Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 5.0s

Alternatives: 15

Speedup: 1.0×

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

\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

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 (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

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 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Initial Program: 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

\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

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 (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

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 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * 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

\[\sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

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 (/ u1 (- 0.5 (- u1 0.5)))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (0.5f - (u1 - 0.5f)))) * sinf((6.28318530718f * u2));
}

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 / (0.5e0 - (u1 - 0.5e0)))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(0.5) - Float32(u1 - Float32(0.5))))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(0.5) - (u1 - single(0.5))))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Step-by-step derivation

3. Applied rewrites 98.3%

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
4. Add Preprocessing

Alternative 2: 96.0% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

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 (<= u2 0.019999999552965164)
  (*
   (sqrt (/ u1 (- 0.5 (- u1 0.5))))
   (fma u2 6.28318530718 (* (* (* u2 u2) -41.341702240407926) u2)))
  (* (sqrt (* u1 (+ 1.0 u1))) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.019999999552965164f) {
		tmp = sqrtf((u1 / (0.5f - (u1 - 0.5f)))) * fmaf(u2, 6.28318530718f, (((u2 * u2) * -41.341702240407926f) * u2));
	} else {
		tmp = sqrtf((u1 * (1.0f + u1))) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.019999999552965164))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(0.5) - Float32(u1 - Float32(0.5))))) * fma(u2, Float32(6.28318530718), Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + u1))) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0199999996

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 88.7%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.8%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right) \]

   if 0.0199999996 < u2

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 86.5%

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\frac{\sqrt{u1}}{u1}}\\ \end{array} \]

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 (<= u2 0.03999999910593033)
  (*
   (sqrt (/ u1 (- 0.5 (- u1 0.5))))
   (fma u2 6.28318530718 (* (* (* u2 u2) -41.341702240407926) u2)))
  (/ (sin (* u2 6.28318530718)) (/ (sqrt u1) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.03999999910593033f) {
		tmp = sqrtf((u1 / (0.5f - (u1 - 0.5f)))) * fmaf(u2, 6.28318530718f, (((u2 * u2) * -41.341702240407926f) * u2));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) / (sqrtf(u1) / u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(0.5) - Float32(u1 - Float32(0.5))))) * fma(u2, Float32(6.28318530718), Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) * u2)));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) / Float32(sqrt(u1) / u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0399999991

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 88.7%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.8%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right) \]

   if 0.0399999991 < u2

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 98.2%

      \[\leadsto \sqrt{\frac{u1}{\frac{1}{\frac{-1}{u1 - 1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. Applied rewrites 98.3%

      \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1 - u1}{u1}}} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\frac{\sqrt{u1}}{u1}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

      \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\frac{\sqrt{u1}}{u1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.0% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u1 \cdot \sin \left(6.28318530718 \cdot u2\right)}{\sqrt{u1}}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= u2 0.03999999910593033)
  (*
   (sqrt (/ u1 (- 0.5 (- u1 0.5))))
   (fma u2 6.28318530718 (* (* (* u2 u2) -41.341702240407926) u2)))
  (/ (* u1 (sin (* 6.28318530718 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.03999999910593033f) {
		tmp = sqrtf((u1 / (0.5f - (u1 - 0.5f)))) * fmaf(u2, 6.28318530718f, (((u2 * u2) * -41.341702240407926f) * u2));
	} else {
		tmp = (u1 * sinf((6.28318530718f * u2))) / sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(0.5) - Float32(u1 - Float32(0.5))))) * fma(u2, Float32(6.28318530718), Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) * u2)));
	else
		tmp = Float32(Float32(u1 * sin(Float32(Float32(6.28318530718) * u2))) / sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0399999991

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 88.7%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.8%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right) \]

   if 0.0399999991 < u2

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 98.2%

      \[\leadsto \sqrt{\frac{u1}{\frac{1}{\frac{-1}{u1 - 1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. Applied rewrites 98.3%

      \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1 - u1}{u1}}} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \frac{u1 \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{u1}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

      \[\leadsto \frac{u1 \cdot \sin \left(6.28318530718 \cdot u2\right)}{\sqrt{u1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.0% 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

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

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 (<= u2 0.03999999910593033)
  (*
   (sqrt (/ u1 (- 0.5 (- u1 0.5))))
   (fma u2 6.28318530718 (* (* (* u2 u2) -41.341702240407926) u2)))
  (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.03999999910593033f) {
		tmp = sqrtf((u1 / (0.5f - (u1 - 0.5f)))) * fmaf(u2, 6.28318530718f, (((u2 * u2) * -41.341702240407926f) * u2));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(0.5) - Float32(u1 - Float32(0.5))))) * fma(u2, Float32(6.28318530718), Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) * u2)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0399999991

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 88.7%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.8%

      \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right) \]

   if 0.0399999991 < u2

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 75.0%

      \[\leadsto \sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.8% 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

\[\sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right) \]

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 (/ u1 (- 0.5 (- u1 0.5))))
 (fma u2 6.28318530718 (* (* (* u2 u2) -41.341702240407926) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (0.5f - (u1 - 0.5f)))) * fmaf(u2, 6.28318530718f, (((u2 * u2) * -41.341702240407926f) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(0.5) - Float32(u1 - Float32(0.5))))) * fma(u2, Float32(6.28318530718), Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) * u2)))
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Step-by-step derivation

3. Applied rewrites 98.3%

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 88.7%

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 88.8%

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \mathsf{fma}\left(u2, 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) \cdot u2\right) \]
9. Add Preprocessing

Alternative 7: 88.7% 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

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \]

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 (/ u1 (- 1.0 u1)))
 (* (fma (* u2 u2) -41.341702240407926 6.28318530718) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (fmaf((u2 * u2), -41.341702240407926f, 6.28318530718f) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(fma(Float32(u2 * u2), Float32(-41.341702240407926), Float32(6.28318530718)) * u2))
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Step-by-step derivation

3. Applied rewrites 98.3%

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 88.7%

   \[\leadsto \sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 88.7%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \]
9. Add Preprocessing

Alternative 8: 80.9% accurate, 2.9× 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

\[\frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}} \]

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)))
  (/ 6.28318530718 (/ (sqrt (/ (- 1.0 u1) u1)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f / (sqrtf(((1.0f - u1) / u1)) / u2);
}

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 = 6.28318530718e0 / (sqrt(((1.0e0 - u1) / u1)) / u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) / Float32(sqrt(Float32(Float32(Float32(1.0) - u1) / u1)) / u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) / (sqrt(((single(1.0) - u1) / u1)) / u2);
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 80.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

6. Applied rewrites 80.8%

   \[\leadsto \frac{1}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2 \cdot 6.28318530718}} \]
7. Step-by-step derivation

8. Applied rewrites 80.9%

   \[\leadsto \frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}} \]
9. Add Preprocessing

Alternative 9: 80.9% accurate, 3.2× 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

\[\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (* u2 6.28318530718) (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * 6.28318530718f) * sqrtf((u1 / (1.0f - u1)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u2 * 6.28318530718e0) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) * sqrt((u1 / (single(1.0) - u1)));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 80.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
5. Step-by-step derivation

6. Applied rewrites 80.9%

   \[\leadsto \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
7. Add Preprocessing

Alternative 10: 80.9% accurate, 3.2× 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

\[\left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2 \]

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)))
  (* (* 6.28318530718 (sqrt (/ u1 (- 1.0 u1)))) u2))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * sqrtf((u1 / (1.0f - u1)))) * u2;
}

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 = (6.28318530718e0 * sqrt((u1 / (1.0e0 - u1)))) * u2
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))) * u2)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * sqrt((u1 / (single(1.0) - u1)))) * u2;
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 80.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
5. Step-by-step derivation

6. Applied rewrites 80.9%

   \[\leadsto \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2 \]
7. Add Preprocessing

Alternative 11: 80.9% accurate, 3.2× 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

\[6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

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)))
  (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - 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 = 6.28318530718e0 * (u2 * sqrt((u1 / (1.0e0 - u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt((u1 / (single(1.0) - u1))));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 80.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
5. Add Preprocessing

Alternative 12: 72.7% accurate, 3.4× 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

\[6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\right) \]

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)))
  (* 6.28318530718 (* u2 (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf((u1 * (1.0f + 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 = 6.28318530718e0 * (u2 * sqrt((u1 * (1.0e0 + u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 * Float32(Float32(1.0) + u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt((u1 * (single(1.0) + u1))));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 80.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

5. Taylor expanded in u1 around 0

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\right) \]
6. Step-by-step derivation

7. Applied rewrites 72.7%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\right) \]
8. Add Preprocessing

Alternative 13: 64.6% accurate, 5.5× 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

\[\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (* u2 6.28318530718) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * 6.28318530718f) * sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u2 * 6.28318530718e0) * sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) * sqrt(u1);
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 80.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

5. Taylor expanded in u1 around 0

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Step-by-step derivation

7. Applied rewrites 64.6%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
8. Step-by-step derivation

9. Applied rewrites 64.6%

   \[\leadsto \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1} \]
10. Add Preprocessing

Alternative 14: 64.6% accurate, 5.5× 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

\[6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]

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)))
  (* 6.28318530718 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (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 = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt(u1));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 80.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

5. Taylor expanded in u1 around 0

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Step-by-step derivation

7. Applied rewrites 64.6%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2026035 +o sampling:rival3
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz 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 (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))