Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 98.3%

Time: 8.9s

Alternatives: 19

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: 58.2% 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

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

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: 98.3% 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 58.2%

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

2. Evaluated real constant 58.2%

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

4. Applied rewrites 98.3%

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

Alternative 2: 96.9% 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

\[\begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;\mathsf{fma}\left(u1 \cdot \sqrt{u1}, 0.25, \sqrt{u1}\right) \cdot \sin \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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 0.0024999999441206455)
    (*
     (fma (* u1 (sqrt 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 <= 0.0024999999441206455f) {
		tmp = fmaf((u1 * sqrtf(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(0.0024999999441206455))
		tmp = Float32(fma(Float32(u1 * sqrt(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))) < 0.00249999994

   1. Initial program 58.2%

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

   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 88.0%

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

   6. Applied rewrites 88.0%

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

   7. Evaluated real constant 88.0%

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

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

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

Alternative 3: 96.8% 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

\[\begin{array}{l} t_0 := \sin \left(6.2831854820251465 \cdot u2\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot t\_0\\ \end{array} \]

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 0.0024999999441206455)
    (* (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 <= 0.0024999999441206455f) {
		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(0.0024999999441206455))
		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))) < 0.00249999994

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 87.7%

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

   7. Applied rewrites 87.7%

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

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

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

Alternative 4: 96.1% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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))))
   (* u2 (fma -41.34170150756836 (* u2 u2) (+ PI PI))))
  (* (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)) * (u2 * fmaf(-41.34170150756836f, (u2 * u2), (((float) M_PI) + ((float) M_PI))));
	} 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)))) * Float32(u2 * fma(Float32(-41.34170150756836), Float32(u2 * u2), Float32(Float32(pi) + Float32(pi)))));
	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 58.2%

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

   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 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   9. Applied rewrites 89.1%

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

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

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 87.7%

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

   7. Applied rewrites 87.7%

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

Alternative 5: 94.4% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.25:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u1 \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}{\sqrt{u1}}\\ \end{array} \]

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) 0.25)
  (*
   (sqrt (- (log1p (- u1))))
   (* u2 (fma -41.34170150756836 (* u2 u2) (+ PI PI))))
  (/ (* u1 (sin (* 2.0 (* u2 PI)))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.25f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-41.34170150756836f, (u2 * u2), (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = (u1 * sinf((2.0f * (u2 * ((float) M_PI))))) / 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(0.25))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-41.34170150756836), Float32(u2 * u2), Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = Float32(Float32(u1 * sin(Float32(Float32(2.0) * Float32(u2 * Float32(pi))))) / sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.25

   1. Initial program 58.2%

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

   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 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   9. Applied rewrites 89.1%

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

   if 0.25 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 87.9%

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

   7. Taylor expanded in u2 around 0

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

   9. Applied rewrites 74.0%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 76.2%

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

Alternative 6: 94.4% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.25:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.25)
  (*
   (sqrt (- (log1p (- u1))))
   (* u2 (fma -41.34170150756836 (* u2 u2) (+ PI PI))))
  (* (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) <= 0.25f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-41.34170150756836f, (u2 * u2), (((float) M_PI) + ((float) M_PI))));
	} 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(0.25))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-41.34170150756836), Float32(u2 * u2), Float32(Float32(pi) + Float32(pi)))));
	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) < 0.25

   1. Initial program 58.2%

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

   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 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   9. Applied rewrites 89.1%

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

   if 0.25 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 76.2%

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

Alternative 7: 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 58.2%

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

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 54.3%

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

6. Applied rewrites 54.3%

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

7. Evaluated real constant 54.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi + \pi\right)\right) \]
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 8: 87.9% 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi\right) + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

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) 0.9975000023841858)
  (*
   (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) <= 0.9975000023841858f) {
		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(0.9975000023841858))
		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) < 0.997500002

   1. Initial program 58.2%

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

   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 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   9. Applied rewrites 54.3%

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

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

   1. Initial program 58.2%

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

   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 54.3%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 80.0%

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

   10. Evaluated real constant 80.0%

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

Alternative 9: 87.9% 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, 6.2831854820251465\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

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) 0.9975000023841858)
  (*
   (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) <= 0.9975000023841858f) {
		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(0.9975000023841858))
		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) < 0.997500002

   1. Initial program 58.2%

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

   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 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   8. Evaluated real constant 54.3%

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

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

   1. Initial program 58.2%

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

   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 54.3%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 80.0%

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

   10. Evaluated real constant 80.0%

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

Alternative 10: 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00865000020712614:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right)\\ \end{array} \]

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) 0.00865000020712614)
  (* (sqrt (- (log1p (- u1)))) (* u2 6.2831854820251465))
  (*
   (sqrt u1)
   (fma
    u2
    (+ PI PI)
    (* (* u2 (* (* u2 u2) -1.3333333333333333)) 31.006277084350586)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00865000020712614f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * 6.2831854820251465f);
	} else {
		tmp = sqrtf(u1) * fmaf(u2, (((float) M_PI) + ((float) M_PI)), ((u2 * ((u2 * u2) * -1.3333333333333333f)) * 31.006277084350586f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00865000020712614))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	else
		tmp = Float32(sqrt(u1) * fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(u2 * Float32(Float32(u2 * u2) * Float32(-1.3333333333333333))) * Float32(31.006277084350586))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00865000021

   1. Initial program 58.2%

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

   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 51.2%

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

   6. Applied rewrites 51.2%

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

   7. Evaluated real constant 51.2%

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

   9. Applied rewrites 81.3%

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

   if 0.00865000021 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.2%

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

   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 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 70.2%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(u2, \pi + \pi, \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right)\right) \cdot 31.006277084350586\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00865000020712614:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right)\\ \end{array} \]

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) 0.00865000020712614)
  (* (sqrt (- (log1p (- u1)))) (* u2 6.2831854820251465))
  (* (sqrt u1) (* u2 (fma (* u2 u2) -41.34170150756836 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00865000020712614f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * 6.2831854820251465f);
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((u2 * u2), -41.34170150756836f, (((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(0.00865000020712614))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(u2 * u2), Float32(-41.34170150756836), 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) < 0.00865000021

   1. Initial program 58.2%

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

   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 51.2%

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

   6. Applied rewrites 51.2%

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

   7. Evaluated real constant 51.2%

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

   9. Applied rewrites 81.3%

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

   if 0.00865000021 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.2%

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

   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 54.3%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 70.2%

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

   10. Evaluated real constant 70.2%

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

Alternative 12: 84.5% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00865000020712614:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right)\\ \end{array} \]

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) 0.00865000020712614)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (* (sqrt u1) (* u2 (fma (* u2 u2) -41.34170150756836 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00865000020712614f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((u2 * u2), -41.34170150756836f, (((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(0.00865000020712614))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(u2 * u2), Float32(-41.34170150756836), 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) < 0.00865000021

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 51.2%

      \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]
   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 0.00865000021 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.2%

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

   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 54.3%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 70.2%

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

   10. Evaluated real constant 70.2%

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

Alternative 13: 83.1% 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \mathbf{elif}\;1 - u1 \leq 0.9999899864196777:\\ \;\;\;\;6.2831854820251465 \cdot \left(\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.34170150756836 \cdot u2, u2, \pi + \pi\right)\right)\\ \end{array} \]

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 (<= (- 1.0 u1) 0.9975000023841858)
  (* 6.2831854820251465 (* u2 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (- 1.0 u1) 0.9999899864196777)
    (* 6.2831854820251465 (* (sqrt (* (fma 0.5 u1 1.0) u1)) u2))
    (*
     (sqrt 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) <= 0.9975000023841858f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-logf((1.0f - u1))));
	} else if ((1.0f - u1) <= 0.9999899864196777f) {
		tmp = 6.2831854820251465f * (sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * u2);
	} 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(1.0) - u1) <= Float32(0.9975000023841858))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	elseif (Float32(Float32(1.0) - u1) <= Float32(0.9999899864196777))
		tmp = Float32(Float32(6.2831854820251465) * Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * u2));
	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 3 regimes

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

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 51.2%

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

   if 0.997500002 < (-.f32 #s(literal 1 binary32) u1) < 0.999989986

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 51.2%

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

   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 73.8%

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

   10. Applied rewrites 73.8%

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

   if 0.999989986 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 58.2%

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

   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 54.3%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 70.2%

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

   10. Evaluated real constant 70.2%

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

Alternative 14: 80.3% 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

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

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)
    (* 6.2831854820251465 (* (sqrt (* (fma 0.5 u1 1.0) u1)) u2))
    (* 6.2831854820251465 (* u2 (sqrt t_0))))))

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 = 6.2831854820251465f * (sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * u2);
	} else {
		tmp = 6.2831854820251465f * (u2 * sqrtf(t_0));
	}
	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(0.0024999999441206455))
		tmp = Float32(Float32(6.2831854820251465) * Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * u2));
	else
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(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 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 51.2%

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

   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 73.8%

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

   10. Applied rewrites 73.8%

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

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

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   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 51.2%

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

Alternative 15: 73.9% 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 58.2%

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

2. Evaluated real constant 58.2%

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

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 51.2%

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

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 73.8%

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

10. Applied rewrites 73.8%

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

Alternative 16: 73.8% 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

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

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)))
  (* 6.2831854820251465 (* u2 (sqrt (fma u1 (* 0.5 u1) u1)))))

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

2. Evaluated real constant 58.2%

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

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 51.2%

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

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 73.8%

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

10. Applied rewrites 73.9%

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

Alternative 17: 65.9% 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

\[\frac{1}{\frac{0.5}{u2 \cdot \left(\pi \cdot \sqrt{u1}\right)}} \]

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)))
  (/ 1.0 (/ 0.5 (* u2 (* PI (sqrt u1))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.2%

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

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 51.2%

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

6. Applied rewrites 51.2%

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

7. Taylor expanded in u2 around 0

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

9. Applied rewrites 51.2%

   \[\leadsto \frac{1}{\frac{0.5}{u2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)}} \]

10. Taylor expanded in u1 around 0

    \[\leadsto \frac{1}{\frac{0.5}{u2 \cdot \left(\pi \cdot \sqrt{u1}\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 65.9%

    \[\leadsto \frac{1}{\frac{0.5}{u2 \cdot \left(\pi \cdot \sqrt{u1}\right)}} \]
13. Add Preprocessing

Alternative 18: 65.9% 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 58.2%

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

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 51.2%

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

6. Applied rewrites 51.2%

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

7. Evaluated real constant 51.2%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 65.9%

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

Alternative 19: 65.9% 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 58.2%

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

2. Evaluated real constant 58.2%

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

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 51.2%

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

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 65.9%

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

Reproduce

herbie shell --seed 2026086 
(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))))