Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 14.4s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((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 = cos((6.28318530718e0 * u2)) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(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 = cos((single(6.28318530718) * u2)) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
end

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. pow1/2 99.0%

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

   3. pow1/2 99.0%

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

   4. pow-prod-down 99.0%

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

   5. clear-num 98.8%

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

   6. clear-num 98.8%

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

   7. inv-pow 98.8%

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

   8. inv-pow 98.8%

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

   9. pow-prod-up 98.7%

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

   10. div-sub 98.8%

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

   11. inv-pow 98.8%

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

   12. pow1 98.8%

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

   13. pow1 98.8%

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

   14. pow-div 98.8%

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

   15. metadata-eval 98.8%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   16. metadata-eval 98.8%

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

   17. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
4. Step-by-step derivation

   1. exp-to-pow 96.5%

      \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. *-commutative 96.5%

      \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   3. log-pow 96.4%

      \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. associate-*r* 96.4%

      \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. metadata-eval 96.4%

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

   6. mul-1-neg 96.4%

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

   7. rec-exp 96.4%

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

   8. rem-exp-log 98.8%

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

   9. unpow-1 98.8%

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

   10. sub-neg 98.8%

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

   11. metadata-eval 98.8%

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

   12. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in u2 around inf 98.8%

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

   1. sub-neg 98.8%

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

   2. metadata-eval 98.8%

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

   3. +-commutative 98.8%

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

   4. unpow-1 98.8%

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

   5. metadata-eval 98.8%

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

   6. pow-sqr 99.0%

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

   7. rem-sqrt-square 99.0%

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

   8. rem-square-sqrt 98.0%

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

   9. fabs-sqr 98.0%

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

   10. rem-square-sqrt 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 2: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9779999852180481:\\ \;\;\;\;t_0 \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9779999852180481)
     (* t_0 (pow (/ 1.0 u1) -0.5))
     (*
      (pow (+ -1.0 (/ 1.0 u1)) -0.5)
      (+ 1.0 (* -19.739208802181317 (* u2 u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9779999852180481f) {
		tmp = t_0 * powf((1.0f / u1), -0.5f);
	} else {
		tmp = powf((-1.0f + (1.0f / u1)), -0.5f) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	}
	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 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9779999852180481e0) then
        tmp = t_0 * ((1.0e0 / u1) ** (-0.5e0))
    else
        tmp = (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9779999852180481))
		tmp = Float32(t_0 * (Float32(Float32(1.0) / u1) ^ Float32(-0.5)));
	else
		tmp = Float32((Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9779999852180481))
		tmp = t_0 * ((single(1.0) / u1) ^ single(-0.5));
	else
		tmp = ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5)) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.977999985

   1. Initial program 96.8%

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

      1. add-sqr-sqrt 96.8%

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

      2. pow1/2 96.8%

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

      3. pow1/2 96.8%

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

      4. pow-prod-down 96.8%

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

      5. clear-num 96.7%

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

      6. clear-num 96.6%

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

      7. inv-pow 96.6%

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

      8. inv-pow 96.6%

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

      9. pow-prod-up 96.7%

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

      10. div-sub 96.6%

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

      11. inv-pow 96.6%

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

      12. pow1 96.6%

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

      13. pow1 96.6%

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

      14. pow-div 96.6%

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

      15. metadata-eval 96.6%

          \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      16. metadata-eval 96.6%

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

      17. metadata-eval 96.6%

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

   3. Applied egg-rr 96.6%

      \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   4. Step-by-step derivation

      1. exp-to-pow 95.0%

         \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      2. *-commutative 95.0%

         \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      3. log-pow 94.9%

         \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      4. associate-*r* 94.9%

         \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      5. metadata-eval 94.9%

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

      6. mul-1-neg 94.9%

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

      7. rec-exp 95.0%

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

      8. rem-exp-log 96.4%

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

      9. unpow-1 96.4%

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

      10. sub-neg 96.4%

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

      11. metadata-eval 96.4%

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

      12. +-commutative 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in u2 around inf 96.4%

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

      1. sub-neg 96.4%

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

      2. metadata-eval 96.4%

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

      3. +-commutative 96.4%

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

      4. unpow-1 96.4%

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

      5. metadata-eval 96.4%

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

      6. pow-sqr 97.0%

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

      7. rem-sqrt-square 97.0%

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

      8. rem-square-sqrt 96.3%

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

      9. fabs-sqr 96.3%

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

      10. rem-square-sqrt 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in u1 around 0 79.2%

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

   if 0.977999985 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow1/2 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-prod-down 99.4%

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

      5. clear-num 99.2%

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

      6. clear-num 99.3%

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

      7. inv-pow 99.3%

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

      8. inv-pow 99.3%

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

      9. pow-prod-up 99.1%

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

      10. div-sub 99.2%

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

      11. inv-pow 99.2%

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

      12. pow1 99.2%

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

      13. pow1 99.2%

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

      14. pow-div 99.2%

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

      15. metadata-eval 99.2%

          \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   4. Step-by-step derivation

      1. exp-to-pow 96.7%

         \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      2. *-commutative 96.7%

         \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      3. log-pow 96.7%

         \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      4. associate-*r* 96.7%

         \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      5. metadata-eval 96.7%

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

      6. mul-1-neg 96.7%

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

      7. rec-exp 96.7%

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

      8. rem-exp-log 99.3%

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

      9. unpow-1 99.3%

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

      10. sub-neg 99.3%

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

      11. metadata-eval 99.3%

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

      12. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in u2 around 0 97.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 97.1%

         \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      2. distribute-rgt1-in 97.1%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. unpow2 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. sub-neg 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

      5. metadata-eval 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      6. +-commutative 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      7. unpow-1 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-1}}} \]

      8. metadata-eval 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      9. pow-sqr 97.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}}} \]

      10. rem-sqrt-square 97.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left|{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right|} \]

      11. rem-square-sqrt 96.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \left|\color{blue}{\sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}}}\right| \]

      12. fabs-sqr 96.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left(\sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}}\right)} \]

      13. rem-square-sqrt 97.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]

   8. Simplified 97.2%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9779999852180481:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 3: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9779999852180481:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9779999852180481)
     (* t_0 (sqrt u1))
     (*
      (pow (+ -1.0 (/ 1.0 u1)) -0.5)
      (+ 1.0 (* -19.739208802181317 (* u2 u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9779999852180481f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf((-1.0f + (1.0f / u1)), -0.5f) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	}
	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 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9779999852180481e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9779999852180481))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32((Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9779999852180481))
		tmp = t_0 * sqrt(u1);
	else
		tmp = ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5)) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.977999985

   1. Initial program 96.8%

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

   2. Taylor expanded in u1 around 0 79.2%

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

   if 0.977999985 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow1/2 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-prod-down 99.4%

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

      5. clear-num 99.2%

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

      6. clear-num 99.3%

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

      7. inv-pow 99.3%

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

      8. inv-pow 99.3%

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

      9. pow-prod-up 99.1%

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

      10. div-sub 99.2%

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

      11. inv-pow 99.2%

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

      12. pow1 99.2%

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

      13. pow1 99.2%

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

      14. pow-div 99.2%

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

      15. metadata-eval 99.2%

          \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   4. Step-by-step derivation

      1. exp-to-pow 96.7%

         \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      2. *-commutative 96.7%

         \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      3. log-pow 96.7%

         \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      4. associate-*r* 96.7%

         \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      5. metadata-eval 96.7%

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

      6. mul-1-neg 96.7%

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

      7. rec-exp 96.7%

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

      8. rem-exp-log 99.3%

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

      9. unpow-1 99.3%

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

      10. sub-neg 99.3%

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

      11. metadata-eval 99.3%

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

      12. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in u2 around 0 97.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 97.1%

         \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      2. distribute-rgt1-in 97.1%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. unpow2 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. sub-neg 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

      5. metadata-eval 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      6. +-commutative 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      7. unpow-1 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-1}}} \]

      8. metadata-eval 97.1%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      9. pow-sqr 97.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}}} \]

      10. rem-sqrt-square 97.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left|{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right|} \]

      11. rem-square-sqrt 96.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \left|\color{blue}{\sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}}}\right| \]

      12. fabs-sqr 96.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left(\sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}}\right)} \]

      13. rem-square-sqrt 97.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]

   8. Simplified 97.2%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9779999852180481:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Final simplification 99.0%

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

Alternative 5: 88.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (+ -1.0 (/ 1.0 u1)) -0.5) (+ 1.0 (* -19.739208802181317 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((-1.0f + (1.0f / u1)), -0.5f) * (1.0f + (-19.739208802181317f * (u2 * 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 = (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5)) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
end

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. pow1/2 99.0%

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

   3. pow1/2 99.0%

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

   4. pow-prod-down 99.0%

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

   5. clear-num 98.8%

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

   6. clear-num 98.8%

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

   7. inv-pow 98.8%

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

   8. inv-pow 98.8%

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

   9. pow-prod-up 98.7%

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

   10. div-sub 98.8%

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

   11. inv-pow 98.8%

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

   12. pow1 98.8%

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

   13. pow1 98.8%

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

   14. pow-div 98.8%

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

   15. metadata-eval 98.8%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   16. metadata-eval 98.8%

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

   17. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
4. Step-by-step derivation

   1. exp-to-pow 96.5%

      \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. *-commutative 96.5%

      \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   3. log-pow 96.4%

      \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. associate-*r* 96.4%

      \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. metadata-eval 96.4%

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

   6. mul-1-neg 96.4%

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

   7. rec-exp 96.4%

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

   8. rem-exp-log 98.8%

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

   9. unpow-1 98.8%

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

   10. sub-neg 98.8%

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

   11. metadata-eval 98.8%

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

   12. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in u2 around 0 87.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* 87.8%

      \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   2. distribute-rgt1-in 87.8%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   3. unpow2 87.8%

      \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

   4. sub-neg 87.8%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

   5. metadata-eval 87.8%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

   6. +-commutative 87.8%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

   7. unpow-1 87.8%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-1}}} \]

   8. metadata-eval 87.8%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   9. pow-sqr 87.9%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}}} \]

   10. rem-sqrt-square 87.9%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left|{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right|} \]

   11. rem-square-sqrt 87.1%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \left|\color{blue}{\sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}}}\right| \]

   12. fabs-sqr 87.1%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left(\sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}}\right)} \]

   13. rem-square-sqrt 87.9%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]

8. Simplified 87.9%

   \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]

9. Final simplification 87.9%

   \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

Alternative 6: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * 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))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 87.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]
3. Step-by-step derivation

   1. unpow2 67.8%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

4. Simplified 87.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

5. Final simplification 87.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

Alternative 7: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0008999999845400453:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0008999999845400453)
   (pow (+ -1.0 (/ 1.0 u1)) -0.5)
   (* (sqrt u1) (+ 1.0 (* -19.739208802181317 (* u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0008999999845400453f) {
		tmp = powf((-1.0f + (1.0f / u1)), -0.5f);
	} else {
		tmp = sqrtf(u1) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	}
	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 <= 0.0008999999845400453e0) then
        tmp = ((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)
    else
        tmp = sqrt(u1) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0008999999845400453))
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5);
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.0008999999845400453))
		tmp = (single(-1.0) + (single(1.0) / u1)) ^ single(-0.5);
	else
		tmp = sqrt(u1) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 8.99999985e-4

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. pow1/2 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-prod-down 99.5%

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

      5. clear-num 99.3%

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

      6. clear-num 99.4%

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

      7. inv-pow 99.4%

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

      8. inv-pow 99.4%

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

      9. pow-prod-up 99.2%

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

      10. div-sub 99.3%

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

      11. inv-pow 99.3%

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

      12. pow1 99.3%

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

      13. pow1 99.3%

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

      14. pow-div 99.3%

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

      15. metadata-eval 99.3%

          \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

      \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   4. Step-by-step derivation

      1. exp-to-pow 96.9%

         \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      2. *-commutative 96.9%

         \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      3. log-pow 96.8%

         \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      4. associate-*r* 96.8%

         \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      5. metadata-eval 96.8%

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

      6. mul-1-neg 96.8%

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

      7. rec-exp 96.8%

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

      8. rem-exp-log 99.4%

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

      9. unpow-1 99.4%

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

      10. sub-neg 99.4%

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

      11. metadata-eval 99.4%

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

      12. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in u2 around 0 97.5%

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

      1. sub-neg 97.5%

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

      2. metadata-eval 97.5%

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

      3. +-commutative 97.5%

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

      4. unpow-1 97.5%

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

      5. metadata-eval 97.5%

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

      6. pow-sqr 97.5%

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

      7. rem-sqrt-square 97.5%

         \[\leadsto \color{blue}{\left|{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right|} \]

      8. rem-square-sqrt 96.6%

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

      9. fabs-sqr 96.6%

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

      10. rem-square-sqrt 97.5%

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

   8. Simplified 97.5%

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

   if 8.99999985e-4 < u2

   1. Initial program 98.0%

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

   2. Taylor expanded in u1 around 0 77.5%

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

   3. Taylor expanded in u2 around 0 57.6%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 57.6%

         \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

   5. Simplified 57.6%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.0008999999845400453:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 8: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = ((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. pow1/2 99.0%

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

   3. pow1/2 99.0%

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

   4. pow-prod-down 99.0%

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

   5. clear-num 98.8%

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

   6. clear-num 98.8%

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

   7. inv-pow 98.8%

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

   8. inv-pow 98.8%

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

   9. pow-prod-up 98.7%

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

   10. div-sub 98.8%

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

   11. inv-pow 98.8%

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

   12. pow1 98.8%

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

   13. pow1 98.8%

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

   14. pow-div 98.8%

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

   15. metadata-eval 98.8%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   16. metadata-eval 98.8%

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

   17. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
4. Step-by-step derivation

   1. exp-to-pow 96.5%

      \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. *-commutative 96.5%

      \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   3. log-pow 96.4%

      \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. associate-*r* 96.4%

      \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. metadata-eval 96.4%

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

   6. mul-1-neg 96.4%

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

   7. rec-exp 96.4%

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

   8. rem-exp-log 98.8%

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

   9. unpow-1 98.8%

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

   10. sub-neg 98.8%

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

   11. metadata-eval 98.8%

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

   12. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in u2 around 0 78.4%

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

   1. sub-neg 78.4%

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

   2. metadata-eval 78.4%

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

   3. +-commutative 78.4%

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

   4. unpow-1 78.3%

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

   5. metadata-eval 78.3%

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

   6. pow-sqr 78.4%

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

   7. rem-sqrt-square 78.4%

      \[\leadsto \color{blue}{\left|{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right|} \]

   8. rem-square-sqrt 77.8%

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

   9. fabs-sqr 77.8%

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

   10. rem-square-sqrt 78.4%

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

8. Simplified 78.4%

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

9. Final simplification 78.4%

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

Alternative 9: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 78.4%

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

3. Final simplification 78.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

Alternative 10: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow (/ 1.0 u1) -0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((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 = (1.0e0 / u1) ** (-0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. pow1/2 99.0%

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

   3. pow1/2 99.0%

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

   4. pow-prod-down 99.0%

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

   5. clear-num 98.8%

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

   6. clear-num 98.8%

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

   7. inv-pow 98.8%

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

   8. inv-pow 98.8%

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

   9. pow-prod-up 98.7%

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

   10. div-sub 98.8%

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

   11. inv-pow 98.8%

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

   12. pow1 98.8%

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

   13. pow1 98.8%

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

   14. pow-div 98.8%

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

   15. metadata-eval 98.8%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   16. metadata-eval 98.8%

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

   17. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
4. Step-by-step derivation

   1. exp-to-pow 96.5%

      \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. *-commutative 96.5%

      \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   3. log-pow 96.4%

      \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. associate-*r* 96.4%

      \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. metadata-eval 96.4%

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

   6. mul-1-neg 96.4%

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

   7. rec-exp 96.4%

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

   8. rem-exp-log 98.8%

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

   9. unpow-1 98.8%

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

   10. sub-neg 98.8%

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

   11. metadata-eval 98.8%

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

   12. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in u2 around 0 78.4%

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

   1. sub-neg 78.4%

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

   2. metadata-eval 78.4%

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

   3. +-commutative 78.4%

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

   4. unpow-1 78.3%

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

   5. metadata-eval 78.3%

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

   6. pow-sqr 78.4%

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

   7. rem-sqrt-square 78.4%

      \[\leadsto \color{blue}{\left|{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right|} \]

   8. rem-square-sqrt 77.8%

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

   9. fabs-sqr 77.8%

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

   10. rem-square-sqrt 78.4%

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

8. Simplified 78.4%

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

9. Taylor expanded in u1 around 0 61.8%

   \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \]

10. Final simplification 61.8%

    \[\leadsto {\left(\frac{1}{u1}\right)}^{-0.5} \]

Alternative 11: 63.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 78.4%

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

3. Taylor expanded in u1 around 0 61.8%

   \[\leadsto \sqrt{\color{blue}{u1}} \]

4. Final simplification 61.8%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023282 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))