Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 14.6s

Alternatives: 7

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 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow1/2 98.8%

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

   3. pow1/2 98.8%

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

   4. pow-prod-down 98.8%

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

   5. swap-sqr 98.8%

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

   6. metadata-eval 98.9%

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

3. Applied egg-rr 98.9%

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

   1. unpow1/2 98.9%

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

   2. unpow2 98.9%

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

   3. *-commutative 98.9%

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

   4. unpow2 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.05999999865889549)
   (* (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) <= 0.05999999865889549f) {
		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) <= 0.05999999865889549e0) 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(0.05999999865889549))
		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(0.05999999865889549))
		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) < 0.0599999987

   1. Initial program 99.3%

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

      1. clear-num 99.2%

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

      2. associate-/r/ 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in u2 around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. *-lft-identity 99.3%

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

      3. associate-*r* 99.3%

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

      4. distribute-rgt-out 99.3%

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

      5. unpow2 99.3%

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

   6. Simplified 99.3%

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

   if 0.0599999987 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.0%

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

      1. clear-num 97.1%

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

      2. associate-/r/ 97.1%

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

   3. Applied egg-rr 97.1%

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

   4. Taylor expanded in u1 around 0 83.0%

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

      1. +-commutative 83.0%

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

   6. Simplified 83.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]

Alternative 3: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.25:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.25)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (* (cos (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.25f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.25e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * sqrt(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(0.25))
		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(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.25))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.25

   1. Initial program 99.3%

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

      1. clear-num 99.2%

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

      2. associate-/r/ 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in u2 around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. *-lft-identity 97.8%

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

      3. associate-*r* 97.8%

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

      4. distribute-rgt-out 97.8%

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

      5. unpow2 97.8%

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

   6. Simplified 97.8%

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

   if 0.25 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 96.3%

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

   2. Taylor expanded in u1 around 0 75.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.25:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 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 98.8%

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

2. Final simplification 98.8%

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

Alternative 5: 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 98.8%

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in u2 around 0 86.6%

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

   1. +-commutative 86.6%

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

   2. *-lft-identity 86.6%

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

   3. associate-*r* 86.6%

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

   4. distribute-rgt-out 86.6%

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

   5. unpow2 86.6%

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

6. Simplified 86.6%

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

7. Final simplification 86.6%

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

Alternative 6: 80.0% 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 98.8%

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

2. Taylor expanded in u2 around 0 78.6%

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

3. Final simplification 78.6%

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

Alternative 7: 63.3% 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 98.8%

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

2. Taylor expanded in u2 around 0 78.6%

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

3. Taylor expanded in u1 around 0 63.7%

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

4. Final simplification 63.7%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023275 
(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))))