Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.4% → 98.5%

Time: 9.9s

Alternatives: 24

Speedup: 6.2×

Specification

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

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

Julia

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

MATLAB

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

Initial Program: 57.4% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.8× speedup

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

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \pi\right), u2, \left(u2 \cdot \pi\right) \cdot 0.5\right)\right) \]

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 (* 3.0 (* 0.5 PI)) u2 (* (* u2 PI) 0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(fmaf((3.0f * (0.5f * ((float) M_PI))), u2, ((u2 * ((float) M_PI)) * 0.5f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(fma(Float32(Float32(3.0) * Float32(Float32(0.5) * Float32(pi))), u2, Float32(Float32(u2 * Float32(pi)) * Float32(0.5)))))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 98.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \pi\right), u2, \left(u2 \cdot \pi\right) \cdot 0.5\right)\right) \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log1p (- u1)))) (sin (* 6.2831854820251465 u2))))

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.4%

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

Alternative 3: 96.2% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u1 \cdot u1, 0.25, u1\right)}{\sqrt{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.05999999865889549)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (* u2 (* -41.341705691712875 (* u2 u2)))))
  (*
   (/ (fma (* u1 u1) 0.25 u1) (sqrt u1))
   (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.05999999865889549f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
	} else {
		tmp = (fmaf((u1 * u1), 0.25f, u1) / sqrtf(u1)) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.05999999865889549))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))));
	else
		tmp = Float32(Float32(fma(Float32(u1 * u1), Float32(0.25), u1) / sqrt(u1)) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0599999987

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.2%

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

   6. Applied rewrites 88.0%

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

   7. Evaluated real constant 88.0%

      \[\leadsto \frac{u1 + \left(u1 \cdot u1\right) \cdot 0.25}{\sqrt{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.0%

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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \sin \left(u2 \cdot 6.2831854820251465\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.05999999865889549)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (* u2 (* -41.341705691712875 (* u2 u2)))))
  (*
   (fma (* (sqrt u1) u1) 0.25 (sqrt u1))
   (sin (* u2 6.2831854820251465)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.05999999865889549f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
	} else {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1)) * sinf((u2 * 6.2831854820251465f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.05999999865889549))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))));
	else
		tmp = Float32(fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1)) * sin(Float32(u2 * Float32(6.2831854820251465))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0599999987

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.2%

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

   6. Applied rewrites 88.2%

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

   7. Evaluated real constant 88.2%

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

Alternative 5: 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.05999999865889549)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (* u2 (* -41.341705691712875 (* u2 u2)))))
  (* (sqrt (fma (* 0.5 u1) u1 u1)) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.05999999865889549f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
	} else {
		tmp = sqrtf(fmaf((0.5f * u1), u1, 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(0.05999999865889549))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))));
	else
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, 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) < 0.0599999987

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 87.9%

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

   7. Evaluated real constant 87.9%

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

Alternative 6: 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.05999999865889549)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (* u2 (* -41.341705691712875 (* u2 u2)))))
  (* (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) <= 0.05999999865889549f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
	} 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(0.05999999865889549))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))));
	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) < 0.0599999987

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 87.9%

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

   5. Evaluated real constant 87.9%

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

   7. Applied rewrites 87.9%

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

Alternative 7: 94.2% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.12999999523162842:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u1}{\sqrt{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.12999999523162842)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (* u2 (* -41.341705691712875 (* u2 u2)))))
  (* (/ u1 (sqrt u1)) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.12999999523162842f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
	} else {
		tmp = (u1 / sqrtf(u1)) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.12999999523162842))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))));
	else
		tmp = Float32(Float32(u1 / sqrt(u1)) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.129999995

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.129999995 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-div N/A

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

      5. pow1 N/A

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

      6. pow1/2 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lower-/.f32 76.6%

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

   6. Applied rewrites 76.5%

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

   7. Evaluated real constant 76.5%

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

Alternative 8: 94.2% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.12999999523162842:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, -6.2831854820251465, \pi\right)\right)\\ \end{array} \]

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) 0.12999999523162842)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (* u2 (* -41.341705691712875 (* u2 u2)))))
  (* (sqrt u1) (sin (fma u2 -6.2831854820251465 PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.12999999523162842f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
	} else {
		tmp = sqrtf(u1) * sinf(fmaf(u2, -6.2831854820251465f, ((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(0.12999999523162842))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))));
	else
		tmp = Float32(sqrt(u1) * sin(fma(u2, Float32(-6.2831854820251465), Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.129999995

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.129999995 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.6%

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

   6. Applied rewrites 49.3%

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

   7. Evaluated real constant 49.3%

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

   9. Applied rewrites 49.4%

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

Alternative 9: 94.1% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.12999999523162842:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.12999999523162842)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (* u2 (* -41.341705691712875 (* u2 u2)))))
  (* (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) <= 0.12999999523162842f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
	} 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(0.12999999523162842))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))));
	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) < 0.129999995

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.129999995 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.6%

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

   5. Evaluated real constant 76.6%

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

Alternative 10: 88.9% 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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]

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))))
 (fma 6.2831854820251465 u2 (* u2 (* -41.341705691712875 (* u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * (-41.341705691712875f * (u2 * u2))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(-41.341705691712875) * Float32(u2 * u2)))))
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 53.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 53.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 88.9%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right)\right) \]
10. Add Preprocessing

Alternative 11: 88.9% 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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\right) \]

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 (- (log1p (- u1))))
 (* (fma -41.341705691712875 (* u2 u2) 6.2831854820251465) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (fmaf(-41.341705691712875f, (u2 * u2), 6.2831854820251465f) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(fma(Float32(-41.341705691712875), Float32(u2 * u2), Float32(6.2831854820251465)) * u2))
end

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 53.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 53.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\right) \]
8. Step-by-step derivation

9. Applied rewrites 88.9%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\right) \]
10. Add Preprocessing

Alternative 12: 87.7% 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (- 1.0 u1) 0.9975000023841858)
  (*
   (sqrt (- (log (- 1.0 u1))))
   (* (fma -41.341705691712875 (* u2 u2) 6.2831854820251465) u2))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (fma
    6.2831854820251465
    u2
    (* (* -41.341705691712875 (* u2 u2)) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9975000023841858f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (fmaf(-41.341705691712875f, (u2 * u2), 6.2831854820251465f) * u2);
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * fmaf(6.2831854820251465f, u2, ((-41.341705691712875f * (u2 * u2)) * u2));
	}
	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(fma(Float32(-41.341705691712875), Float32(u2 * u2), Float32(6.2831854820251465)) * u2));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * fma(Float32(6.2831854820251465), u2, Float32(Float32(Float32(-41.341705691712875) * Float32(u2 * u2)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

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

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

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 87.7% 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

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0
        (*
         (fma -41.341705691712875 (* u2 u2) 6.2831854820251465)
         u2)))
  (if (<= (- 1.0 u1) 0.9975000023841858)
    (* (sqrt (- (log (- 1.0 u1)))) t_0)
    (* t_0 (sqrt (* (fma 0.5 u1 1.0) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(-41.341705691712875f, (u2 * u2), 6.2831854820251465f) * u2;
	float tmp;
	if ((1.0f - u1) <= 0.9975000023841858f) {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	} else {
		tmp = t_0 * sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(fma(Float32(-41.341705691712875), Float32(u2 * u2), Float32(6.2831854820251465)) * u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9975000023841858))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	else
		tmp = Float32(t_0 * sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

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

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

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.1%

       \[\leadsto \left(\mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 86.3% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9975000023841858:\\ \;\;\;\;u2 \cdot \left(6.2831854820251465 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (- 1.0 u1) 0.9975000023841858)
  (* u2 (* 6.2831854820251465 (sqrt (- (log1p (- u1))))))
  (*
   (* (fma -41.341705691712875 (* u2 u2) 6.2831854820251465) u2)
   (sqrt (* (fma 0.5 u1 1.0) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9975000023841858f) {
		tmp = u2 * (6.2831854820251465f * sqrtf(-log1pf(-u1)));
	} else {
		tmp = (fmaf(-41.341705691712875f, (u2 * u2), 6.2831854820251465f) * u2) * sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9975000023841858))
		tmp = Float32(u2 * Float32(Float32(6.2831854820251465) * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(Float32(fma(Float32(-41.341705691712875), Float32(u2 * u2), Float32(6.2831854820251465)) * u2) * sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u2 around 0

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

   8. Applied rewrites 50.1%

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

   10. Applied rewrites 81.0%

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

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

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.1%

       \[\leadsto \left(\mathsf{fma}\left(-41.341705691712875, u2 \cdot u2, 6.2831854820251465\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 84.2% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(\sqrt{u1}, 6.2831854820251465, \left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}\right) \cdot -41.341705691712875\right)\\ \end{array} \]

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) 0.017999999225139618)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (*
   u2
   (fma
    (sqrt u1)
    6.2831854820251465
    (* (* (* u2 u2) (sqrt u1)) -41.341705691712875)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.017999999225139618f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = u2 * fmaf(sqrtf(u1), 6.2831854820251465f, (((u2 * u2) * sqrtf(u1)) * -41.341705691712875f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.017999999225139618))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(u2 * fma(sqrt(u1), Float32(6.2831854820251465), Float32(Float32(Float32(u2 * u2) * sqrt(u1)) * Float32(-41.341705691712875))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0179999992

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 80.9%

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

   if 0.0179999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \mathsf{fma}\left(-41.341705691712875, {u2}^{2} \cdot \sqrt{u1}, 6.2831854820251465 \cdot \sqrt{u1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto u2 \cdot \mathsf{fma}\left(-41.341705691712875, {u2}^{2} \cdot \sqrt{u1}, 6.2831854820251465 \cdot \sqrt{u1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1}, 6.2831854820251465, \left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}\right) \cdot -41.341705691712875\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 84.2% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-41.341705691712875 \cdot \left(u2 \cdot u2\right), u2, 6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.017999999225139618)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (*
   (sqrt u1)
   (fma
    (* -41.341705691712875 (* u2 u2))
    u2
    (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.017999999225139618f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * fmaf((-41.341705691712875f * (u2 * u2)), u2, (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(0.017999999225139618))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(-41.341705691712875) * Float32(u2 * u2)), u2, 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) < 0.0179999992

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 80.9%

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

   if 0.0179999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(-41.341705691712875 \cdot \left(u2 \cdot u2\right), u2, 6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 82.9% 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.998199999332428:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{elif}\;1 - u1 \leq 0.9999985098838806:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-41.341705691712875 \cdot \left(u2 \cdot u2\right), u2, 6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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 (<= (- 1.0 u1) 0.998199999332428)
  (* (sqrt (- (log (- 1.0 u1)))) (* u2 6.2831854820251465))
  (if (<= (- 1.0 u1) 0.9999985098838806)
    (* 6.2831854820251465 (* u2 (sqrt (fma (* 0.5 u1) u1 u1))))
    (*
     (sqrt u1)
     (fma
      (* -41.341705691712875 (* u2 u2))
      u2
      (* 6.2831854820251465 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.998199999332428f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (u2 * 6.2831854820251465f);
	} else if ((1.0f - u1) <= 0.9999985098838806f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(fmaf((0.5f * u1), u1, u1)));
	} else {
		tmp = sqrtf(u1) * fmaf((-41.341705691712875f * (u2 * u2)), u2, (6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.998199999332428))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	elseif (Float32(Float32(1.0) - u1) <= Float32(0.9999985098838806))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(fma(Float32(Float32(0.5) * u1), u1, u1))));
	else
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(-41.341705691712875) * Float32(u2 * u2)), u2, Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.998199999

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u2 around 0

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

   8. Applied rewrites 50.1%

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

   if 0.998199999 < (-.f32 #s(literal 1 binary32) u1) < 0.99999851

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   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 73.9%

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

   10. Applied rewrites 73.9%

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

   if 0.99999851 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(-41.341705691712875 \cdot \left(u2 \cdot u2\right), u2, 6.2831854820251465 \cdot u2\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 82.9% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.998199999332428:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot 6.2831854820251465\right)\\ \mathbf{elif}\;1 - u1 \leq 0.9999985098838806:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right) - -6.2831854820251465\right)\right)\\ \end{array} \]

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) 0.998199999332428)
  (* (sqrt (- (log (- 1.0 u1)))) (* u2 6.2831854820251465))
  (if (<= (- 1.0 u1) 0.9999985098838806)
    (* 6.2831854820251465 (* u2 (sqrt (fma (* 0.5 u1) u1 u1))))
    (*
     (sqrt u1)
     (*
      u2
      (- (* -41.341705691712875 (* u2 u2)) -6.2831854820251465))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.998199999332428f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (u2 * 6.2831854820251465f);
	} else if ((1.0f - u1) <= 0.9999985098838806f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(fmaf((0.5f * u1), u1, u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * ((-41.341705691712875f * (u2 * u2)) - -6.2831854820251465f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.998199999332428))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(u2 * Float32(6.2831854820251465)));
	elseif (Float32(Float32(1.0) - u1) <= Float32(0.9999985098838806))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(fma(Float32(Float32(0.5) * u1), u1, u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(Float32(-41.341705691712875) * Float32(u2 * u2)) - Float32(-6.2831854820251465))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.998199999

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u2 around 0

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

   8. Applied rewrites 50.1%

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

   if 0.998199999 < (-.f32 #s(literal 1 binary32) u1) < 0.99999851

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   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 73.9%

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

   10. Applied rewrites 73.9%

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

   if 0.99999851 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right) - -6.2831854820251465\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 82.9% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.998199999332428:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \mathbf{elif}\;1 - u1 \leq 0.9999985098838806:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right) - -6.2831854820251465\right)\right)\\ \end{array} \]

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) 0.998199999332428)
  (* 6.2831854820251465 (* u2 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (- 1.0 u1) 0.9999985098838806)
    (* 6.2831854820251465 (* u2 (sqrt (fma (* 0.5 u1) u1 u1))))
    (*
     (sqrt u1)
     (*
      u2
      (- (* -41.341705691712875 (* u2 u2)) -6.2831854820251465))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.998199999332428f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-logf((1.0f - u1))));
	} else if ((1.0f - u1) <= 0.9999985098838806f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(fmaf((0.5f * u1), u1, u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * ((-41.341705691712875f * (u2 * u2)) - -6.2831854820251465f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.998199999332428))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	elseif (Float32(Float32(1.0) - u1) <= Float32(0.9999985098838806))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(fma(Float32(Float32(0.5) * u1), u1, u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(Float32(-41.341705691712875) * Float32(u2 * u2)) - Float32(-6.2831854820251465))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.998199999

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   if 0.998199999 < (-.f32 #s(literal 1 binary32) u1) < 0.99999851

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   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 73.9%

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

   10. Applied rewrites 73.9%

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

   if 0.99999851 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right) - -6.2831854820251465\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 77.4% accurate, 2.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right) - -6.2831854820251465\right)\right)\\ \end{array} \]

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) 0.017999999225139618)
  (* 6.2831854820251465 (* u2 (sqrt (fma (* 0.5 u1) u1 u1))))
  (*
   (sqrt u1)
   (* u2 (- (* -41.341705691712875 (* u2 u2)) -6.2831854820251465)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.017999999225139618f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(fmaf((0.5f * u1), u1, u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * ((-41.341705691712875f * (u2 * u2)) - -6.2831854820251465f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.017999999225139618))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(fma(Float32(Float32(0.5) * u1), u1, u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(Float32(-41.341705691712875) * Float32(u2 * u2)) - Float32(-6.2831854820251465))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0179999992

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   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 73.9%

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

   10. Applied rewrites 73.9%

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

   if 0.0179999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(-41.341705691712875 \cdot \left(u2 \cdot u2\right) - -6.2831854820251465\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 77.4% accurate, 2.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341705691712875 \cdot u2, u2, 6.2831854820251465\right)\right)\\ \end{array} \]

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) 0.017999999225139618)
  (* 6.2831854820251465 (* u2 (sqrt (fma (* 0.5 u1) u1 u1))))
  (*
   (sqrt u1)
   (* u2 (fma (* -41.341705691712875 u2) u2 6.2831854820251465)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.017999999225139618f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(fmaf((0.5f * u1), u1, u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((-41.341705691712875f * u2), u2, 6.2831854820251465f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.017999999225139618))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(fma(Float32(Float32(0.5) * u1), u1, u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(Float32(-41.341705691712875) * u2), u2, Float32(6.2831854820251465))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0179999992

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 50.1%

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

   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 73.9%

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

   10. Applied rewrites 73.9%

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

   if 0.0179999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.4%

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

   2. Evaluated real constant 57.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341705691712875 \cdot u2, u2, 6.2831854820251465\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 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

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

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 (* 0.5 u1) u1 u1)))))

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 50.1%

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

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 73.9%

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

10. Applied rewrites 73.9%

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

Alternative 23: 66.1% accurate, 3.5× 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

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

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)))
  (* 2.0 (* (* u2 PI) (/ u1 (sqrt u1)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 66.1%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-div N/A

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

   5. pow1 N/A

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

   6. pow1/2 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. mult-flip N/A

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

   9. lift-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 66.0%

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

9. Applied rewrites 66.0%

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

11. Applied rewrites 66.1%

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

Alternative 24: 66.1% accurate, 6.2× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* 6.2831854820251465 (* u2 (sqrt u1))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Evaluated real constant 57.4%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 50.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 66.1%

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

Reproduce

herbie shell --seed 2026070 
(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))))