Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.0% → 99.2%

Time: 50.2s

Alternatives: 18

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: 58.0% 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, 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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]

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))))
 (sin (fma -6.2831854820251465 u2 1.5707963705062866))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(fmaf(-6.2831854820251465f, u2, 1.5707963705062866f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(fma(Float32(-6.2831854820251465), u2, Float32(1.5707963705062866))))
end

Derivation

1. Initial program 58.0%

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

3. Applied rewrites 58.0%

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

4. Evaluated real constant 58.0%

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

5. Evaluated real constant 58.0%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]

7. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.0% 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 58.0%

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

2. Evaluated real constant 58.0%

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

4. Applied rewrites 99.0%

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

Alternative 3: 97.0% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.07000000029802322:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right)\\ \end{array} \]

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) 0.07000000029802322)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* (* u2 u2) 9.869604110717773) -2.0 1.0))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (sin (fma -6.2831854820251465 u2 1.5707963705062866)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.07000000029802322f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(((u2 * u2) * 9.869604110717773f), -2.0f, 1.0f);
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf(fmaf(-6.2831854820251465f, u2, 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(0.07000000029802322))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(u2 * u2) * Float32(9.869604110717773)), Float32(-2.0), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(fma(Float32(-6.2831854820251465), u2, 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) < 0.0700000003

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   9. Applied rewrites 88.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   if 0.0700000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   3. Applied rewrites 58.0%

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

   4. Evaluated real constant 58.0%

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

   5. Evaluated real constant 58.0%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]

   8. Applied rewrites 88.2%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.9% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.07000000029802322:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 0.5 \cdot u1, u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.07000000029802322)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* (* u2 u2) 9.869604110717773) -2.0 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) <= 0.07000000029802322f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(((u2 * u2) * 9.869604110717773f), -2.0f, 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(0.07000000029802322))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(u2 * u2) * Float32(9.869604110717773)), Float32(-2.0), 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) < 0.0700000003

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   9. Applied rewrites 88.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   if 0.0700000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.1%

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

   6. Applied rewrites 88.2%

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

   7. Evaluated real constant 88.2%

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

Alternative 5: 96.9% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.07000000029802322:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.07000000029802322)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* (* u2 u2) 9.869604110717773) -2.0 1.0))
  (* (sqrt (* u1 (fma u1 0.5 1.0))) (cos (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.07000000029802322f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(((u2 * u2) * 9.869604110717773f), -2.0f, 1.0f);
	} else {
		tmp = sqrtf((u1 * fmaf(u1, 0.5f, 1.0f))) * 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(0.07000000029802322))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(u2 * u2) * Float32(9.869604110717773)), Float32(-2.0), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0)))) * 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) < 0.0700000003

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   9. Applied rewrites 88.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   if 0.0700000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   2. Evaluated real constant 58.0%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 88.1%

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

Alternative 6: 94.8% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.19499999284744263:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right)\\ \end{array} \]

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) 0.19499999284744263)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* (* u2 u2) 9.869604110717773) -2.0 1.0))
  (* (sqrt u1) (sin (fma -6.2831854820251465 u2 1.5707963705062866)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.19499999284744263f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(((u2 * u2) * 9.869604110717773f), -2.0f, 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf(fmaf(-6.2831854820251465f, u2, 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(0.19499999284744263))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(u2 * u2) * Float32(9.869604110717773)), Float32(-2.0), Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(fma(Float32(-6.2831854820251465), u2, 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) < 0.194999993

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   9. Applied rewrites 88.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   if 0.194999993 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   3. Applied rewrites 58.0%

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

   4. Evaluated real constant 58.0%

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

   5. Evaluated real constant 58.0%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]

   8. Applied rewrites 76.4%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.2831854820251465, u2, 1.5707963705062866\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.7% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.19499999284744263:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.19499999284744263)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* (* u2 u2) 9.869604110717773) -2.0 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) <= 0.19499999284744263f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(((u2 * u2) * 9.869604110717773f), -2.0f, 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(0.19499999284744263))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(u2 * u2) * Float32(9.869604110717773)), Float32(-2.0), 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) < 0.194999993

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   9. Applied rewrites 88.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   if 0.194999993 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   2. Evaluated real constant 58.0%

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

   3. Taylor expanded in u1 around 0

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

   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 8: 88.2% 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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

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 (* (* u2 u2) 9.869604110717773) -2.0 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * fmaf(((u2 * u2) * 9.869604110717773f), -2.0f, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(u2 * u2) * Float32(9.869604110717773)), Float32(-2.0), Float32(1.0)))
end

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 53.4%

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

5. Evaluated real constant 53.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

7. Applied rewrites 53.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

9. Applied rewrites 88.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]
10. Add Preprocessing

Alternative 9: 86.9% accurate, 0.6× 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

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.05400000140070915:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 0.5 \cdot u1, u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot u2\right) \cdot u2, \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right)\\ \end{array} \]

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 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.05400000140070915)
    (*
     (sqrt (fma u1 (* 0.5 u1) u1))
     (fma (* (* -2.0 u2) u2) (* PI PI) 1.0))
    (* t_0 (+ 1.0 (* (* u2 u2) -19.739208221435547))))))

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))) <= 0.05400000140070915f) {
		tmp = sqrtf(fmaf(u1, (0.5f * u1), u1)) * fmaf(((-2.0f * u2) * u2), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = t_0 * (1.0f + ((u2 * u2) * -19.739208221435547f));
	}
	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(0.05400000140070915))
		tmp = Float32(sqrt(fma(u1, Float32(Float32(0.5) * u1), u1)) * fma(Float32(Float32(Float32(-2.0) * u2) * u2), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = Float32(t_0 * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208221435547))));
	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))) < 0.0540000014

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 79.0%

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

   11. Applied rewrites 79.0%

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

   if 0.0540000014 < (*.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 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   6. Applied rewrites 53.4%

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

   7. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot u2\right) \cdot u2, 9.869604110717773, 1\right) \]

   9. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.8% 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

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054499998688697815:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right)\\ \end{array} \]

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 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.054499998688697815)
    (*
     (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
     (fma (* (* u2 u2) 9.869604110717773) -2.0 1.0))
    (* t_0 (+ 1.0 (* (* u2 u2) -19.739208221435547))))))

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))) <= 0.054499998688697815f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * fmaf(((u2 * u2) * 9.869604110717773f), -2.0f, 1.0f);
	} else {
		tmp = t_0 * (1.0f + ((u2 * u2) * -19.739208221435547f));
	}
	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(0.054499998688697815))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * fma(Float32(Float32(u2 * u2) * Float32(9.869604110717773)), Float32(-2.0), Float32(1.0)));
	else
		tmp = Float32(t_0 * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208221435547))));
	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))) < 0.0544999987

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   10. Applied rewrites 79.0%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 9.869604110717773, -2, 1\right) \]

   if 0.0544999987 < (*.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 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   6. Applied rewrites 53.4%

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

   7. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot u2\right) \cdot u2, 9.869604110717773, 1\right) \]

   9. Applied rewrites 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.5% 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

\[\begin{array}{l} t_0 := \sqrt{\sqrt{u1}}\\ \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, -19.739209900765786 \cdot \left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0 (sqrt (sqrt u1))))
  (if (<= (* (* 2.0 PI) u2) 0.006000000052154064)
    (sqrt (- (log1p (- u1))))
    (fma t_0 t_0 (* -19.739209900765786 (* (* u2 u2) (sqrt u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(sqrtf(u1));
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.006000000052154064f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = fmaf(t_0, t_0, (-19.739209900765786f * ((u2 * u2) * sqrtf(u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(sqrt(u1))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.006000000052154064))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = fma(t_0, t_0, Float32(Float32(-19.739209900765786) * Float32(Float32(u2 * u2) * sqrt(u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00600000005

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.0%

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

   6. Applied rewrites 79.7%

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

   if 0.00600000005 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   2. Evaluated real constant 58.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 53.3%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 68.9%

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

Alternative 12: 83.5% accurate, 2.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right)\\ \end{array} \]

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) 0.006000000052154064)
  (sqrt (- (log1p (- u1))))
  (* (sqrt u1) (+ 1.0 (* (* u2 u2) -19.739208221435547)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.006000000052154064f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(u1) * (1.0f + ((u2 * u2) * -19.739208221435547f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.006000000052154064))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208221435547))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00600000005

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.0%

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

   6. Applied rewrites 79.7%

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

   if 0.00600000005 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   8. Applied rewrites 69.1%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   10. Applied rewrites 69.1%

       \[\leadsto \sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.6% 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

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.024000000208616257:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

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))) 0.024000000208616257)
    (* (sqrt u1) (+ 1.0 (* (* u2 u2) -19.739208221435547)))
    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))) <= 0.024000000208616257f) {
		tmp = sqrtf(u1) * (1.0f + ((u2 * u2) * -19.739208221435547f));
	} 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(0.024000000208616257))
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208221435547))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt(-log((single(1.0) - u1)));
	tmp = single(0.0);
	if ((t_0 * cos(((single(2.0) * single(pi)) * u2))) <= single(0.024000000208616257))
		tmp = sqrt(u1) * (single(1.0) + ((u2 * u2) * single(-19.739208221435547)));
	else
		tmp = t_0;
	end
	tmp_2 = 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))) < 0.0240000002

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   8. Applied rewrites 69.1%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   10. Applied rewrites 69.1%

       \[\leadsto \sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208221435547\right) \]

   if 0.0240000002 < (*.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 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.0%

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

Alternative 14: 79.6% 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

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.024000000208616257:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(u2 \cdot u2, -19.739208221435547, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

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))) 0.024000000208616257)
    (* (sqrt u1) (fma (* u2 u2) -19.739208221435547 1.0))
    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))) <= 0.024000000208616257f) {
		tmp = sqrtf(u1) * fmaf((u2 * u2), -19.739208221435547f, 1.0f);
	} 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(0.024000000208616257))
		tmp = Float32(sqrt(u1) * fma(Float32(u2 * u2), Float32(-19.739208221435547), Float32(1.0)));
	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))) < 0.0240000002

   1. Initial program 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 53.4%

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

   5. Evaluated real constant 53.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   8. Applied rewrites 69.1%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -2 \cdot \left({u2}^{2} \cdot 9.869604110717773\right)\right) \]

   10. Applied rewrites 69.1%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(u2 \cdot u2, -19.739208221435547, 1\right) \]

   if 0.0240000002 < (*.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 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.0%

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

Alternative 15: 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

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

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 0.5 (* u1 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(0.5f, (u1 * 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(Float32(0.5), Float32(u1 * 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 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]

   7. Applied rewrites 72.2%

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

   9. Applied rewrites 72.2%

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

   11. Applied rewrites 72.3%

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

   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 58.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 50.0%

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

Alternative 16: 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

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

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 0.5 (* u1 u1) u1)))

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 50.0%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]

7. Applied rewrites 72.2%

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

9. Applied rewrites 72.2%

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

11. Applied rewrites 72.3%

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

Alternative 17: 72.2% 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

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

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 50.0%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]

7. Applied rewrites 72.2%

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

9. Applied rewrites 72.2%

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

Alternative 18: 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 58.0%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 50.0%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]

7. Applied rewrites 64.3%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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))))