Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.1% → 99.2%

Time: 7.0s

Alternatives: 24

Speedup: 18.5×

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

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

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)))) (cos (* (* 2.0 PI) u2))))

C

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

Julia

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

MATLAB

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

Initial Program: 58.1% 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

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

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)))) (cos (* (* 2.0 PI) u2))))

C

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

Julia

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

MATLAB

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

Alternative 1: 99.2% 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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right) \]

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 (* (fma -2.0 u2 0.5) PI))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((fmaf(-2.0f, u2, 0.5f) * ((float) M_PI)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(fma(Float32(-2.0), u2, Float32(0.5)) * Float32(pi))))
end

Derivation

1. Initial program 58.1%

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

3. Applied rewrites 99.1%

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

5. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right) \]
6. Add Preprocessing

Alternative 2: 99.1% 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

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

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)))) (cos (* 6.2831854820251465 u2))))

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

4. Applied rewrites 99.1%

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

Alternative 3: 96.8% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \cos \left(u2 \cdot 6.2831854820251465\right)\\ \end{array} \]

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)))
  (if (<= (* (* 2.0 PI) u2) 0.05000000074505806)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (*
   (fma (* (sqrt u1) u1) 0.25 (sqrt u1))
   (cos (* u2 6.2831854820251465)))))

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((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1)) * cosf((u2 * 6.2831854820251465f));
	}
	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(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1)) * cos(Float32(u2 * Float32(6.2831854820251465))));
	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 58.1%

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

   2. Evaluated real constant 58.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

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

   1. Initial program 58.1%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.3%

      \[\leadsto \left(\sqrt{u1} + 0.25 \cdot \frac{{u1}^{2}}{\sqrt{u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Applied rewrites 88.3%

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

   7. Evaluated real constant 88.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \cos \left(u2 \cdot 6.2831854820251465\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.7% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 0.5 \cdot u1, u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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 (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt (fma u1 (* 0.5 u1) u1)) (cos (* 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((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(fmaf(u1, (0.5f * u1), u1)) * cosf((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(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(fma(u1, Float32(Float32(0.5) * u1), u1)) * cos(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 58.1%

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

   2. Evaluated real constant 58.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

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

   1. Initial program 58.1%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.0%

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

   6. Applied rewrites 88.1%

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

   7. Evaluated real constant 88.1%

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

Alternative 5: 94.7% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11500000208616257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 1.5707963705062866\right)\right)\\ \end{array} \]

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) 0.11500000208616257)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt u1) (sin (fma (* -2.0 u2) PI 1.5707963705062866)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11500000208616257f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf(fmaf((-2.0f * u2), ((float) M_PI), 1.5707963705062866f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.11500000208616257))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(fma(Float32(Float32(-2.0) * u2), Float32(pi), Float32(1.5707963705062866))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.115000002

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.115000002 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   3. Applied rewrites 58.1%

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

   4. Evaluated real constant 58.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 1.5707963705062866\right)\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 1.5707963705062866\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 76.4%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 1.5707963705062866\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.7% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11500000208616257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right)\\ \end{array} \]

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) 0.11500000208616257)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt u1) (sin (* (fma -2.0 u2 0.5) PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11500000208616257f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf((fmaf(-2.0f, u2, 0.5f) * ((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(0.11500000208616257))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(fma(Float32(-2.0), u2, Float32(0.5)) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.115000002

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.115000002 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.3%

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

   6. Applied rewrites 76.4%

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

Alternative 7: 94.6% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11500000208616257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(-2, u2 \cdot \pi, 1.5707963705062866\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

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) 0.11500000208616257)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sin (fma -2.0 (* u2 PI) 1.5707963705062866)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11500000208616257f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sinf(fmaf(-2.0f, (u2 * ((float) M_PI)), 1.5707963705062866f)) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.11500000208616257))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sin(fma(Float32(-2.0), Float32(u2 * Float32(pi)), Float32(1.5707963705062866))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.115000002

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.115000002 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   3. Applied rewrites 58.1%

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

   4. Taylor expanded in u1 around 0

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

   6. Applied rewrites 76.3%

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

   7. Evaluated real constant 76.3%

      \[\leadsto \sin \left(\mathsf{fma}\left(-2, u2 \cdot \pi, 1.5707963705062866\right)\right) \cdot \sqrt{u1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.6% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11500000208616257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.11500000208616257)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt u1) (cos (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11500000208616257f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf((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(0.11500000208616257))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(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) < 0.115000002

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.115000002 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 76.3%

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

Alternative 9: 88.3% accurate, 1.4× 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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

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))))
 (+ 1.0 (* -19.739209900765786 (pow u2 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (1.0f + (-19.739209900765786f * powf(u2, 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(1.0) + Float32(Float32(-19.739209900765786) * (u2 ^ Float32(2.0)))))
end

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
6. Step-by-step derivation

7. Applied rewrites 88.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
8. Add Preprocessing

Alternative 10: 88.3% accurate, 2.1× 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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

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 (* -19.739209900765786 u2) u2 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)))
end

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
6. Step-by-step derivation

7. Applied rewrites 88.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
8. Step-by-step derivation

9. Applied rewrites 88.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]
10. Add Preprocessing

Alternative 11: 86.8% 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \end{array} \]

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)
  (*
   (sqrt (- (log (- 1.0 u1))))
   (+ 1.0 (* -19.739209900765786 (* u2 u2))))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9975000023841858f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (1.0f + (-19.739209900765786f * (u2 * u2)));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9975000023841858))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(1.0) + Float32(Float32(-19.739209900765786) * Float32(u2 * u2))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot \left(u2 \cdot u2\right)\right) \]

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

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 86.8% 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

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot t\_0\\ \end{array} \]

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 (fma (* -19.739209900765786 u2) u2 1.0))
       (t_1 (- (log (- 1.0 u1)))))
  (if (<= t_1 0.0024999999441206455)
    (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) t_0)
    (* (sqrt t_1) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf((-19.739209900765786f * u2), u2, 1.0f);
	float t_1 = -logf((1.0f - u1));
	float tmp;
	if (t_1 <= 0.0024999999441206455f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * t_0;
	} else {
		tmp = sqrtf(t_1) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0))
	t_1 = Float32(-log(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.0024999999441206455))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * 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))) < 0.00249999994

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

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

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 86.8% accurate, 1.9× 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \end{array} \]

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)
  (*
   (sqrt (- (log (- 1.0 u1))))
   (fma -19.739209900765786 (* u2 u2) 1.0))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9975000023841858f) {
		tmp = sqrtf(-logf((1.0f - u1))) * fmaf(-19.739209900765786f, (u2 * u2), 1.0f);
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9975000023841858))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * fma(Float32(-19.739209900765786), Float32(u2 * u2), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \]

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

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 85.5% accurate, 1.9× 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9989500045776367:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \end{array} \]

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) 0.9989500045776367)
  (sqrt (- (log1p (- u1))))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9989500045776367f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9989500045776367))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.998950005

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   6. Applied rewrites 79.7%

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

   if 0.998950005 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 85.5% accurate, 2.0× 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9989500045776367:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \end{array} \]

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) 0.9989500045776367)
  (sqrt (- (log1p (- u1))))
  (*
   (fma -19.739209900765786 (* u2 u2) 1.0)
   (sqrt (* (fma 0.5 u1 1.0) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9989500045776367f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = fmaf(-19.739209900765786f, (u2 * u2), 1.0f) * sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9989500045776367))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(fma(Float32(-19.739209900765786), Float32(u2 * u2), Float32(1.0)) * sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.998950005

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   6. Applied rewrites 79.7%

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

   if 0.998950005 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 83.2% accurate, 0.7× 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

\[\begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.0007324200123548508:\\ \;\;\;\;\mathsf{fma}\left(-19.739209900765786 \cdot \left(u2 \cdot u2\right), \sqrt{u1}, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

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 (<=
     (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
     0.0007324200123548508)
  (fma (* -19.739209900765786 (* u2 u2)) (sqrt u1) (sqrt u1))
  (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.0007324200123548508f) {
		tmp = fmaf((-19.739209900765786f * (u2 * u2)), sqrtf(u1), sqrtf(u1));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.0007324200123548508))
		tmp = fma(Float32(Float32(-19.739209900765786) * Float32(u2 * u2)), sqrt(u1), sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 7.32420012e-4

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{-173627926472025}{8796093022208} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{u1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto \sqrt{u1} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{u1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.3%

       \[\leadsto \mathsf{fma}\left(-19.739209900765786 \cdot \left(u2 \cdot u2\right), \sqrt{u1}, \sqrt{u1}\right) \]

   if 7.32420012e-4 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   6. Applied rewrites 79.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 83.2% accurate, 0.7× 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

\[\begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.0007324200123548508:\\ \;\;\;\;\mathsf{fma}\left(-19.739209900765786, \left(u2 \cdot u2\right) \cdot \sqrt{u1}, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

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 (<=
     (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
     0.0007324200123548508)
  (fma -19.739209900765786 (* (* u2 u2) (sqrt u1)) (sqrt u1))
  (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.0007324200123548508f) {
		tmp = fmaf(-19.739209900765786f, ((u2 * u2) * sqrtf(u1)), sqrtf(u1));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.0007324200123548508))
		tmp = fma(Float32(-19.739209900765786), Float32(Float32(u2 * u2) * sqrt(u1)), sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 7.32420012e-4

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{-173627926472025}{8796093022208} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(-19.739209900765786, \left(u2 \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}, \sqrt{-\log \left(1 - u1\right)}\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{fma}\left(-19.739209900765786, \left(u2 \cdot u2\right) \cdot \sqrt{u1}, \sqrt{u1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.3%

       \[\leadsto \mathsf{fma}\left(-19.739209900765786, \left(u2 \cdot u2\right) \cdot \sqrt{u1}, \sqrt{u1}\right) \]

   if 7.32420012e-4 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   6. Applied rewrites 79.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 83.2% 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

\[\begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.0007324200123548508:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

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 (<=
     (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
     0.0007324200123548508)
  (* (sqrt u1) (fma (* -19.739209900765786 u2) u2 1.0))
  (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.0007324200123548508f) {
		tmp = sqrtf(u1) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.0007324200123548508))
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 7.32420012e-4

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.3%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 7.32420012e-4 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   6. Applied rewrites 79.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 82.1% accurate, 0.4× 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

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_1 \leq 0.0007324200123548508:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

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 (sqrt (- (log (- 1.0 u1)))))
       (t_1 (* t_0 (cos (* (* 2.0 PI) u2)))))
  (if (<= t_1 0.0007324200123548508)
    (* (sqrt u1) (fma (* -19.739209900765786 u2) u2 1.0))
    (if (<= t_1 0.05000000074505806)
      (fma (* (sqrt u1) u1) 0.25 (sqrt u1))
      t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float t_1 = t_0 * cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_1 <= 0.0007324200123548508f) {
		tmp = sqrtf(u1) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else if (t_1 <= 0.05000000074505806f) {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_1 = Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.0007324200123548508))
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	elseif (t_1 <= Float32(0.05000000074505806))
		tmp = fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 7.32420012e-4

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.3%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 7.32420012e-4 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} + \frac{1}{4} \cdot \frac{{u1}^{2}}{\sqrt{u1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.4%

      \[\leadsto \sqrt{u1} + 0.25 \cdot \frac{{u1}^{2}}{\sqrt{u1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.4%

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

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

Alternative 20: 82.1% accurate, 0.4× 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

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_1 \leq 0.0007324200123548508:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

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 (sqrt (- (log (- 1.0 u1)))))
       (t_1 (* t_0 (cos (* (* 2.0 PI) u2)))))
  (if (<= t_1 0.0007324200123548508)
    (* (sqrt u1) (fma -19.739209900765786 (* u2 u2) 1.0))
    (if (<= t_1 0.05000000074505806)
      (fma (* (sqrt u1) u1) 0.25 (sqrt u1))
      t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float t_1 = t_0 * cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_1 <= 0.0007324200123548508f) {
		tmp = sqrtf(u1) * fmaf(-19.739209900765786f, (u2 * u2), 1.0f);
	} else if (t_1 <= 0.05000000074505806f) {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_1 = Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.0007324200123548508))
		tmp = Float32(sqrt(u1) * fma(Float32(-19.739209900765786), Float32(u2 * u2), Float32(1.0)));
	elseif (t_1 <= Float32(0.05000000074505806))
		tmp = fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 7.32420012e-4

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.3%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \]

   if 7.32420012e-4 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} + \frac{1}{4} \cdot \frac{{u1}^{2}}{\sqrt{u1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.4%

      \[\leadsto \sqrt{u1} + 0.25 \cdot \frac{{u1}^{2}}{\sqrt{u1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.4%

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

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

Alternative 21: 78.7% 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

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 0.5 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

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 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.05000000074505806)
    (sqrt (fma u1 (* 0.5 u1) u1))
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.05000000074505806f) {
		tmp = sqrtf(fmaf(u1, (0.5f * u1), u1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.05000000074505806))
		tmp = sqrt(fma(u1, Float32(Float32(0.5) * u1), u1));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 72.3%

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

   9. Applied rewrites 72.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(1, u1, \left(0.5 \cdot u1\right) \cdot u1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.3%

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

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 49.8%

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

Alternative 22: 72.3% accurate, 5.1× 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

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

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)))
  (sqrt (fma u1 (* 0.5 u1) u1)))

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 49.8%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 72.3%

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

9. Applied rewrites 72.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(1, u1, \left(0.5 \cdot u1\right) \cdot u1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 72.3%

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

Alternative 23: 72.3% accurate, 5.1× 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

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

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)))
  (sqrt (* (fma 0.5 u1 1.0) u1)))

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 49.8%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 72.3%

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

9. Applied rewrites 72.3%

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

Alternative 24: 64.3% accurate, 18.5× 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

\[\sqrt{u1} \]

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)))
  (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 49.8%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 64.3%

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

Reproduce

herbie shell --seed 2026086 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))