Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 11.2s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (sqrt (* (* u2 u2) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf(sqrtf(((u2 * u2) * 39.47841760436263f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos(sqrt(((u2 * u2) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(sqrt(Float32(Float32(u2 * u2) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos(sqrt(((u2 * u2) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.7%

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

   1. add-sqr-sqrt 98.6%

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

   2. pow1/2 98.6%

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

   3. pow1/2 98.6%

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

   4. pow-prod-down 98.7%

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

   5. swap-sqr 98.7%

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

   6. metadata-eval 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.8%

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

   2. unpow2 98.8%

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

   3. *-commutative 98.8%

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

   4. unpow2 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

Alternative 2: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9984999895095825:\\ \;\;\;\;t_0 \cdot \sqrt{u1 + u1 \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9984999895095825)
     (* t_0 (sqrt (+ u1 (* u1 u1))))
     (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* u2 (* u2 -19.739208802181317)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9984999895095825f) {
		tmp = t_0 * sqrtf((u1 + (u1 * u1)));
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
	}
	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((u2 * 6.28318530718e0))
    if (t_0 <= 0.9984999895095825e0) then
        tmp = t_0 * sqrt((u1 + (u1 * u1)))
    else
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9984999895095825))
		tmp = Float32(t_0 * sqrt(Float32(u1 + Float32(u1 * u1))));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.9984999895095825))
		tmp = t_0 * sqrt((u1 + (u1 * u1)));
	else
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.99849999

   1. Initial program 97.2%

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

      1. clear-num 97.2%

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

      2. inv-pow 97.2%

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

      3. div-sub 96.9%

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

      4. pow1 96.9%

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

      5. pow1 96.9%

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

      6. pow-div 96.9%

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

      7. metadata-eval 96.9%

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

      8. metadata-eval 96.9%

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

   3. Applied egg-rr 96.9%

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

      1. unpow-1 96.9%

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

      2. sub-neg 96.9%

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

      3. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in u1 around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. unpow2 88.5%

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

   8. Simplified 88.5%

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

   if 0.99849999 < (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. add-sqr-sqrt 99.3%

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

      2. pow1/2 99.3%

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

      3. pow1/2 99.3%

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

      4. pow-prod-down 99.3%

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

      5. swap-sqr 99.3%

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

      6. metadata-eval 99.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
   4. Step-by-step derivation

      1. unpow1/2 99.3%

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

      2. unpow2 99.3%

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

      3. *-commutative 99.3%

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

      4. unpow2 99.3%

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

   5. Simplified 99.3%

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

      1. clear-num 93.0%

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

      2. associate-/r/ 93.0%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in u2 around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. *-lft-identity 99.2%

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

      3. associate-*r* 99.2%

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

      4. distribute-rgt-out 99.1%

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

      5. unpow2 99.1%

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

      6. associate-*r* 99.1%

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

      7. *-commutative 99.1%

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

      8. *-commutative 99.1%

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

   10. Simplified 99.1%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9984999895095825:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(u2 \cdot {39.47841760436263}^{0.5}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* u2 (pow 39.47841760436263 0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((u2 * powf(39.47841760436263f, 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 = sqrt((u1 / (1.0e0 - u1))) * cos((u2 * (39.47841760436263e0 ** 0.5e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(u2 * (Float32(39.47841760436263) ^ Float32(0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((u2 * (single(39.47841760436263) ^ single(0.5))));
end

Derivation

1. Initial program 98.7%

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

   1. add-sqr-sqrt 98.6%

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

   2. pow1/2 98.6%

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

   3. pow1/2 98.6%

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

   4. pow-prod-down 98.7%

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

   5. swap-sqr 98.7%

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

   6. metadata-eval 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.8%

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

   2. unpow2 98.8%

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

   3. *-commutative 98.8%

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

   4. unpow2 98.8%

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

5. Simplified 98.8%

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

   1. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(\left(u2 \cdot u2\right) \cdot 39.47841760436263\right)}^{0.5}\right)} \]

   2. *-commutative 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\color{blue}{\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]

   3. unpow-prod-down 98.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({39.47841760436263}^{0.5} \cdot {\left(u2 \cdot u2\right)}^{0.5}\right)} \]

   4. pow1/2 98.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot \color{blue}{\sqrt{u2 \cdot u2}}\right) \]

   5. sqrt-prod 98.6%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot \color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)}\right) \]

   6. add-sqr-sqrt 98.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot \color{blue}{u2}\right) \]

7. Applied egg-rr 98.7%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({39.47841760436263}^{0.5} \cdot u2\right)} \]

8. Final simplification 98.7%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(u2 \cdot {39.47841760436263}^{0.5}\right) \]

Alternative 4: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9955000281333923:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9955000281333923)
     (* t_0 (sqrt u1))
     (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* u2 (* u2 -19.739208802181317)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9955000281333923f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
	}
	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((u2 * 6.28318530718e0))
    if (t_0 <= 0.9955000281333923e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9955000281333923))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.9955000281333923))
		tmp = t_0 * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.995500028

   1. Initial program 96.9%

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

   2. Taylor expanded in u1 around 0 77.5%

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

   if 0.995500028 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow1/2 99.2%

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

      3. pow1/2 99.2%

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

      4. pow-prod-down 99.2%

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

      5. swap-sqr 99.2%

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

      6. metadata-eval 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

   3. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
   4. Step-by-step derivation

      1. unpow1/2 99.2%

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

      2. unpow2 99.2%

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

      3. *-commutative 99.2%

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

      4. unpow2 99.2%

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

   5. Simplified 99.2%

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

      1. clear-num 90.8%

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

      2. associate-/r/ 90.8%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in u2 around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. *-lft-identity 98.5%

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

      3. associate-*r* 98.5%

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

      4. distribute-rgt-out 98.4%

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

      5. unpow2 98.4%

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

      6. associate-*r* 98.4%

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

      7. *-commutative 98.4%

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

      8. *-commutative 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9955000281333923:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((u2 * 6.28318530718f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Final simplification 98.7%

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

Alternative 6: 88.5% 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 98.7%

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

   1. add-sqr-sqrt 98.6%

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

   2. pow1/2 98.6%

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

   3. pow1/2 98.6%

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

   4. pow-prod-down 98.7%

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

   5. swap-sqr 98.7%

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

   6. metadata-eval 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.8%

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

   2. unpow2 98.8%

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

   3. *-commutative 98.8%

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

   4. unpow2 98.8%

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

5. Simplified 98.8%

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

   1. clear-num 77.4%

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

   2. associate-/r/ 77.4%

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

7. Applied egg-rr 98.7%

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

8. Taylor expanded in u2 around 0 86.4%

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

   1. +-commutative 86.4%

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

   2. *-lft-identity 86.4%

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

   3. associate-*r* 86.4%

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

   4. distribute-rgt-out 86.4%

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

   5. unpow2 86.4%

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

   6. associate-*r* 86.4%

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

   7. *-commutative 86.4%

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

   8. *-commutative 86.4%

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

10. Simplified 86.4%

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

11. Final simplification 86.4%

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

Alternative 7: 88.5% accurate, 1.8× speedup

Math

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

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(Float32(u2 * 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 98.7%

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

   1. add-sqr-sqrt 98.6%

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

   2. pow1/2 98.6%

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

   3. pow1/2 98.6%

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

   4. pow-prod-down 98.7%

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

   5. swap-sqr 98.7%

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

   6. metadata-eval 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.8%

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

   2. unpow2 98.8%

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

   3. *-commutative 98.8%

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

   4. unpow2 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in u2 around 0 86.4%

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

   1. +-commutative 86.4%

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

   2. *-lft-identity 86.4%

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

   3. associate-*r* 86.4%

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

   4. distribute-rgt-out 86.4%

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

   5. *-commutative 86.4%

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

   6. unpow2 86.4%

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

8. Simplified 86.4%

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

9. Final simplification 86.4%

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

Alternative 8: 71.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 * (u1 + 1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 77.6%

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

   1. clear-num 77.4%

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

   2. associate-/r/ 77.4%

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

4. Applied egg-rr 77.4%

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

5. Taylor expanded in u1 around 0 69.9%

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

   1. +-commutative 69.9%

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

7. Simplified 69.9%

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

8. Final simplification 69.9%

   \[\leadsto \sqrt{u1 \cdot \left(u1 + 1\right)} \]

Alternative 9: 71.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 + (u1 * 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 + (u1 * u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 + Float32(u1 * u1)))
end

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 77.6%

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

   1. clear-num 77.4%

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

   2. associate-/r/ 77.4%

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

4. Applied egg-rr 77.4%

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

5. Taylor expanded in u1 around 0 69.9%

   \[\leadsto \sqrt{\color{blue}{{u1}^{2} + u1}} \]
6. Step-by-step derivation

   1. +-commutative 69.9%

      \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]

   2. unpow2 69.9%

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

7. Simplified 69.9%

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

8. Final simplification 69.9%

   \[\leadsto \sqrt{u1 + u1 \cdot u1} \]

Alternative 10: 80.3% 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 98.7%

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

2. Taylor expanded in u2 around 0 77.6%

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

3. Final simplification 77.6%

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

Alternative 11: 63.4% 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 98.7%

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

2. Taylor expanded in u2 around 0 77.6%

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

3. Taylor expanded in u1 around 0 62.0%

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

4. Final simplification 62.0%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023188 
(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))))