Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.2s

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 99.2%

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

2. Final simplification 99.2%

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

Alternative 2: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.07999999821186066:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.07999999821186066)
   (*
    (+ 1.0 (* u2 (* u2 -19.739208802181317)))
    (sqrt (/ 1.0 (+ -1.0 (/ 1.0 u1)))))
   (* (cos (* 6.28318530718 u2)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.07999999821186066f) {
		tmp = (1.0f + (u2 * (u2 * -19.739208802181317f))) * sqrtf((1.0f / (-1.0f + (1.0f / u1))));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 + (u1 * 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) <= 0.07999999821186066e0) then
        tmp = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * sqrt((1.0e0 / ((-1.0e0) + (1.0e0 / u1))))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 + (u1 * 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(0.07999999821186066))
		tmp = Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))) * sqrt(Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.07999999821186066))
		tmp = (single(1.0) + (u2 * (u2 * single(-19.739208802181317)))) * sqrt((single(1.0) / (single(-1.0) + (single(1.0) / u1))));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0799999982

   1. Initial program 99.4%

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

      1. pow1/2 99.4%

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

      2. clear-num 99.4%

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

      3. inv-pow 99.4%

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

      4. pow-pow 99.5%

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

      5. div-sub 99.4%

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

      6. pow1 99.4%

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

      7. pow1 99.4%

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

      8. pow-div 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in u2 around 0 99.2%

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

      1. associate-*r* 99.2%

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

      2. distribute-lft1-in 99.2%

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

      3. *-commutative 99.2%

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

      4. unpow2 99.2%

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

      5. associate-*l* 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   if 0.0799999982 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.6%

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

   2. Taylor expanded in u1 around 0 90.3%

      \[\leadsto \sqrt{\color{blue}{{u1}^{2} + \left({u1}^{3} + u1\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   3. Step-by-step derivation

      1. unpow2 90.3%

         \[\leadsto \sqrt{\color{blue}{u1 \cdot u1} + \left({u1}^{3} + u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      2. unpow3 90.3%

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

      3. unpow2 90.3%

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

      4. distribute-lft1-in 90.3%

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

      5. distribute-rgt-in 90.1%

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

      6. associate-+l+ 90.1%

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

      7. +-commutative 90.1%

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

      8. unpow2 90.1%

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

      9. fma-udef 90.1%

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

      10. +-commutative 90.1%

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

      11. distribute-rgt-in 90.3%

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

      12. *-lft-identity 90.3%

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

   4. Simplified 90.3%

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

   5. Taylor expanded in u1 around 0 83.4%

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

      1. unpow2 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.07999999821186066:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0949999988079071:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0949999988079071)
   (*
    (+ 1.0 (* u2 (* u2 -19.739208802181317)))
    (sqrt (/ 1.0 (+ -1.0 (/ 1.0 u1)))))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0949999988079071f) {
		tmp = (1.0f + (u2 * (u2 * -19.739208802181317f))) * sqrtf((1.0f / (-1.0f + (1.0f / u1))));
	} 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) <= 0.0949999988079071e0) then
        tmp = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * sqrt((1.0e0 / ((-1.0e0) + (1.0e0 / u1))))
    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(0.0949999988079071))
		tmp = Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))) * sqrt(Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))));
	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(0.0949999988079071))
		tmp = (single(1.0) + (u2 * (u2 * single(-19.739208802181317)))) * sqrt((single(1.0) / (single(-1.0) + (single(1.0) / u1))));
	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) < 0.0949999988

   1. Initial program 99.5%

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

      1. pow1/2 99.5%

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

      2. clear-num 99.4%

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

      3. inv-pow 99.4%

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

      4. pow-pow 99.5%

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

      5. div-sub 99.4%

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

      6. pow1 99.4%

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

      7. pow1 99.4%

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

      8. pow-div 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in u2 around 0 99.1%

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

      1. associate-*r* 99.1%

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

      2. distribute-lft1-in 99.1%

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

      3. *-commutative 99.1%

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

      4. unpow2 99.1%

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

      5. associate-*l* 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

      8. +-commutative 99.1%

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

   6. Simplified 99.1%

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

   if 0.0949999988 < (*.f32 314159265359/50000000000 u2)

   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 68.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0949999988079071:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 87.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * sqrt((1.0e0 / ((-1.0e0) + (1.0e0 / u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. pow1/2 99.2%

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

   2. clear-num 99.1%

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

   3. inv-pow 99.1%

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

   4. pow-pow 99.1%

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

   5. div-sub 99.1%

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

   6. pow1 99.1%

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

   7. pow1 99.1%

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

   8. pow-div 99.1%

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

   9. metadata-eval 99.1%

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

   10. metadata-eval 99.1%

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

   11. metadata-eval 99.1%

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

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 90.2%

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

   1. associate-*r* 90.2%

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

   2. distribute-lft1-in 90.2%

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

   3. *-commutative 90.2%

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

   4. unpow2 90.2%

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

   5. associate-*l* 90.2%

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

   6. sub-neg 90.2%

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

   7. metadata-eval 90.2%

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

   8. +-commutative 90.2%

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

6. Simplified 90.2%

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

7. Final simplification 90.2%

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

Alternative 5: 79.6% 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 99.2%

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

   1. pow1/2 99.2%

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

   2. clear-num 99.1%

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

   3. inv-pow 99.1%

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

   4. pow-pow 99.1%

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

   5. div-sub 99.1%

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

   6. pow1 99.1%

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

   7. pow1 99.1%

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

   8. pow-div 99.1%

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

   9. metadata-eval 99.1%

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

   10. metadata-eval 99.1%

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

   11. metadata-eval 99.1%

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

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 84.0%

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

   1. inv-pow 84.0%

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

   2. sqrt-pow1 84.0%

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

   3. sub-neg 84.0%

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

   4. metadata-eval 84.0%

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

   5. metadata-eval 84.0%

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

6. Applied egg-rr 84.0%

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

7. Final simplification 84.0%

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

Alternative 6: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. pow1/2 99.2%

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

   2. clear-num 99.1%

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

   3. inv-pow 99.1%

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

   4. pow-pow 99.1%

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

   5. div-sub 99.1%

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

   6. pow1 99.1%

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

   7. pow1 99.1%

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

   8. pow-div 99.1%

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

   9. metadata-eval 99.1%

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

   10. metadata-eval 99.1%

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

   11. metadata-eval 99.1%

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

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 84.0%

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

   1. expm1-log1p-u 83.9%

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

   2. expm1-udef 70.4%

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

   3. inv-pow 70.4%

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

   4. sqrt-pow1 70.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{u1} - 1\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

   5. sub-neg 70.4%

      \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} + \left(-1\right)\right)}}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

   6. metadata-eval 70.4%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

   7. metadata-eval 70.4%

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

6. Applied egg-rr 70.4%

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

   1. expm1-def 83.9%

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

   2. expm1-log1p 84.0%

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

   3. metadata-eval 84.0%

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

   4. *-inverses 84.0%

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

   5. sub-neg 84.0%

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

   6. div-sub 84.0%

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

8. Simplified 84.0%

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

9. Final simplification 84.0%

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

Alternative 7: 79.7% 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 99.2%

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

2. Taylor expanded in u2 around 0 84.0%

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

3. Final simplification 84.0%

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

Alternative 8: 63.5% 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 99.2%

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

   1. pow1/2 99.2%

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

   2. clear-num 99.1%

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

   3. inv-pow 99.1%

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

   4. pow-pow 99.1%

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

   5. div-sub 99.1%

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

   6. pow1 99.1%

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

   7. pow1 99.1%

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

   8. pow-div 99.1%

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

   9. metadata-eval 99.1%

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

   10. metadata-eval 99.1%

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

   11. metadata-eval 99.1%

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

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 u1 around 0 72.9%

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

5. Taylor expanded in u2 around 0 65.2%

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

6. Final simplification 65.2%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023187 
(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))))