465 (sqrt (* u1 (+ 1.0 (* 0.5 u1))))))
    (* (sqrt (fabs t_0)) (* u2 6.2831854820251465)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (-t_0 <= 0.0024999999441206455f) {
		tmp = u2 * (6.2831854820251465f * sqrtf((u1 * (1.0f + (0.5f * u1)))));
	} else {
		tmp = sqrtf(fabsf(t_0)) * (u2 * 6.2831854820251465f);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = log((1.0e0 - u1))
    if (-t_0 <= 0.0024999999441206455e0) then
        tmp = u2 * (6.2831854820251465e0 * sqrt((u1 * (1.0e0 + (0.5e0 * u1)))))
    else
        tmp = sqrt(abs(t_0)) * (u2 * 6.2831854820251465e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (Float32(-t_0) <= Float32(0.0024999999441206455))
		tmp = Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1))))));
	else
		tmp = Float32(sqrt(abs(t_0)) * Float32(u2 * Float32(6.2831854820251465)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (-t_0 <= single(0.0024999999441206455))
		tmp = u2 * (single(6.2831854820251465) * sqrt((u1 * (single(1.0) + (single(0.5) * u1)))));
	else
		tmp = sqrt(abs(t_0)) * (u2 * single(6.2831854820251465));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.00249999994

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + {u2}^{2} \cdot \left(\frac{-762619864465648886625}{18446744073709551616} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 54.4%

      \[\leadsto u2 \cdot \mathsf{fma}\left(6.2831854820251465, \sqrt{-\log \left(1 - u1\right)}, {u2}^{2} \cdot \mathsf{fma}\left(-41.341705691712875, \sqrt{-\log \left(1 - u1\right)}, 81.6052606304404 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\right) \]

   6. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 74.1%

       \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \]

   if 0.00249999994 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot \left(\pi + \pi\right)\right) \]

   7. Evaluated real constant 50.4%

      \[\leadsto \sqrt{\left|\log \left(1 - u1\right)\right|} \cdot \left(u2 \cdot 6.2831854820251465\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 80.5% accurate, 2.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (- 1.0 u1) 0.9975000023841858)
  (* u2 (* 6.2831854820251465 (sqrt (- (log (- 1.0 u1))))))
  (* u2 (* 6.2831854820251465 (sqrt (* u1 (+ 1.0 (* 0.5 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9975000023841858f) {
		tmp = u2 * (6.2831854820251465f * sqrtf(-logf((1.0f - u1))));
	} else {
		tmp = u2 * (6.2831854820251465f * sqrtf((u1 * (1.0f + (0.5f * u1)))));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((1.0e0 - u1) <= 0.9975000023841858e0) then
        tmp = u2 * (6.2831854820251465e0 * sqrt(-log((1.0e0 - u1))))
    else
        tmp = u2 * (6.2831854820251465e0 * sqrt((u1 * (1.0e0 + (0.5e0 * u1)))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9975000023841858))
		tmp = Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	else
		tmp = Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9975000023841858))
		tmp = u2 * (single(6.2831854820251465) * sqrt(-log((single(1.0) - u1))));
	else
		tmp = u2 * (single(6.2831854820251465) * sqrt((u1 * (single(1.0) + (single(0.5) * u1)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997500002

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + {u2}^{2} \cdot \left(\frac{-762619864465648886625}{18446744073709551616} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 54.4%

      \[\leadsto u2 \cdot \mathsf{fma}\left(6.2831854820251465, \sqrt{-\log \left(1 - u1\right)}, {u2}^{2} \cdot \mathsf{fma}\left(-41.341705691712875, \sqrt{-\log \left(1 - u1\right)}, 81.6052606304404 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\right) \]

   6. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   if 0.997500002 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + {u2}^{2} \cdot \left(\frac{-762619864465648886625}{18446744073709551616} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 54.4%

      \[\leadsto u2 \cdot \mathsf{fma}\left(6.2831854820251465, \sqrt{-\log \left(1 - u1\right)}, {u2}^{2} \cdot \mathsf{fma}\left(-41.341705691712875, \sqrt{-\log \left(1 - u1\right)}, 81.6052606304404 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\right) \]

   6. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 74.1%

       \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 80.5% accurate, 2.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (if (<= (- 1.0 u1) 0.9975000023841858)
  (* 6.2831854820251465 (* u2 (sqrt (- (log (- 1.0 u1))))))
  (* u2 (* 6.2831854820251465 (sqrt (* u1 (+ 1.0 (* 0.5 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9975000023841858f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-logf((1.0f - u1))));
	} else {
		tmp = u2 * (6.2831854820251465f * sqrtf((u1 * (1.0f + (0.5f * u1)))));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((1.0e0 - u1) <= 0.9975000023841858e0) then
        tmp = 6.2831854820251465e0 * (u2 * sqrt(-log((1.0e0 - u1))))
    else
        tmp = u2 * (6.2831854820251465e0 * sqrt((u1 * (1.0e0 + (0.5e0 * u1)))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9975000023841858))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	else
		tmp = Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9975000023841858))
		tmp = single(6.2831854820251465) * (u2 * sqrt(-log((single(1.0) - u1))));
	else
		tmp = u2 * (single(6.2831854820251465) * sqrt((u1 * (single(1.0) + (single(0.5) * u1)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997500002

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 50.4%

      \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   if 0.997500002 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + {u2}^{2} \cdot \left(\frac{-762619864465648886625}{18446744073709551616} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 54.4%

      \[\leadsto u2 \cdot \mathsf{fma}\left(6.2831854820251465, \sqrt{-\log \left(1 - u1\right)}, {u2}^{2} \cdot \mathsf{fma}\left(-41.341705691712875, \sqrt{-\log \left(1 - u1\right)}, 81.6052606304404 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\right) \]

   6. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 74.1%

       \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 74.1% accurate, 3.2× speedup

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

Math

\[u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (* u2 (* 6.2831854820251465 (sqrt (* u1 (+ 1.0 (* 0.5 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (6.2831854820251465f * sqrtf((u1 * (1.0f + (0.5f * u1)))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u2 * (6.2831854820251465e0 * sqrt((u1 * (1.0e0 + (0.5e0 * u1)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(6.2831854820251465) * sqrt((u1 * (single(1.0) + (single(0.5) * u1)))));
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

3. Taylor expanded in u2 around 0

   \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + {u2}^{2} \cdot \left(\frac{-762619864465648886625}{18446744073709551616} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 54.4%

   \[\leadsto u2 \cdot \mathsf{fma}\left(6.2831854820251465, \sqrt{-\log \left(1 - u1\right)}, {u2}^{2} \cdot \mathsf{fma}\left(-41.341705691712875, \sqrt{-\log \left(1 - u1\right)}, 81.6052606304404 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\right) \]

6. Taylor expanded in u2 around 0

   \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
7. Step-by-step derivation

8. Applied rewrites 50.4%

   \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

9. Taylor expanded in u1 around 0

   \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}\right) \]
10. Step-by-step derivation

11. Applied rewrites 74.1%

    \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \]
12. Add Preprocessing

Alternative 24: 74.1% accurate, 3.3× speedup

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

Math

\[\left(6.2831854820251465 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (* (* 6.2831854820251465 u2) (sqrt (* (fma 0.5 u1 1.0) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.2831854820251465f * u2) * sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.2831854820251465) * u2) * sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)))
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

3. Taylor expanded in u2 around 0

   \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
4. Step-by-step derivation

5. Applied rewrites 50.4%

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

6. Taylor expanded in u1 around 0

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}\right) \]
7. Step-by-step derivation

8. Applied rewrites 74.1%

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 74.1%

    \[\leadsto \left(6.2831854820251465 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 25: 74.1% accurate, 3.3× speedup

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

Math

\[6.2831854820251465 \cdot \left(\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (* 6.2831854820251465 (* (sqrt (* (fma 0.5 u1 1.0) u1)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.2831854820251465f * (sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.2831854820251465) * Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * u2))
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

3. Taylor expanded in u2 around 0

   \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
4. Step-by-step derivation

5. Applied rewrites 50.4%

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

6. Taylor expanded in u1 around 0

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}\right) \]
7. Step-by-step derivation

8. Applied rewrites 74.1%

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 74.1%

    \[\leadsto 6.2831854820251465 \cdot \left(\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot u2\right) \]
11. Add Preprocessing

Alternative 26: 66.4% accurate, 3.3× speedup

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

Math

\[2 \cdot \frac{\left(u1 \cdot u2\right) \cdot \pi}{\sqrt{\left|u1\right|}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (* 2.0 (/ (* (* u1 u2) PI) (sqrt (fabs u1)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 50.4%

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

6. Applied rewrites 50.4%

   \[\leadsto 2 \cdot \left(u2 \cdot \left(\pi \cdot \frac{1}{\sqrt{\left|\frac{-1}{\log \left(1 - u1\right)}\right|}}\right)\right) \]

7. Taylor expanded in u1 around 0

   \[\leadsto 2 \cdot \frac{u1 \cdot \left(u2 \cdot \pi\right)}{\sqrt{\left|u1\right|}} \]
8. Step-by-step derivation

9. Applied rewrites 66.4%

   \[\leadsto 2 \cdot \frac{u1 \cdot \left(u2 \cdot \pi\right)}{\sqrt{\left|u1\right|}} \]
10. Step-by-step derivation

11. Applied rewrites 66.4%

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

Alternative 27: 66.4% accurate, 6.2× speedup

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

Math

\[\sqrt{u1} \cdot \left(u2 \cdot 6.2831854820251465\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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 u1) (* u2 6.2831854820251465)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * (u2 * 6.2831854820251465f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1) * (u2 * 6.2831854820251465e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * Float32(u2 * Float32(6.2831854820251465)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * (u2 * single(6.2831854820251465));
end

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u1 around 0

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

4. Applied rewrites 76.8%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 70.8%

   \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 66.4%

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

11. Evaluated real constant 66.4%

    \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot 6.2831854820251465\right) \]
12. Add Preprocessing

Alternative 28: 66.4% accurate, 6.2× speedup

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

Math

\[u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1}\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (* u2 (* 6.2831854820251465 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (6.2831854820251465f * sqrtf(u1));
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u2 * (6.2831854820251465e0 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(6.2831854820251465) * sqrt(u1));
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

3. Taylor expanded in u2 around 0

   \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + {u2}^{2} \cdot \left(\frac{-762619864465648886625}{18446744073709551616} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 54.4%

   \[\leadsto u2 \cdot \mathsf{fma}\left(6.2831854820251465, \sqrt{-\log \left(1 - u1\right)}, {u2}^{2} \cdot \mathsf{fma}\left(-41.341705691712875, \sqrt{-\log \left(1 - u1\right)}, 81.6052606304404 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\right) \]

6. Taylor expanded in u2 around 0

   \[\leadsto u2 \cdot \left(\frac{13176795}{2097152} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
7. Step-by-step derivation

8. Applied rewrites 50.4%

   \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

9. Taylor expanded in u1 around 0

   \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1}\right) \]
10. Step-by-step derivation

11. Applied rewrites 66.4%

    \[\leadsto u2 \cdot \left(6.2831854820251465 \cdot \sqrt{u1}\right) \]
12. Add Preprocessing

Alternative 29: 66.4% accurate, 6.2× speedup

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

Math

\[6.2831854820251465 \cdot \left(u2 \cdot \sqrt{u1}\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (* 6.2831854820251465 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.2831854820251465f * (u2 * sqrtf(u1));
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.2831854820251465e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.2831854820251465) * (u2 * sqrt(u1));
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

3. Taylor expanded in u2 around 0

   \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
4. Step-by-step derivation

5. Applied rewrites 50.4%

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

6. Taylor expanded in u1 around 0

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
7. Step-by-step derivation

8. Applied rewrites 66.4%

   \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2026084 
(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))))