Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.1% → 98.3%

Time: 58.1s

Alternatives: 21

Speedup: 6.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.1% accurate, 1.0× speedup

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

4. Applied rewrites 98.3%

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

Alternative 2: 97.1% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.3799999952316284:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + \left(0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.3799999952316284)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (*
     u2
     (*
      (* u2 u2)
      (fma (* u2 u2) 81.6052606304404 -41.341705691712875)))))
  (* (sqrt (+ u1 (* (* 0.5 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) <= 0.3799999952316284f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
	} else {
		tmp = sqrtf((u1 + ((0.5f * 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(0.3799999952316284))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))));
	else
		tmp = Float32(sqrt(Float32(u1 + Float32(Float32(Float32(0.5) * 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) < 0.379999995

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]

   if 0.379999995 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 87.8%

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

   6. Applied rewrites 87.8%

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

   7. Evaluated real constant 87.8%

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

   9. Applied rewrites 87.8%

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

Alternative 3: 97.1% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.3799999952316284:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 0.5 \cdot u1, u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.3799999952316284)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (*
     u2
     (*
      (* u2 u2)
      (fma (* u2 u2) 81.6052606304404 -41.341705691712875)))))
  (* (sqrt (fma u1 (* 0.5 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) <= 0.3799999952316284f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
	} else {
		tmp = sqrtf(fmaf(u1, (0.5f * 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(0.3799999952316284))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))));
	else
		tmp = Float32(sqrt(fma(u1, Float32(Float32(0.5) * 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) < 0.379999995

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]

   if 0.379999995 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 87.8%

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

   6. Applied rewrites 87.8%

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

   7. Evaluated real constant 87.8%

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

Alternative 4: 97.1% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.3799999952316284:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.3799999952316284)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (*
     u2
     (*
      (* u2 u2)
      (fma (* u2 u2) 81.6052606304404 -41.341705691712875)))))
  (* (sqrt (* u1 (fma u1 0.5 1.0))) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.3799999952316284f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
	} else {
		tmp = sqrtf((u1 * fmaf(u1, 0.5f, 1.0f))) * 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(0.3799999952316284))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))));
	else
		tmp = Float32(sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0)))) * 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) < 0.379999995

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]

   if 0.379999995 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 87.8%

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

Alternative 5: 95.9% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.699999988079071:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u1 \cdot \frac{\sqrt{u1}}{u1}\right) \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.699999988079071)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (*
     u2
     (*
      (* u2 u2)
      (fma (* u2 u2) 81.6052606304404 -41.341705691712875)))))
  (* (* u1 (/ (sqrt 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) <= 0.699999988079071f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
	} else {
		tmp = (u1 * (sqrtf(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(0.699999988079071))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))));
	else
		tmp = Float32(Float32(u1 * Float32(sqrt(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) < 0.699999988

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]

   if 0.699999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in u1 around inf

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

   7. Applied rewrites 76.2%

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

   8. Evaluated real constant 76.2%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 76.2%

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

Alternative 6: 95.9% 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

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.699999988079071:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

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) 0.699999988079071)
  (*
   (sqrt (- (log1p (- u1))))
   (fma
    6.2831854820251465
    u2
    (*
     u2
     (*
      (* u2 u2)
      (fma (* u2 u2) 81.6052606304404 -41.341705691712875)))))
  (* (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) <= 0.699999988079071f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
	} 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(0.699999988079071))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))));
	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) < 0.699999988

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]

   if 0.699999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 76.3%

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

Alternative 7: 91.7% accurate, 1.4× speedup

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

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]

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
   (*
    (* u2 u2)
    (fma (* u2 u2) 81.6052606304404 -41.341705691712875))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * fmaf(6.2831854820251465f, u2, (u2 * ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))))
end

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

5. Applied rewrites 55.2%

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

7. Applied rewrites 55.2%

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

9. Applied rewrites 91.7%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 91.7% accurate, 1.5× speedup

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

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right), u2 \cdot u2, 6.2831854820251465\right)\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (*
 (sqrt (- (log1p (- u1))))
 (*
  u2
  (fma
   (fma (* u2 u2) 81.6052606304404 -41.341705691712875)
   (* u2 u2)
   6.2831854820251465))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (u2 * fmaf(fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f), (u2 * u2), 6.2831854820251465f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875)), Float32(u2 * u2), Float32(6.2831854820251465))))
end

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

5. Applied rewrites 55.2%

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

7. Applied rewrites 55.2%

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

9. Applied rewrites 91.7%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right), u2 \cdot u2, 6.2831854820251465\right)\right) \]
10. Add Preprocessing

Alternative 9: 89.0% accurate, 1.7× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (*
 (sqrt (- (log1p (- u1))))
 (* u2 (fma (* u2 u2) -41.34170150756836 (+ PI PI)))))

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 54.3%

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

6. Applied rewrites 54.3%

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

7. Evaluated real constant 54.3%

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

9. Applied rewrites 89.0%

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

Alternative 10: 89.0% accurate, 1.7× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (*
 (sqrt (- (log1p (- u1))))
 (* u2 (+ (fma -41.34170150756836 (* u2 u2) PI) PI))))

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 54.3%

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

6. Applied rewrites 54.3%

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

7. Evaluated real constant 54.3%

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

9. Applied rewrites 89.0%

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

Alternative 11: 87.9% accurate, 1.3× 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

\[\begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.002950000111013651:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \left(u2 \cdot \left(\mathsf{fma}\left(-41.34170150756836, u2 \cdot u2, \pi\right) + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0 (- (log (- 1.0 u1)))))
  (if (<= t_0 0.002950000111013651)
    (*
     (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
     (* u2 (fma (* u2 u2) -41.34170150756836 (+ PI PI))))
    (*
     (sqrt t_0)
     (* u2 (+ (fma -41.34170150756836 (* u2 u2) PI) PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = -logf((1.0f - u1));
	float tmp;
	if (t_0 <= 0.002950000111013651f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (u2 * fmaf((u2 * u2), -41.34170150756836f, (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = sqrtf(t_0) * (u2 * (fmaf(-41.34170150756836f, (u2 * u2), ((float) M_PI)) + ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(-log(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.002950000111013651))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(u2 * fma(Float32(u2 * u2), Float32(-41.34170150756836), Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = Float32(sqrt(t_0) * Float32(u2 * Float32(fma(Float32(-41.34170150756836), Float32(u2 * u2), Float32(pi)) + Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.00295000011

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right) \]

   10. Applied rewrites 79.9%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right) \]

   if 0.00295000011 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

Alternative 12: 87.9% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9970499873161316:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, 6.2831854820251465\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (- 1.0 u1) 0.9970499873161316)
  (*
   (sqrt (- (log (- 1.0 u1))))
   (* u2 (fma (* u2 u2) -41.34170150756836 6.2831854820251465)))
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (* u2 (fma (* u2 u2) -41.34170150756836 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9970499873161316f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (u2 * fmaf((u2 * u2), -41.34170150756836f, 6.2831854820251465f));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (u2 * fmaf((u2 * u2), -41.34170150756836f, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9970499873161316))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(u2 * fma(Float32(u2 * u2), Float32(-41.34170150756836), Float32(6.2831854820251465))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(u2 * fma(Float32(u2 * u2), Float32(-41.34170150756836), Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997049987

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   8. Evaluated real constant 54.3%

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

   if 0.997049987 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right) \]

   10. Applied rewrites 79.9%

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

Alternative 13: 86.1% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.01600000075995922:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right)\\ \end{array} \]

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 (<= (sin (* (* 2.0 PI) u2)) 0.01600000075995922)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (*
   (sqrt u1)
   (fma
    6.2831854820251465
    u2
    (*
     u2
     (*
      (* u2 u2)
      (fma (* u2 u2) 81.6052606304404 -41.341705691712875)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (sinf(((2.0f * ((float) M_PI)) * u2)) <= 0.01600000075995922f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * fmaf(6.2831854820251465f, u2, (u2 * ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.01600000075995922))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * fma(Float32(6.2831854820251465), u2, Float32(u2 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (sin.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.0160000008

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 81.2%

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

   if 0.0160000008 < (sin.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + {u2}^{2} \cdot \left(81.6052606304404 \cdot {u2}^{2} - 41.341705691712875\right)\right)\right) \]

   10. Applied rewrites 71.8%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(6.2831854820251465, u2, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 86.1% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.01600000075995922:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right)\\ \end{array} \]

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 (<= (sin (* (* 2.0 PI) u2)) 0.01600000075995922)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (*
   (sqrt u1)
   (*
    u2
    (+
     6.2831854820251465
     (*
      (* u2 u2)
      (fma (* u2 u2) 81.6052606304404 -41.341705691712875)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (sinf(((2.0f * ((float) M_PI)) * u2)) <= 0.01600000075995922f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * (6.2831854820251465f + ((u2 * u2) * fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.01600000075995922))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(6.2831854820251465) + Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (sin.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.0160000008

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 81.2%

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

   if 0.0160000008 < (sin.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + {u2}^{2} \cdot \left(81.6052606304404 \cdot {u2}^{2} - 41.341705691712875\right)\right)\right) \]

   10. Applied rewrites 71.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 86.1% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.01600000075995922:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right), u2 \cdot u2, 6.2831854820251465\right)\right)\\ \end{array} \]

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 (<= (sin (* (* 2.0 PI) u2)) 0.01600000075995922)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (*
   (sqrt u1)
   (*
    u2
    (fma
     (fma (* u2 u2) 81.6052606304404 -41.341705691712875)
     (* u2 u2)
     6.2831854820251465)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (sinf(((2.0f * ((float) M_PI)) * u2)) <= 0.01600000075995922f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf(fmaf((u2 * u2), 81.6052606304404f, -41.341705691712875f), (u2 * u2), 6.2831854820251465f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.01600000075995922))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(fma(Float32(u2 * u2), Float32(81.6052606304404), Float32(-41.341705691712875)), Float32(u2 * u2), Float32(6.2831854820251465))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (sin.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.0160000008

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 81.2%

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

   if 0.0160000008 < (sin.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + {u2}^{2} \cdot \left(81.6052606304404 \cdot {u2}^{2} - 41.341705691712875\right)\right)\right) \]

   10. Applied rewrites 71.8%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, 81.6052606304404, -41.341705691712875\right), u2 \cdot u2, 6.2831854820251465\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 84.5% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.01600000075995922:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.01600000075995922)
  (* 6.2831854820251465 (* u2 (sqrt (- (log1p (- u1))))))
  (* (sqrt u1) (* u2 (fma (* u2 u2) -41.34170150756836 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.01600000075995922f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-log1pf(-u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((u2 * u2), -41.34170150756836f, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.01600000075995922))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log1p(Float32(-u1))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(u2 * u2), Float32(-41.34170150756836), Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0160000008

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 81.2%

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

   if 0.0160000008 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 70.1%

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

Alternative 17: 80.7% 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

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9994000196456909:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -41.34170150756836, \pi + \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (- 1.0 u1) 0.9994000196456909)
  (* 6.2831854820251465 (* u2 (sqrt (- (log (- 1.0 u1))))))
  (* (sqrt u1) (* u2 (fma (* u2 u2) -41.34170150756836 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9994000196456909f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-logf((1.0f - u1))));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((u2 * u2), -41.34170150756836f, (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9994000196456909))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(u2 * u2), Float32(-41.34170150756836), Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99940002

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

   5. Applied rewrites 51.3%

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

   if 0.99940002 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 58.1%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 54.3%

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

   6. Applied rewrites 54.3%

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

   7. Evaluated real constant 54.3%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 70.1%

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

Alternative 18: 80.4% accurate, 2.2× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

   5. Applied rewrites 51.3%

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

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

   1. Initial program 58.1%

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

   2. Evaluated real constant 58.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 73.7%

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

   10. Applied rewrites 73.7%

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

Alternative 19: 73.7% 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

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

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

5. Applied rewrites 51.3%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 73.7%

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

10. Applied rewrites 73.7%

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

Alternative 20: 65.7% accurate, 6.2× speedup

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

3. Taylor expanded in u1 around 0

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

5. Applied rewrites 76.3%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + {u2}^{2} \cdot \left(\frac{26482421150709471190661161091833125}{324518553658426726783156020576256} \cdot {u2}^{2} - \frac{762619864465648886625}{18446744073709551616}\right)\right)\right) \]

8. Applied rewrites 71.8%

   \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(6.2831854820251465 + {u2}^{2} \cdot \left(81.6052606304404 \cdot {u2}^{2} - 41.341705691712875\right)\right)\right) \]

9. Taylor expanded in u2 around 0

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

11. Applied rewrites 65.7%

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

Alternative 21: 65.7% accurate, 6.2× speedup

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Evaluated real constant 58.1%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

5. Applied rewrites 51.3%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 65.7%

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

Reproduce

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