Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 8.7s

Alternatives: 10

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: 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

Derivation

1. Initial program 99.1%

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

2. Final simplification 99.1%

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

Alternative 2: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.949999988079071:\\ \;\;\;\;\frac{t_0}{{u1}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.949999988079071)
     (/ t_0 (pow u1 -0.5))
     (*
      (+ 1.0 (* u2 (* u2 -19.739208802181317)))
      (pow (+ -1.0 (/ 1.0 u1)) -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.949999988079071f) {
		tmp = t_0 / powf(u1, -0.5f);
	} else {
		tmp = (1.0f + (u2 * (u2 * -19.739208802181317f))) * powf((-1.0f + (1.0f / u1)), -0.5f);
	}
	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 <= 0.949999988079071e0) then
        tmp = t_0 / (u1 ** (-0.5e0))
    else
        tmp = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
    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(0.949999988079071))
		tmp = Float32(t_0 / (u1 ^ Float32(-0.5)));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)));
	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(0.949999988079071))
		tmp = t_0 / (u1 ^ single(-0.5));
	else
		tmp = (single(1.0) + (u2 * (u2 * single(-19.739208802181317)))) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.949999988

   1. Initial program 97.5%

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

      1. *-commutative 97.5%

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

      2. sqrt-div 97.0%

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

      3. associate-*r/ 97.2%

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

   3. Applied egg-rr 97.2%

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

      1. associate-/l* 97.3%

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

   5. Simplified 97.3%

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

      1. clear-num 97.4%

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

      2. sqrt-div 97.4%

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

      3. pow1/2 97.4%

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

      4. pow-flip 97.6%

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

      5. metadata-eval 97.6%

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

   7. Applied egg-rr 97.6%

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

   8. Taylor expanded in u1 around 0 78.3%

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

   if 0.949999988 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.3%

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

      1. pow1/2 91.3%

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

      2. clear-num 91.3%

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

      3. inv-pow 91.2%

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

      4. pow-pow 91.4%

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

      5. div-sub 91.4%

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

      6. pow1 91.4%

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

      7. pow1 91.4%

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

      8. pow-div 91.4%

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

      9. metadata-eval 91.4%

         \[\leadsto {\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \]

      10. metadata-eval 91.4%

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

      11. metadata-eval 91.4%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in u2 around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. distribute-lft1-in 97.7%

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

      3. unpow2 97.7%

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

      4. associate-*r* 97.7%

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

      5. sub-neg 97.7%

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

      6. metadata-eval 97.7%

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

      7. unpow1/2 97.7%

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

      8. rem-exp-log 95.2%

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

      9. exp-neg 95.2%

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

      10. exp-prod 95.0%

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

      11. distribute-lft-neg-out 95.0%

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

      12. distribute-rgt-neg-in 95.0%

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

      13. metadata-eval 95.0%

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

      14. exp-to-pow 97.8%

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

      15. +-commutative 97.8%

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

   6. Simplified 97.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.949999988079071:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{{u1}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.949999988079071:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.949999988079071)
     (* t_0 (sqrt u1))
     (*
      (+ 1.0 (* u2 (* u2 -19.739208802181317)))
      (pow (+ -1.0 (/ 1.0 u1)) -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.949999988079071f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = (1.0f + (u2 * (u2 * -19.739208802181317f))) * powf((-1.0f + (1.0f / u1)), -0.5f);
	}
	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 <= 0.949999988079071e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
    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(0.949999988079071))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)));
	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(0.949999988079071))
		tmp = t_0 * sqrt(u1);
	else
		tmp = (single(1.0) + (u2 * (u2 * single(-19.739208802181317)))) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.949999988

   1. Initial program 97.5%

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

   2. Taylor expanded in u1 around 0 78.1%

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

   if 0.949999988 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.3%

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

      1. pow1/2 91.3%

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

      2. clear-num 91.3%

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

      3. inv-pow 91.2%

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

      4. pow-pow 91.4%

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

      5. div-sub 91.4%

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

      6. pow1 91.4%

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

      7. pow1 91.4%

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

      8. pow-div 91.4%

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

      9. metadata-eval 91.4%

         \[\leadsto {\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \]

      10. metadata-eval 91.4%

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

      11. metadata-eval 91.4%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in u2 around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. distribute-lft1-in 97.7%

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

      3. unpow2 97.7%

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

      4. associate-*r* 97.7%

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

      5. sub-neg 97.7%

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

      6. metadata-eval 97.7%

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

      7. unpow1/2 97.7%

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

      8. rem-exp-log 95.2%

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

      9. exp-neg 95.2%

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

      10. exp-prod 95.0%

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

      11. distribute-lft-neg-out 95.0%

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

      12. distribute-rgt-neg-in 95.0%

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

      13. metadata-eval 95.0%

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

      14. exp-to-pow 97.8%

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

      15. +-commutative 97.8%

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

   6. Simplified 97.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.949999988079071:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]

Alternative 4: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3199999928474426:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.3199999928474426)
   (* (+ 1.0 (* u2 (* u2 -19.739208802181317))) (pow (+ -1.0 (/ 1.0 u1)) -0.5))
   (* (cos (* 6.28318530718 u2)) (pow (/ 1.0 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.3199999928474426f) {
		tmp = (1.0f + (u2 * (u2 * -19.739208802181317f))) * powf((-1.0f + (1.0f / u1)), -0.5f);
	} else {
		tmp = cosf((6.28318530718f * u2)) * powf((1.0f / u1), -0.5f);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.3199999928474426e0) then
        tmp = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
    else
        tmp = cos((6.28318530718e0 * u2)) * ((1.0e0 / u1) ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.3199999928474426))
		tmp = Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * (Float32(Float32(1.0) / u1) ^ Float32(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.3199999928474426))
		tmp = (single(1.0) + (u2 * (u2 * single(-19.739208802181317)))) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
	else
		tmp = cos((single(6.28318530718) * u2)) * ((single(1.0) / u1) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.319999993

   1. Initial program 99.3%

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

      1. pow1/2 91.3%

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

      2. clear-num 91.3%

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

      3. inv-pow 91.2%

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

      4. pow-pow 91.4%

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

      5. div-sub 91.4%

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

      6. pow1 91.4%

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

      7. pow1 91.4%

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

      8. pow-div 91.4%

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

      9. metadata-eval 91.4%

         \[\leadsto {\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \]

      10. metadata-eval 91.4%

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

      11. metadata-eval 91.4%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in u2 around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. distribute-lft1-in 97.7%

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

      3. unpow2 97.7%

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

      4. associate-*r* 97.7%

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

      5. sub-neg 97.7%

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

      6. metadata-eval 97.7%

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

      7. unpow1/2 97.7%

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

      8. rem-exp-log 95.2%

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

      9. exp-neg 95.2%

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

      10. exp-prod 95.0%

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

      11. distribute-lft-neg-out 95.0%

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

      12. distribute-rgt-neg-in 95.0%

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

      13. metadata-eval 95.0%

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

      14. exp-to-pow 97.8%

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

      15. +-commutative 97.8%

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

   6. Simplified 97.8%

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

   if 0.319999993 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.5%

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

      1. pow1/2 26.1%

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

      2. clear-num 26.1%

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

      3. inv-pow 26.1%

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

      4. pow-pow 26.1%

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

      5. div-sub 26.1%

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

      6. pow1 26.1%

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

      7. pow1 26.1%

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

      8. pow-div 26.1%

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

      9. metadata-eval 26.1%

         \[\leadsto {\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \]

      10. metadata-eval 26.1%

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

      11. metadata-eval 26.1%

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

   3. Applied egg-rr 97.1%

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

   4. Taylor expanded in u1 around 0 78.4%

      \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3199999928474426:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]

Alternative 5: 88.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

   1. pow1/2 83.7%

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

   2. clear-num 83.6%

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

   3. inv-pow 83.6%

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

   4. pow-pow 83.7%

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

   5. div-sub 83.7%

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

   6. pow1 83.7%

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

   7. pow1 83.7%

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

   8. pow-div 83.7%

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

   9. metadata-eval 83.7%

      \[\leadsto {\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \]

   10. metadata-eval 83.7%

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

   11. metadata-eval 83.7%

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

3. Applied egg-rr 99.1%

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

4. Taylor expanded in u2 around 0 90.7%

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

   1. associate-*r* 90.7%

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

   2. distribute-lft1-in 90.6%

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

   3. unpow2 90.6%

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

   4. associate-*r* 90.6%

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

   5. sub-neg 90.6%

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

   6. metadata-eval 90.6%

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

   7. unpow1/2 90.6%

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

   8. rem-exp-log 88.4%

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

   9. exp-neg 88.4%

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

   10. exp-prod 88.2%

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

   11. distribute-lft-neg-out 88.2%

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

   12. distribute-rgt-neg-in 88.2%

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

   13. metadata-eval 88.2%

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

   14. exp-to-pow 90.7%

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

   15. +-commutative 90.7%

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

6. Simplified 90.7%

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

7. Final simplification 90.7%

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

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 99.1%

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

2. Taylor expanded in u2 around 0 90.8%

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

   1. +-commutative 90.8%

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

   2. *-lft-identity 90.8%

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

   3. associate-*r* 90.8%

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

   4. distribute-rgt-out 90.7%

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

   5. unpow2 90.7%

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

4. Simplified 90.7%

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

5. Final simplification 90.7%

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

Alternative 7: 88.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
}

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 + (u2 * (u2 * (-19.739208802181317e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. sqrt-div 98.8%

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

   3. associate-*r/ 98.8%

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

3. Applied egg-rr 98.8%

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

   1. associate-/l* 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in u2 around 0 90.8%

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

   1. +-commutative 90.8%

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

   2. *-lft-identity 90.8%

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

   3. associate-*r* 90.8%

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

   4. distribute-rgt-out 90.7%

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

   5. unpow2 90.7%

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

   6. associate-*r* 90.7%

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

8. Simplified 90.7%

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

9. Final simplification 90.7%

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

Alternative 8: 79.7% 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 99.1%

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

2. Taylor expanded in u2 around 0 83.7%

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

   1. pow1/2 83.7%

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

   2. clear-num 83.6%

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

   3. inv-pow 83.6%

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

   4. pow-pow 83.7%

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

   5. div-sub 83.7%

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

   6. pow1 83.7%

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

   7. pow1 83.7%

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

   8. pow-div 83.7%

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

   9. metadata-eval 83.7%

      \[\leadsto {\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \]

   10. metadata-eval 83.7%

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

   11. metadata-eval 83.7%

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

4. Applied egg-rr 83.7%

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

5. Final simplification 83.7%

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

Alternative 9: 79.8% 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 99.1%

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

2. Taylor expanded in u2 around 0 83.7%

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

3. Final simplification 83.7%

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

Alternative 10: 62.7% 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 99.1%

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

2. Taylor expanded in u2 around 0 83.7%

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

3. Taylor expanded in u1 around 0 65.4%

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

4. Final simplification 65.4%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023178 
(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))))