Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 12.6s

Alternatives: 10

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

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((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))) * sin((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((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))) * sin((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* (/ u1 (- 1.0 u1)) (log E))) (sin (* 6.28318530718 u2))))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * log(Float32(exp(1))))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 / (single(1.0) - u1)) * log(single(2.71828182845904523536)))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.1%

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

   1. add-log-exp 54.4%

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

   2. *-un-lft-identity 54.4%

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

   3. exp-prod 54.4%

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

   4. log-pow 98.3%

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \log e} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 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))) * sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.1%

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

   3. *-commutative 98.1%

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

   4. *-commutative 98.1%

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

   5. swap-sqr 97.9%

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

   6. pow2 97.9%

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

   7. metadata-eval 98.3%

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \]
6. Add Preprocessing

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.008999999612569809:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\sin \left(6.28318530718 \cdot u2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.008999999612569809)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (/ (sqrt u1) (/ 1.0 (sin (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.008999999612569809f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) / (1.0f / sinf((6.28318530718f * u2)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.008999999612569809e0) then
        tmp = (6.28318530718e0 * u2) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(u1) / (1.0e0 / sin((6.28318530718e0 * u2)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.008999999612569809))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) / Float32(Float32(1.0) / sin(Float32(Float32(6.28318530718) * u2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.008999999612569809))
		tmp = (single(6.28318530718) * u2) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(u1) / (single(1.0) / sin((single(6.28318530718) * u2)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00899999961

   1. Initial program 98.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*r* 96.3%

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

   5. Simplified 96.3%

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

   if 0.00899999961 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.4%

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

      1. *-commutative 97.4%

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

      2. sqrt-div 97.2%

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

      3. associate-*r/ 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-/l* 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in u1 around 0 74.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.008999999612569809:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\sin \left(6.28318530718 \cdot u2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02199999988079071:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.02199999988079071)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.02199999988079071f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.02199999988079071e0) then
        tmp = (6.28318530718e0 * u2) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.02199999988079071))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.02199999988079071))
		tmp = (single(6.28318530718) * u2) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0219999999

   1. Initial program 98.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-*r* 95.4%

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

   5. Simplified 95.4%

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

   if 0.0219999999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 74.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02199999988079071:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * 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 = sin((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Final simplification 98.1%

   \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
4. Add Preprocessing

Alternative 6: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * 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 = 6.28318530718e0 * (u2 * sqrt((u1 / (1.0e0 - u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 77.6%

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

4. Final simplification 77.6%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * 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 = (6.28318530718e0 * u2) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 77.6%

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

   1. *-commutative 77.6%

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

   2. associate-*r* 77.6%

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

5. Simplified 77.6%

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

6. Final simplification 77.6%

   \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
7. Add Preprocessing

Alternative 8: 64.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * 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 = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(u1)))
end

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 77.6%

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

4. Taylor expanded in u1 around 0 63.3%

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

5. Final simplification 63.3%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Add Preprocessing

Alternative 9: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{u1 \cdot 39.47841760436263} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (sqrt (* u1 39.47841760436263))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf((u1 * 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 = u2 * sqrt((u1 * 39.47841760436263e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * sqrt(Float32(u1 * Float32(39.47841760436263))))
end

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 77.6%

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

4. Taylor expanded in u1 around 0 63.3%

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

   1. *-commutative 63.3%

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

   2. associate-*l* 63.4%

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

   3. metadata-eval 63.4%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\frac{1}{0.15915494309188485}}\right) \]

   4. div-inv 63.4%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\frac{u2}{0.15915494309188485}} \]

   5. clear-num 63.3%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\frac{1}{\frac{0.15915494309188485}{u2}}} \]

   6. div-inv 63.3%

      \[\leadsto \color{blue}{\frac{\sqrt{u1}}{\frac{0.15915494309188485}{u2}}} \]

6. Applied egg-rr 63.3%

   \[\leadsto \color{blue}{\frac{\sqrt{u1}}{\frac{0.15915494309188485}{u2}}} \]
7. Step-by-step derivation

   1. associate-/r/ 63.4%

      \[\leadsto \color{blue}{\frac{\sqrt{u1}}{0.15915494309188485} \cdot u2} \]

8. Simplified 63.4%

   \[\leadsto \color{blue}{\frac{\sqrt{u1}}{0.15915494309188485} \cdot u2} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 63.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\sqrt{u1}}{0.15915494309188485}} \cdot \sqrt{\frac{\sqrt{u1}}{0.15915494309188485}}\right)} \cdot u2 \]

   2. sqrt-unprod 63.4%

      \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{u1}}{0.15915494309188485} \cdot \frac{\sqrt{u1}}{0.15915494309188485}}} \cdot u2 \]

   3. div-inv 63.4%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{u1} \cdot \frac{1}{0.15915494309188485}\right)} \cdot \frac{\sqrt{u1}}{0.15915494309188485}} \cdot u2 \]

   4. metadata-eval 63.4%

      \[\leadsto \sqrt{\left(\sqrt{u1} \cdot \color{blue}{6.28318530718}\right) \cdot \frac{\sqrt{u1}}{0.15915494309188485}} \cdot u2 \]

   5. div-inv 63.4%

      \[\leadsto \sqrt{\left(\sqrt{u1} \cdot 6.28318530718\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{1}{0.15915494309188485}\right)}} \cdot u2 \]

   6. metadata-eval 63.4%

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

   7. swap-sqr 63.3%

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

   8. add-sqr-sqrt 63.4%

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

   9. metadata-eval 63.3%

      \[\leadsto \sqrt{u1 \cdot \color{blue}{39.47841760436263}} \cdot u2 \]

10. Applied egg-rr 63.3%

    \[\leadsto \color{blue}{\sqrt{u1 \cdot 39.47841760436263}} \cdot u2 \]

11. Final simplification 63.3%

    \[\leadsto u2 \cdot \sqrt{u1 \cdot 39.47841760436263} \]
12. Add Preprocessing

Alternative 10: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \frac{u2}{0.15915494309188485} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (/ u2 0.15915494309188485)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * (u2 / 0.15915494309188485f);
}

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) * (u2 / 0.15915494309188485e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * Float32(u2 / Float32(0.15915494309188485)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * (u2 / single(0.15915494309188485));
end

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 77.6%

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

4. Taylor expanded in u1 around 0 63.3%

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

   1. *-commutative 63.3%

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

   2. associate-*l* 63.4%

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

   3. metadata-eval 63.4%

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\frac{1}{0.15915494309188485}}\right) \]

   4. div-inv 63.4%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\frac{u2}{0.15915494309188485}} \]

   5. clear-num 63.3%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\frac{1}{\frac{0.15915494309188485}{u2}}} \]

   6. div-inv 63.3%

      \[\leadsto \color{blue}{\frac{\sqrt{u1}}{\frac{0.15915494309188485}{u2}}} \]

6. Applied egg-rr 63.3%

   \[\leadsto \color{blue}{\frac{\sqrt{u1}}{\frac{0.15915494309188485}{u2}}} \]
7. Step-by-step derivation

   1. associate-/r/ 63.4%

      \[\leadsto \color{blue}{\frac{\sqrt{u1}}{0.15915494309188485} \cdot u2} \]

8. Simplified 63.4%

   \[\leadsto \color{blue}{\frac{\sqrt{u1}}{0.15915494309188485} \cdot u2} \]
9. Step-by-step derivation

   1. *-commutative 63.4%

      \[\leadsto \color{blue}{u2 \cdot \frac{\sqrt{u1}}{0.15915494309188485}} \]

   2. clear-num 63.3%

      \[\leadsto u2 \cdot \color{blue}{\frac{1}{\frac{0.15915494309188485}{\sqrt{u1}}}} \]

   3. un-div-inv 63.4%

      \[\leadsto \color{blue}{\frac{u2}{\frac{0.15915494309188485}{\sqrt{u1}}}} \]

10. Applied egg-rr 63.4%

    \[\leadsto \color{blue}{\frac{u2}{\frac{0.15915494309188485}{\sqrt{u1}}}} \]
11. Step-by-step derivation

    1. associate-/r/ 63.4%

       \[\leadsto \color{blue}{\frac{u2}{0.15915494309188485} \cdot \sqrt{u1}} \]

12. Simplified 63.4%

    \[\leadsto \color{blue}{\frac{u2}{0.15915494309188485} \cdot \sqrt{u1}} \]

13. Final simplification 63.4%

    \[\leadsto \sqrt{u1} \cdot \frac{u2}{0.15915494309188485} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :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))) (sin (* 6.28318530718 u2))))