Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 10.2s

Alternatives: 8

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: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17759999632835388:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \frac{1}{{u1}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.17759999632835388)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2))))
   (* (cos (* 6.28318530718 u2)) (/ 1.0 (pow u1 -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.17759999632835388f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * (1.0f / powf(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) <= 0.17759999632835388e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    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(0.17759999632835388))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	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(0.17759999632835388))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	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) < 0.177599996

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. *-lft-identity 97.8%

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

      3. associate-*r* 97.8%

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

      4. distribute-rgt-out 97.7%

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

      5. unpow2 97.7%

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

   4. Simplified 97.7%

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

   if 0.177599996 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

      1. clear-num 98.2%

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

      2. sqrt-div 98.1%

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

      3. metadata-eval 98.1%

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

      4. div-sub 98.1%

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

      5. pow1 98.1%

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

      6. pow1 98.1%

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

      7. pow-div 98.1%

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

      8. metadata-eval 98.1%

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

      9. metadata-eval 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 78.9%

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

      3. pow1/2 78.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{1}{u1} - 1\right)}^{0.5}}}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      4. pow-flip 78.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{u1} - 1\right)}^{\left(-0.5\right)}}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      5. sub-neg 78.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} + \left(-1\right)\right)}}^{\left(-0.5\right)}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      6. metadata-eval 78.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\left(-0.5\right)}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      7. metadata-eval 78.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} - 1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   6. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      2. expm1-log1p 97.9%

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

      3. metadata-eval 97.9%

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

      4. *-inverses 97.9%

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

      5. sub-neg 97.9%

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

      6. div-sub 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in u1 around 0 74.1%

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

      1. add-sqr-sqrt 74.2%

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

      2. unpow-prod-down 74.3%

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

      3. inv-pow 74.3%

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

      4. sqrt-pow1 74.3%

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

      5. metadata-eval 74.3%

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

      6. inv-pow 74.3%

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

      7. sqrt-pow1 74.3%

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

      8. metadata-eval 74.3%

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

   10. Applied egg-rr 74.3%

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

       1. pow-sqr 74.5%

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

       2. metadata-eval 74.5%

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

       3. unpow-1 74.5%

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

   12. Simplified 74.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17759999632835388:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \frac{1}{{u1}^{-0.5}}\\ \end{array} \]

Alternative 3: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17759999632835388:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.17759999632835388)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2))))
   (/ (cos (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.17759999632835388f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	} else {
		tmp = cosf((6.28318530718f * u2)) / sqrtf((1.0f / u1));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.17759999632835388e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    else
        tmp = cos((6.28318530718e0 * u2)) / sqrt((1.0e0 / u1))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.17759999632835388))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.17759999632835388))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	else
		tmp = cos((single(6.28318530718) * u2)) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.177599996

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. *-lft-identity 97.8%

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

      3. associate-*r* 97.8%

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

      4. distribute-rgt-out 97.7%

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

      5. unpow2 97.7%

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

   4. Simplified 97.7%

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

   if 0.177599996 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

      1. clear-num 98.2%

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

      2. sqrt-div 98.1%

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

      3. metadata-eval 98.1%

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

      4. div-sub 98.1%

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

      5. pow1 98.1%

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

      6. pow1 98.1%

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

      7. pow-div 98.1%

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

      8. metadata-eval 98.1%

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

      9. metadata-eval 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. expm1-log1p-u 98.0%

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

      2. expm1-udef 78.4%

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

      3. associate-*l/ 78.4%

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

      4. *-un-lft-identity 78.4%

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

      5. *-commutative 78.4%

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

      6. sub-neg 78.4%

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

      7. metadata-eval 78.4%

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

   5. Applied egg-rr 78.4%

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

      1. expm1-def 98.2%

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

      2. expm1-log1p 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-inverses 98.2%

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

      5. sub-neg 98.2%

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

      6. div-sub 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in u1 around 0 74.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17759999632835388:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]

Alternative 4: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17759999632835388:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.17759999632835388)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2))))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.17759999632835388f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.17759999632835388e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.17759999632835388))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.17759999632835388))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.177599996

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. *-lft-identity 97.8%

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

      3. associate-*r* 97.8%

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

      4. distribute-rgt-out 97.7%

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

      5. unpow2 97.7%

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

   4. Simplified 97.7%

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

   if 0.177599996 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

   2. Taylor expanded in u1 around 0 74.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17759999632835388:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 89.2% 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 88.6%

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

   1. +-commutative 88.6%

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

   2. *-lft-identity 88.6%

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

   3. associate-*r* 88.6%

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

   4. distribute-rgt-out 88.5%

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

   5. unpow2 88.5%

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

4. Simplified 88.5%

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

5. Final simplification 88.5%

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

Alternative 6: 84.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0006799999973736703:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0006799999973736703)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (+ 1.0 (* -19.739208802181317 (* u2 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0006799999973736703f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = (1.0f + (-19.739208802181317f * (u2 * u2))) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if (u2 <= 0.0006799999973736703e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = (1.0e0 + ((-19.739208802181317e0) * (u2 * u2))) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0006799999973736703))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.0006799999973736703))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = (single(1.0) + (single(-19.739208802181317) * (u2 * u2))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 6.79999997e-4

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 98.1%

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

   if 6.79999997e-4 < u2

   1. Initial program 98.6%

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

      1. clear-num 98.6%

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

      2. sqrt-div 98.4%

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

      3. metadata-eval 98.4%

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

      4. div-sub 98.4%

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

      5. pow1 98.4%

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

      6. pow1 98.4%

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

      7. pow-div 98.4%

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

      8. metadata-eval 98.4%

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

      9. metadata-eval 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in u2 around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. distribute-lft1-in 69.1%

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

      3. *-commutative 69.1%

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

      4. unpow2 69.1%

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

      5. *-inverses 69.1%

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

      6. div-sub 69.1%

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

   6. Simplified 69.1%

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

   7. Taylor expanded in u1 around 0 59.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.0006799999973736703:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 7: 81.3% 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 79.6%

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

3. Final simplification 79.6%

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

Alternative 8: 63.9% 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 79.6%

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

3. Taylor expanded in u1 around 0 64.4%

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

4. Final simplification 64.4%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023258 
(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))))