Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 11.1s

Alternatives: 14

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 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. pow1/2 99.0%

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

   3. pow1/2 99.0%

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

   4. pow-prod-down 99.0%

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

   5. swap-sqr 99.0%

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

   6. metadata-eval 99.1%

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

3. Applied egg-rr 99.1%

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

   1. unpow1/2 99.1%

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

   2. unpow2 99.1%

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

   3. *-commutative 99.1%

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

   4. unpow2 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 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 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. pow1/2 99.0%

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

   3. pow1/2 99.0%

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

   4. pow-prod-down 99.0%

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

   5. swap-sqr 99.0%

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

   6. metadata-eval 99.1%

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

3. Applied egg-rr 99.1%

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

   1. unpow1/2 99.1%

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

   2. unpow2 99.1%

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

   3. *-commutative 99.1%

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

   4. unpow2 99.1%

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

5. Simplified 99.1%

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

   1. pow1/2 99.1%

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

   2. *-commutative 99.1%

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

   3. metadata-eval 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{\color{blue}{\left(0.16666666666666666 \cdot 3\right)}}\right) \]

   4. unpow-prod-down 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({39.47841760436263}^{\left(0.16666666666666666 \cdot 3\right)} \cdot {\left(u2 \cdot u2\right)}^{\left(0.16666666666666666 \cdot 3\right)}\right)} \]

   5. metadata-eval 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({39.47841760436263}^{\color{blue}{0.5}} \cdot {\left(u2 \cdot u2\right)}^{\left(0.16666666666666666 \cdot 3\right)}\right) \]

   6. metadata-eval 99.0%

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

   7. pow1/2 99.0%

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

   8. sqrt-prod 99.0%

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

   9. add-sqr-sqrt 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 3: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9599999785423279:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9599999785423279)
     (* 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 <= 0.9599999785423279f) {
		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 <= 0.9599999785423279e0) 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(0.9599999785423279))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * 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(0.9599999785423279))
		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)) < 0.959999979

   1. Initial program 96.7%

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

   2. Taylor expanded in u1 around 0 74.9%

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

   if 0.959999979 < (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. Taylor expanded in u2 around 0 97.5%

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

      1. *-rgt-identity 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*l* 97.5%

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

      4. distribute-lft-out 97.4%

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

      5. unpow2 97.4%

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

   4. Simplified 97.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9599999785423279:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.11999999731779099)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (* (cos (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.11999999731779099f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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 * 6.28318530718e0) <= 0.11999999731779099e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.11999999731779099))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.11999999731779099))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.119999997

   1. Initial program 99.3%

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

   2. Taylor expanded in u2 around 0 98.7%

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

      1. *-rgt-identity 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*l* 98.7%

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

      4. distribute-lft-out 98.6%

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

      5. unpow2 98.6%

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

   4. Simplified 98.6%

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

   if 0.119999997 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.5%

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

      1. clear-num 97.5%

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

      2. associate-/r/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in u1 around 0 87.2%

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

      1. +-commutative 87.2%

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

   6. Simplified 87.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]

Alternative 5: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.039000000804662704:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.039000000804662704)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (* (cos (* u2 6.28318530718)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.039000000804662704f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.039000000804662704e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * sqrt((u1 + (u1 * u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.039000000804662704))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.039000000804662704))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0390000008

   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 99.3%

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

      1. *-rgt-identity 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l* 99.3%

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

      4. distribute-lft-out 99.2%

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

      5. unpow2 99.2%

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

   4. Simplified 99.2%

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

   if 0.0390000008 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.0%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 97.8%

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

   3. Applied egg-rr 97.8%

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

   4. Taylor expanded in u1 around 0 87.8%

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

      1. +-commutative 87.8%

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

   6. Simplified 87.8%

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

      1. distribute-lft1-in 88.2%

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

   8. Applied egg-rr 88.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.039000000804662704:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 6: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.30000001192092896:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\cos \left(u2 \cdot 6.28318530718\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.30000001192092896)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (/ (sqrt u1) (/ 1.0 (cos (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.30000001192092896f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = sqrtf(u1) / (1.0f / cosf((u2 * 6.28318530718f)));
	}
	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 * 6.28318530718e0) <= 0.30000001192092896e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = sqrt(u1) / (1.0e0 / cos((u2 * 6.28318530718e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.30000001192092896))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = Float32(sqrt(u1) / Float32(Float32(1.0) / cos(Float32(u2 * Float32(6.28318530718)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.30000001192092896))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = sqrt(u1) / (single(1.0) / cos((u2 * single(6.28318530718))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.300000012

   1. Initial program 99.3%

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

   2. Taylor expanded in u2 around 0 97.5%

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

      1. *-rgt-identity 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*l* 97.5%

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

      4. distribute-lft-out 97.4%

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

      5. unpow2 97.4%

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

   4. Simplified 97.4%

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

   if 0.300000012 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 96.7%

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

      1. *-commutative 96.7%

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

      2. sqrt-div 96.4%

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

      3. associate-*r/ 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-/l* 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in u1 around 0 75.2%

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

      1. *-commutative 75.2%

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

   8. Simplified 75.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.30000001192092896:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\cos \left(u2 \cdot 6.28318530718\right)}}\\ \end{array} \]

Alternative 7: 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 99.0%

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

2. Final simplification 99.0%

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

Alternative 8: 88.3% 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 99.0%

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

2. Taylor expanded in u2 around 0 91.7%

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

   1. *-rgt-identity 91.7%

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

   2. *-commutative 91.7%

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

   3. associate-*l* 91.7%

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

   4. distribute-lft-out 91.7%

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

   5. unpow2 91.7%

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

4. Simplified 91.7%

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

5. Final simplification 91.7%

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

Alternative 9: 72.1% 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 99.0%

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

   1. clear-num 99.0%

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

   2. associate-/r/ 98.9%

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

3. Applied egg-rr 98.9%

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

4. Taylor expanded in u1 around 0 88.4%

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

   1. +-commutative 88.4%

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

6. Simplified 88.4%

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

7. Taylor expanded in u2 around 0 74.6%

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

8. Final simplification 74.6%

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

Alternative 10: 80.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 82.6%

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

3. Final simplification 82.6%

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

Alternative 11: 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 99.0%

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

2. Taylor expanded in u2 around 0 82.6%

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

3. Taylor expanded in u1 around 0 66.8%

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

4. Final simplification 66.8%

   \[\leadsto \sqrt{u1} \]

Alternative 12: 20.5% accurate, 19.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (1.0f + ((u2 * u2) * -19.739208802181317f)) * (u1 + 0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))) * (u1 + 0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. clear-num 99.0%

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

   2. associate-/r/ 98.9%

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

3. Applied egg-rr 98.9%

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

4. Taylor expanded in u1 around 0 88.4%

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

   1. +-commutative 88.4%

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

6. Simplified 88.4%

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

7. Taylor expanded in u1 around inf 20.9%

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

   1. distribute-rgt-out 20.9%

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

   2. +-commutative 20.9%

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

9. Simplified 20.9%

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

10. Taylor expanded in u2 around 0 20.7%

    \[\leadsto \color{blue}{\left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \cdot \left(u1 + 0.5\right) \]
11. Step-by-step derivation

    1. unpow2 20.7%

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

12. Simplified 20.7%

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

13. Final simplification 20.7%

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

Alternative 13: 20.1% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ u1 + 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (+ u1 0.5))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 + Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 + single(0.5);
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 82.6%

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

3. Taylor expanded in u1 around 0 74.6%

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

   1. +-commutative 74.6%

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

   2. unpow2 74.6%

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

   3. fma-udef 74.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

5. Simplified 74.6%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

6. Taylor expanded in u1 around inf 20.2%

   \[\leadsto \color{blue}{0.5 + u1} \]
7. Step-by-step derivation

   1. +-commutative 20.2%

      \[\leadsto \color{blue}{u1 + 0.5} \]

8. Simplified 20.2%

   \[\leadsto \color{blue}{u1 + 0.5} \]

9. Final simplification 20.2%

   \[\leadsto u1 + 0.5 \]

Alternative 14: 19.1% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 82.6%

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

3. Taylor expanded in u1 around 0 74.6%

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

   1. +-commutative 74.6%

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

   2. unpow2 74.6%

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

   3. fma-udef 74.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

5. Simplified 74.6%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

6. Taylor expanded in u1 around inf 19.0%

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

7. Final simplification 19.0%

   \[\leadsto u1 \]

Reproduce

herbie shell --seed 2023293 
(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))))