Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 14.5s

Alternatives: 9

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 98.4%

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

   1. add-sqr-sqrt 97.9%

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

   2. sqrt-unprod 98.4%

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

   3. *-commutative 98.4%

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

   4. *-commutative 98.4%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   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. Final simplification 98.4%

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

Alternative 2: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.010999999940395355)
   (* u2 (/ 6.28318530718 (sqrt (+ -1.0 (/ 1.0 u1)))))
   (/ (sin (* u2 6.28318530718)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.010999999940395355f) {
		tmp = u2 * (6.28318530718f / sqrtf((-1.0f + (1.0f / u1))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) / sqrtf((1.0f / 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) <= 0.010999999940395355e0) then
        tmp = u2 * (6.28318530718e0 / sqrt(((-1.0e0) + (1.0e0 / u1))))
    else
        tmp = sin((u2 * 6.28318530718e0)) / sqrt((1.0e0 / 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(0.010999999940395355))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) / sqrt(Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.010999999940395355))
		tmp = u2 * (single(6.28318530718) / sqrt((single(-1.0) + (single(1.0) / u1))));
	else
		tmp = sin((u2 * single(6.28318530718))) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0109999999

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. sqrt-div 96.5%

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

      3. associate-*r/ 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-/l* 96.5%

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

   7. Simplified 96.5%

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

      1. associate-*r/ 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-/l* 96.4%

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

      4. sqrt-undiv 96.7%

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

   9. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\frac{u2}{\frac{\sqrt{\frac{1 - u1}{u1}}}{6.28318530718}}} \]
   10. Step-by-step derivation

       1. associate-/l* 96.5%

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

       2. *-commutative 96.5%

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

       3. *-rgt-identity 96.5%

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

       4. times-frac 96.8%

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

       5. /-rgt-identity 96.8%

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

       6. div-sub 96.8%

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

       7. *-inverses 96.8%

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

       8. sub-neg 96.8%

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

       9. metadata-eval 96.8%

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

       10. +-commutative 96.8%

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

   11. Simplified 96.8%

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

   if 0.0109999999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

      1. add-sqr-sqrt 97.4%

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

      2. sqrt-unprod 97.8%

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

      3. *-commutative 97.8%

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

      4. *-commutative 97.8%

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

      5. swap-sqr 97.6%

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

      6. pow2 97.6%

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

      7. metadata-eval 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. sqrt-prod 97.1%

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

      3. metadata-eval 97.8%

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

      4. unpow2 97.8%

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

      5. sqrt-prod 97.2%

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

      6. add-sqr-sqrt 97.8%

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

      7. sqrt-div 97.4%

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

      8. clear-num 97.2%

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

      9. associate-/r/ 97.5%

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

      10. sqrt-undiv 97.6%

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

      11. *-commutative 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. associate-/r/ 97.3%

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

      2. associate-*l/ 97.8%

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

      3. *-lft-identity 97.8%

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

      4. div-sub 97.8%

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

      5. sub-neg 97.8%

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

      6. *-inverses 97.8%

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

      7. metadata-eval 97.8%

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

   8. Simplified 97.8%

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

      1. add-sqr-sqrt 97.2%

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

      2. pow2 97.2%

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

      3. pow1/2 97.2%

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

      4. sqrt-pow1 97.4%

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

      5. metadata-eval 97.4%

         \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{{\left({\left(\frac{1}{u1} + -1\right)}^{\color{blue}{0.25}}\right)}^{2}} \]

   10. Applied egg-rr 97.4%

       \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\color{blue}{{\left({\left(\frac{1}{u1} + -1\right)}^{0.25}\right)}^{2}}} \]

   11. Taylor expanded in u1 around 0 73.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\sin \left(u2 \cdot 6.28318530718\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.010999999940395355)
   (* u2 (/ 6.28318530718 (sqrt (+ -1.0 (/ 1.0 u1)))))
   (/ (sqrt u1) (/ 1.0 (sin (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.010999999940395355f) {
		tmp = u2 * (6.28318530718f / sqrtf((-1.0f + (1.0f / u1))));
	} else {
		tmp = sqrtf(u1) / (1.0f / sinf((u2 * 6.28318530718f)));
	}
	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) <= 0.010999999940395355e0) then
        tmp = u2 * (6.28318530718e0 / sqrt(((-1.0e0) + (1.0e0 / u1))))
    else
        tmp = sqrt(u1) / (1.0e0 / sin((u2 * 6.28318530718e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.010999999940395355))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) / sqrt(Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))));
	else
		tmp = Float32(sqrt(u1) / Float32(Float32(1.0) / sin(Float32(u2 * Float32(6.28318530718)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.010999999940395355))
		tmp = u2 * (single(6.28318530718) / sqrt((single(-1.0) + (single(1.0) / u1))));
	else
		tmp = sqrt(u1) / (single(1.0) / sin((u2 * single(6.28318530718))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0109999999

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. sqrt-div 96.5%

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

      3. associate-*r/ 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-/l* 96.5%

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

   7. Simplified 96.5%

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

      1. associate-*r/ 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-/l* 96.4%

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

      4. sqrt-undiv 96.7%

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

   9. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\frac{u2}{\frac{\sqrt{\frac{1 - u1}{u1}}}{6.28318530718}}} \]
   10. Step-by-step derivation

       1. associate-/l* 96.5%

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

       2. *-commutative 96.5%

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

       3. *-rgt-identity 96.5%

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

       4. times-frac 96.8%

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

       5. /-rgt-identity 96.8%

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

       6. div-sub 96.8%

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

       7. *-inverses 96.8%

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

       8. sub-neg 96.8%

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

       9. metadata-eval 96.8%

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

       10. +-commutative 96.8%

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

   11. Simplified 96.8%

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

   if 0.0109999999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

      1. add-sqr-sqrt 97.4%

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

      2. sqrt-unprod 97.8%

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

      3. *-commutative 97.8%

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

      4. *-commutative 97.8%

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

      5. swap-sqr 97.6%

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

      6. pow2 97.6%

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

      7. metadata-eval 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. sqrt-div 97.4%

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

      3. associate-*r/ 97.6%

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

      4. sqrt-prod 96.9%

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

      5. metadata-eval 97.8%

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

      6. unpow2 97.8%

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

      7. sqrt-prod 97.2%

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

      8. add-sqr-sqrt 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-/l* 97.7%

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

   8. Simplified 97.7%

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

   9. Taylor expanded in u1 around 0 73.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\sin \left(u2 \cdot 6.28318530718\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.010999999940395355)
   (* u2 (/ 6.28318530718 (sqrt (+ -1.0 (/ 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) <= 0.010999999940395355f) {
		tmp = u2 * (6.28318530718f / sqrtf((-1.0f + (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) <= 0.010999999940395355e0) then
        tmp = u2 * (6.28318530718e0 / sqrt(((-1.0e0) + (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(0.010999999940395355))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) / sqrt(Float32(Float32(-1.0) + 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(0.010999999940395355))
		tmp = u2 * (single(6.28318530718) / sqrt((single(-1.0) + (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 314159265359/50000000000 u2) < 0.0109999999

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. sqrt-div 96.5%

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

      3. associate-*r/ 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-/l* 96.5%

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

   7. Simplified 96.5%

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

      1. associate-*r/ 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-/l* 96.4%

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

      4. sqrt-undiv 96.7%

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

   9. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\frac{u2}{\frac{\sqrt{\frac{1 - u1}{u1}}}{6.28318530718}}} \]
   10. Step-by-step derivation

       1. associate-/l* 96.5%

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

       2. *-commutative 96.5%

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

       3. *-rgt-identity 96.5%

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

       4. times-frac 96.8%

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

       5. /-rgt-identity 96.8%

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

       6. div-sub 96.8%

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

       7. *-inverses 96.8%

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

       8. sub-neg 96.8%

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

       9. metadata-eval 96.8%

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

       10. +-commutative 96.8%

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

   11. Simplified 96.8%

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

   if 0.0109999999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 73.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 6: 81.6% 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 98.4%

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

3. Taylor expanded in u2 around 0 80.2%

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

4. Final simplification 80.2%

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

Alternative 7: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 97.9%

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

   2. sqrt-unprod 98.4%

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

   3. *-commutative 98.4%

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

   4. *-commutative 98.4%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   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. Taylor expanded in u2 around 0 80.2%

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

   1. *-commutative 80.2%

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

   2. *-commutative 80.2%

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

   3. associate-*r* 80.2%

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

   4. *-commutative 80.2%

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

7. Simplified 80.2%

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

8. Final simplification 80.2%

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

Alternative 8: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 80.2%

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

   1. *-commutative 80.2%

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

   2. sqrt-div 80.1%

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

   3. associate-*r/ 80.0%

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

5. Applied egg-rr 80.0%

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

   1. associate-/l* 80.1%

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

7. Simplified 80.1%

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

   1. associate-*r/ 80.0%

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

   2. *-commutative 80.0%

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

   3. associate-/l* 80.0%

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

   4. sqrt-undiv 80.2%

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

9. Applied egg-rr 80.2%

   \[\leadsto \color{blue}{\frac{u2}{\frac{\sqrt{\frac{1 - u1}{u1}}}{6.28318530718}}} \]
10. Step-by-step derivation

    1. associate-/l* 80.1%

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

    2. *-commutative 80.1%

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

    3. *-rgt-identity 80.1%

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

    4. times-frac 80.3%

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

    5. /-rgt-identity 80.3%

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

    6. div-sub 80.3%

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

    7. *-inverses 80.3%

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

    8. sub-neg 80.3%

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

    9. metadata-eval 80.3%

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

    10. +-commutative 80.3%

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

11. Simplified 80.3%

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

12. Final simplification 80.3%

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

Alternative 9: 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 98.4%

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

3. Taylor expanded in u2 around 0 80.2%

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

4. Taylor expanded in u1 around 0 65.6%

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

5. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2024039 
(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))))