Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 10.0s

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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

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(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.2%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 97.8%

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

   6. pow2 97.8%

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

   7. metadata-eval 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot \color{blue}{39.47841760436263}}\right) \]

4. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.005499999970197678:\\ \;\;\;\;u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.005499999970197678)
   (* u2 (/ (sqrt (* u1 39.47841760436263)) (sqrt (- 1.0 u1))))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.005499999970197678f) {
		tmp = u2 * (sqrtf((u1 * 39.47841760436263f)) / sqrtf((1.0f - u1)));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.005499999970197678e0) then
        tmp = u2 * (sqrt((u1 * 39.47841760436263e0)) / sqrt((1.0e0 - u1)))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.005499999970197678))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 * Float32(39.47841760436263))) / sqrt(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * 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 ((u2 * single(6.28318530718)) <= single(0.005499999970197678))
		tmp = u2 * (sqrt((u1 * single(39.47841760436263))) / sqrt((single(1.0) - u1)));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00549999997

   1. Initial program 98.4%

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

      1. add-sqr-sqrt 97.8%

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

      2. sqrt-unprod 98.4%

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

      3. swap-sqr 98.3%

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

      4. add-sqr-sqrt 98.5%

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

      5. pow2 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. associate-*l/ 98.7%

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

      2. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in u2 around 0 98.0%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 98.0%

         \[\leadsto \color{blue}{1 \cdot \sqrt{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]

      2. associate-*r/ 97.9%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{39.47841760436263 \cdot \left(u1 \cdot {u2}^{2}\right)}{1 - u1}}} \]

      3. sqrt-div 97.7%

         \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{39.47841760436263 \cdot \left(u1 \cdot {u2}^{2}\right)}}{\sqrt{1 - u1}}} \]

   9. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{1 \cdot \frac{u2 \cdot \sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 97.6%

          \[\leadsto \color{blue}{\frac{u2 \cdot \sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]

       2. associate-/l* 97.6%

          \[\leadsto \color{blue}{u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]

   11. Simplified 97.6%

       \[\leadsto \color{blue}{u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]

   if 0.00549999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0 83.2%

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

      1. +-commutative 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.005499999970197678:\\ \;\;\;\;u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009999999776482582)
   (* u2 (/ (sqrt (* u1 39.47841760436263)) (sqrt (- 1.0 u1))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009999999776482582f) {
		tmp = u2 * (sqrtf((u1 * 39.47841760436263f)) / sqrtf((1.0f - u1)));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.009999999776482582e0) then
        tmp = u2 * (sqrt((u1 * 39.47841760436263e0)) / sqrt((1.0e0 - u1)))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.009999999776482582))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 * Float32(39.47841760436263))) / sqrt(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.009999999776482582))
		tmp = u2 * (sqrt((u1 * single(39.47841760436263))) / sqrt((single(1.0) - u1)));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 98.4%

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

      1. add-sqr-sqrt 97.7%

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

      2. sqrt-unprod 98.4%

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

      3. swap-sqr 98.3%

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

      4. add-sqr-sqrt 98.5%

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

      5. pow2 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. associate-*l/ 98.7%

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

      2. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in u2 around 0 96.8%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 96.8%

         \[\leadsto \color{blue}{1 \cdot \sqrt{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]

      2. associate-*r/ 96.7%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{39.47841760436263 \cdot \left(u1 \cdot {u2}^{2}\right)}{1 - u1}}} \]

      3. sqrt-div 96.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{39.47841760436263 \cdot \left(u1 \cdot {u2}^{2}\right)}}{\sqrt{1 - u1}}} \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{1 \cdot \frac{u2 \cdot \sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 96.4%

          \[\leadsto \color{blue}{\frac{u2 \cdot \sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]

       2. associate-/l* 96.4%

          \[\leadsto \color{blue}{u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]

   11. Simplified 96.4%

       \[\leadsto \color{blue}{u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}} \]

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

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

   3. Taylor expanded in u1 around 0 69.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \frac{\sqrt{u1 \cdot 39.47841760436263}}{\sqrt{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009999999776482582)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (+ (- 2.0 u1) -1.0)))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009999999776482582f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / ((2.0f - u1) + -1.0f))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.009999999776482582e0) then
        tmp = 6.28318530718e0 * (u2 * sqrt((u1 / ((2.0e0 - u1) + (-1.0e0)))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.009999999776482582))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0))))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.009999999776482582))
		tmp = single(6.28318530718) * (u2 * sqrt((u1 / ((single(2.0) - u1) + single(-1.0)))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 96.2%

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

      1. expm1-log1p-u 96.1%

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

   5. Applied egg-rr 96.1%

      \[\leadsto 6.28318530718 \cdot \left(\sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot u2\right) \]
   6. Step-by-step derivation

      1. expm1-undefine 96.1%

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

      2. sub-neg 96.1%

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

      3. log1p-undefine 96.2%

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

      4. rem-exp-log 96.2%

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

      5. associate-+r- 96.2%

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

      6. metadata-eval 96.2%

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

      7. metadata-eval 96.2%

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

   7. Simplified 96.2%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

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

   3. Taylor expanded in u1 around 0 69.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. clear-num 98.1%

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

   2. associate-/r/ 98.1%

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

4. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. clear-num 98.1%

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

   4. sqrt-div 98.0%

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

   5. metadata-eval 98.0%

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

   6. associate-*l/ 98.1%

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

   7. *-un-lft-identity 98.1%

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

   8. clear-num 98.3%

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

   9. div-sub 98.1%

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

   10. *-inverses 98.1%

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

   11. sub-neg 98.1%

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

   12. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in u1 around 0 98.3%

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

   1. neg-mul-1 98.3%

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

   2. sub-neg 98.3%

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

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Final simplification 98.2%

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

Alternative 7: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 = 6.28318530718e0 * (u2 * sqrt((u1 / ((2.0e0 - u1) + (-1.0e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 81.7%

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

   1. expm1-log1p-u 81.7%

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

5. Applied egg-rr 81.7%

   \[\leadsto 6.28318530718 \cdot \left(\sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot u2\right) \]
6. Step-by-step derivation

   1. expm1-undefine 81.7%

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

   2. sub-neg 81.7%

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

   3. log1p-undefine 81.7%

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

   4. rem-exp-log 81.7%

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

   5. associate-+r- 81.7%

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

   6. metadata-eval 81.7%

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

   7. metadata-eval 81.7%

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

7. Simplified 81.7%

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

8. Final simplification 81.7%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\right) \]
9. Add Preprocessing

Alternative 8: 81.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 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)
    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(6.28318530718) * Float32(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 98.2%

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

3. Taylor expanded in u2 around 0 81.7%

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

Alternative 9: 73.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 81.7%

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

4. Taylor expanded in u1 around 0 72.7%

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

   1. +-commutative 84.9%

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

6. Simplified 72.7%

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

7. Final simplification 72.7%

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

Alternative 10: 65.1% 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. Add Preprocessing

3. Taylor expanded in u2 around 0 81.7%

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

4. Taylor expanded in u1 around 0 65.8%

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

5. Final simplification 65.8%

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

Alternative 11: -0.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{-39.47841760436263} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (sqrt -39.47841760436263)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf(-39.47841760436263f);
}

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 = u2 * sqrt((-39.47841760436263e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * sqrt(Float32(-39.47841760436263)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * sqrt(single(-39.47841760436263));
end

Derivation

1. Initial program 98.2%

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

   1. add-sqr-sqrt 94.2%

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

   2. sqrt-unprod 95.0%

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

   3. swap-sqr 94.7%

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

   4. add-sqr-sqrt 95.0%

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

   5. pow2 95.0%

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

4. Applied egg-rr 95.0%

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

   1. associate-*l/ 95.2%

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

   2. associate-/l* 95.0%

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

6. Simplified 95.0%

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

7. Taylor expanded in u2 around 0 82.1%

   \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]

8. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto \color{blue}{u2 \cdot \sqrt{-39.47841760436263}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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))))