Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 13.3s

Alternatives: 11

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

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (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)
    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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (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)
    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: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1 + 1}{\frac{1 - {u1}^{2}}{u1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ (+ u1 1.0) (/ (- 1.0 (pow u1 2.0)) u1)))
  (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 + 1.0f) / ((1.0f - powf(u1, 2.0f)) / u1))) * sinf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((u1 + 1.0e0) / ((1.0e0 - (u1 ** 2.0e0)) / u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(u1 + Float32(1.0)) / Float32(Float32(Float32(1.0) - (u1 ^ Float32(2.0))) / u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 + single(1.0)) / ((single(1.0) - (u1 ^ single(2.0))) / u1))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.2%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. pow2 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{{u1}^{2}}} \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. +-commutative 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

      \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right) \cdot \frac{u1}{1 - {u1}^{2}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. clear-num 98.4%

      \[\leadsto \sqrt{\left(u1 + 1\right) \cdot \color{blue}{\frac{1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. un-div-inv 98.4%

      \[\leadsto \sqrt{\color{blue}{\frac{u1 + 1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\color{blue}{\frac{u1 + 1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Final simplification 98.4%

   \[\leadsto \sqrt{\frac{u1 + 1}{\frac{1 - {u1}^{2}}{u1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\left(u1 + 1\right) \cdot \frac{u1}{1 - {u1}^{2}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* 6.28318530718 u2))
  (sqrt (* (+ u1 1.0) (/ u1 (- 1.0 (pow u1 2.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf(((u1 + 1.0f) * (u1 / (1.0f - powf(u1, 2.0f)))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((6.28318530718e0 * u2)) * sqrt(((u1 + 1.0e0) * (u1 / (1.0e0 - (u1 ** 2.0e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(Float32(u1 + Float32(1.0)) * Float32(u1 / Float32(Float32(1.0) - (u1 ^ Float32(2.0)))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) * sqrt(((u1 + single(1.0)) * (u1 / (single(1.0) - (u1 ^ single(2.0))))));
end

Derivation

1. Initial program 98.2%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. pow2 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{{u1}^{2}}} \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. +-commutative 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Final simplification 98.4%

   \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\left(u1 + 1\right) \cdot \frac{u1}{1 - {u1}^{2}}} \]

Alternative 3: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.002964999992400408:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.002964999992400408)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1)))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.002964999992400408f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.002964999992400408e0) then
        tmp = 6.28318530718e0 * (u2 * sqrt((u1 / (1.0e0 - u1))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.002964999992400408))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.002964999992400408))
		tmp = single(6.28318530718) * (u2 * sqrt((u1 / (single(1.0) - u1))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00296499999

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 97.8%

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

   if 0.00296499999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

      1. flip-- 98.2%

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

      2. associate-/r/ 98.2%

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

      3. metadata-eval 98.2%

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

      4. pow2 98.2%

         \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{{u1}^{2}}} \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      5. +-commutative 98.2%

         \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

         \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right) \cdot \frac{u1}{1 - {u1}^{2}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. clear-num 98.1%

         \[\leadsto \sqrt{\left(u1 + 1\right) \cdot \color{blue}{\frac{1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      3. un-div-inv 98.1%

         \[\leadsto \sqrt{\color{blue}{\frac{u1 + 1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. Applied egg-rr 98.1%

      \[\leadsto \sqrt{\color{blue}{\frac{u1 + 1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. Taylor expanded in u1 around 0 87.7%

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

      1. unpow2 87.7%

         \[\leadsto \sqrt{u1 + \color{blue}{u1 \cdot u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. distribute-rgt1-in 87.8%

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

      3. *-commutative 87.8%

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

   8. Simplified 87.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.002964999992400408:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{-1 - u1}{u1 + \frac{-1}{u1}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 6.28318530718 u2)) (sqrt (/ (- -1.0 u1) (+ u1 (/ -1.0 u1))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. pow2 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{{u1}^{2}}} \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. +-commutative 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

      \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right) \cdot \frac{u1}{1 - {u1}^{2}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. clear-num 98.4%

      \[\leadsto \sqrt{\left(u1 + 1\right) \cdot \color{blue}{\frac{1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. un-div-inv 98.4%

      \[\leadsto \sqrt{\color{blue}{\frac{u1 + 1}{\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Applied egg-rr 98.4%

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

   1. frac-2neg 98.4%

      \[\leadsto \sqrt{\color{blue}{\frac{-\left(u1 + 1\right)}{-\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. div-inv 98.4%

      \[\leadsto \sqrt{\color{blue}{\left(-\left(u1 + 1\right)\right) \cdot \frac{1}{-\frac{1 - {u1}^{2}}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. div-sub 98.3%

      \[\leadsto \sqrt{\left(-\left(u1 + 1\right)\right) \cdot \frac{1}{-\color{blue}{\left(\frac{1}{u1} - \frac{{u1}^{2}}{u1}\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. pow1 98.3%

      \[\leadsto \sqrt{\left(-\left(u1 + 1\right)\right) \cdot \frac{1}{-\left(\frac{1}{u1} - \frac{{u1}^{2}}{\color{blue}{{u1}^{1}}}\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. pow-div 98.3%

      \[\leadsto \sqrt{\left(-\left(u1 + 1\right)\right) \cdot \frac{1}{-\left(\frac{1}{u1} - \color{blue}{{u1}^{\left(2 - 1\right)}}\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. metadata-eval 98.3%

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

   7. pow1 98.3%

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

7. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.4%

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

   2. *-rgt-identity 98.4%

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

   3. neg-sub0 98.4%

      \[\leadsto \sqrt{\frac{\color{blue}{0 - \left(u1 + 1\right)}}{-\left(\frac{1}{u1} - u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. +-commutative 98.4%

      \[\leadsto \sqrt{\frac{0 - \color{blue}{\left(1 + u1\right)}}{-\left(\frac{1}{u1} - u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. associate--r+ 98.4%

      \[\leadsto \sqrt{\frac{\color{blue}{\left(0 - 1\right) - u1}}{-\left(\frac{1}{u1} - u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. metadata-eval 98.4%

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

   7. neg-sub0 98.4%

      \[\leadsto \sqrt{\frac{-1 - u1}{\color{blue}{0 - \left(\frac{1}{u1} - u1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. associate--r- 98.4%

      \[\leadsto \sqrt{\frac{-1 - u1}{\color{blue}{\left(0 - \frac{1}{u1}\right) + u1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. neg-sub0 98.4%

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

   10. distribute-neg-frac 98.4%

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

   11. metadata-eval 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(6.28318530718 \cdot u2\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 6.28318530718 u2)) (pow (+ -1.0 (/ 1.0 u1)) -0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * powf((-1.0f + (1.0f / u1)), -0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((6.28318530718e0 * u2)) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
end

Derivation

1. Initial program 98.2%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. pow2 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{{u1}^{2}}} \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. +-commutative 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

   2. expm1-udef 41.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} - 1} \]

5. Applied egg-rr 41.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 98.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}\right)\right)} \]

   2. expm1-log1p 98.3%

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

   3. div-sub 98.3%

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

   4. sub-neg 98.3%

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

   5. *-inverses 98.3%

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

   6. metadata-eval 98.3%

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

7. Simplified 98.3%

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

   1. div-inv 98.2%

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

   2. pow1/2 98.2%

      \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot \frac{1}{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{0.5}}} \]

   3. pow-flip 98.3%

      \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(-0.5\right)}} \]

   4. metadata-eval 98.3%

      \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \]

9. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]

10. Final simplification 98.3%

    \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \]

Alternative 6: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005249999929219484:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005249999929219484)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.005249999929219484f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.005249999929219484e0) then
        tmp = 6.28318530718e0 * (u2 * sqrt((u1 / (1.0e0 - u1))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.005249999929219484))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.005249999929219484))
		tmp = single(6.28318530718) * (u2 * sqrt((u1 / (single(1.0) - u1))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00524999993

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 97.3%

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

   if 0.00524999993 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

   2. Taylor expanded in u1 around 0 74.4%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005249999929219484:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 6.28318530718 u2)) (sqrt (/ u1 (- 1.0 u1)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Final simplification 98.2%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{-1 + \frac{1}{u1}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ (sin (* 6.28318530718 u2)) (sqrt (+ -1.0 (/ 1.0 u1)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. pow2 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{{u1}^{2}}} \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. +-commutative 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

   2. expm1-udef 41.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} - 1} \]

5. Applied egg-rr 41.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 98.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}\right)\right)} \]

   2. expm1-log1p 98.3%

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

   3. div-sub 98.3%

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

   4. sub-neg 98.3%

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

   5. *-inverses 98.3%

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

   6. metadata-eval 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 9: 86.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot 1.0471975511966667 + 0.15915494309188485 \cdot \frac{1}{u2}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.002964999992400408:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{-0.5 \cdot \left(u1 \cdot t_0\right) + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (+ (* u2 1.0471975511966667) (* 0.15915494309188485 (/ 1.0 u2)))))
   (if (<= (* 6.28318530718 u2) 0.002964999992400408)
     (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1)))))
     (/ (sqrt u1) (+ (* -0.5 (* u1 t_0)) t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (u2 * 1.0471975511966667f) + (0.15915494309188485f * (1.0f / u2));
	float tmp;
	if ((6.28318530718f * u2) <= 0.002964999992400408f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sqrtf(u1) / ((-0.5f * (u1 * t_0)) + t_0);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (u2 * 1.0471975511966667e0) + (0.15915494309188485e0 * (1.0e0 / u2))
    if ((6.28318530718e0 * u2) <= 0.002964999992400408e0) then
        tmp = 6.28318530718e0 * (u2 * sqrt((u1 / (1.0e0 - u1))))
    else
        tmp = sqrt(u1) / (((-0.5e0) * (u1 * t_0)) + t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(u2 * Float32(1.0471975511966667)) + Float32(Float32(0.15915494309188485) * Float32(Float32(1.0) / u2)))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.002964999992400408))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sqrt(u1) / Float32(Float32(Float32(-0.5) * Float32(u1 * t_0)) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = (u2 * single(1.0471975511966667)) + (single(0.15915494309188485) * (single(1.0) / u2));
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.002964999992400408))
		tmp = single(6.28318530718) * (u2 * sqrt((u1 / (single(1.0) - u1))));
	else
		tmp = sqrt(u1) / ((single(-0.5) * (u1 * t_0)) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00296499999

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 97.8%

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

   if 0.00296499999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

      1. flip-- 98.2%

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

      2. associate-/r/ 98.2%

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

      3. metadata-eval 98.2%

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

      4. pow2 98.2%

         \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{{u1}^{2}}} \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      5. +-commutative 98.2%

         \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. Applied egg-rr 98.2%

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

      1. add-sqr-sqrt 97.2%

         \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. associate-*l* 97.3%

         \[\leadsto \color{blue}{\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

      3. pow1/2 97.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)\right)}^{0.5}}} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      4. sqrt-pow1 97.5%

         \[\leadsto \color{blue}{{\left(\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      5. associate-/r/ 97.6%

         \[\leadsto {\color{blue}{\left(\frac{u1}{\frac{1 - {u1}^{2}}{u1 + 1}}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      6. metadata-eval 97.6%

         \[\leadsto {\left(\frac{u1}{\frac{\color{blue}{1 \cdot 1} - {u1}^{2}}{u1 + 1}}\right)}^{\left(\frac{0.5}{2}\right)} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      7. unpow2 97.6%

         \[\leadsto {\left(\frac{u1}{\frac{1 \cdot 1 - \color{blue}{u1 \cdot u1}}{u1 + 1}}\right)}^{\left(\frac{0.5}{2}\right)} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      8. +-commutative 97.6%

         \[\leadsto {\left(\frac{u1}{\frac{1 \cdot 1 - u1 \cdot u1}{\color{blue}{1 + u1}}}\right)}^{\left(\frac{0.5}{2}\right)} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      9. flip-- 97.5%

         \[\leadsto {\left(\frac{u1}{\color{blue}{1 - u1}}\right)}^{\left(\frac{0.5}{2}\right)} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      10. sqrt-pow1 97.3%

          \[\leadsto \color{blue}{\sqrt{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

      11. pow1/2 97.3%

          \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{u1}{1 - u1}}}} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \]

   5. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-/l* 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in u2 around 0 71.8%

      \[\leadsto \frac{\sqrt{u1}}{\color{blue}{0.15915494309188485 \cdot \left(\frac{1}{u2} \cdot \sqrt{1 - u1}\right) + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{1 - u1}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 71.9%

         \[\leadsto \frac{\sqrt{u1}}{\color{blue}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{1 - u1}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{1 - u1}\right)} \]

      2. associate-*r* 71.9%

         \[\leadsto \frac{\sqrt{u1}}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{1 - u1} + \color{blue}{\left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{1 - u1}}} \]

      3. distribute-rgt-out 71.9%

         \[\leadsto \frac{\sqrt{u1}}{\color{blue}{\sqrt{1 - u1} \cdot \left(0.15915494309188485 \cdot \frac{1}{u2} + 1.0471975511966667 \cdot u2\right)}} \]

      4. associate-*r/ 72.0%

         \[\leadsto \frac{\sqrt{u1}}{\sqrt{1 - u1} \cdot \left(\color{blue}{\frac{0.15915494309188485 \cdot 1}{u2}} + 1.0471975511966667 \cdot u2\right)} \]

      5. metadata-eval 72.0%

         \[\leadsto \frac{\sqrt{u1}}{\sqrt{1 - u1} \cdot \left(\frac{\color{blue}{0.15915494309188485}}{u2} + 1.0471975511966667 \cdot u2\right)} \]

      6. *-commutative 72.0%

         \[\leadsto \frac{\sqrt{u1}}{\sqrt{1 - u1} \cdot \left(\frac{0.15915494309188485}{u2} + \color{blue}{u2 \cdot 1.0471975511966667}\right)} \]

   10. Simplified 72.0%

       \[\leadsto \frac{\sqrt{u1}}{\color{blue}{\sqrt{1 - u1} \cdot \left(\frac{0.15915494309188485}{u2} + u2 \cdot 1.0471975511966667\right)}} \]

   11. Taylor expanded in u1 around 0 68.4%

       \[\leadsto \frac{\sqrt{u1}}{\color{blue}{-0.5 \cdot \left(u1 \cdot \left(1.0471975511966667 \cdot u2 + 0.15915494309188485 \cdot \frac{1}{u2}\right)\right) + \left(0.15915494309188485 \cdot \frac{1}{u2} + 1.0471975511966667 \cdot u2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.002964999992400408:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{-0.5 \cdot \left(u1 \cdot \left(u2 \cdot 1.0471975511966667 + 0.15915494309188485 \cdot \frac{1}{u2}\right)\right) + \left(u2 \cdot 1.0471975511966667 + 0.15915494309188485 \cdot \frac{1}{u2}\right)}\\ \end{array} \]

Alternative 10: 81.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 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)
    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 98.2%

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

2. Taylor expanded in u2 around 0 80.8%

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

3. Final simplification 80.8%

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

Alternative 11: 64.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 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)
    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 98.2%

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

2. Taylor expanded in u2 around 0 80.8%

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

3. Taylor expanded in u1 around 0 66.5%

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

4. Final simplification 66.5%

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

Reproduce

herbie shell --seed 2023340 
(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))))