Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 11.5s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

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, 0.5× 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 98.3%

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

   1. add-log-exp 55.8%

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

   2. *-un-lft-identity 55.8%

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

   3. exp-prod 55.8%

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

   4. log-pow 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02500000037252903:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.02500000037252903)
   (* u2 (* 6.28318530718 (sqrt (/ u1 (- 1.0 u1)))))
   (/ (sin (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.02500000037252903f) {
		tmp = u2 * (6.28318530718f * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sinf((6.28318530718f * u2)) / sqrtf((1.0f / 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) <= 0.02500000037252903e0) then
        tmp = u2 * (6.28318530718e0 * sqrt((u1 / (1.0e0 - u1))))
    else
        tmp = sin((6.28318530718e0 * u2)) / sqrt((1.0e0 / 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(0.02500000037252903))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.02500000037252903))
		tmp = u2 * (single(6.28318530718) * sqrt((u1 / (single(1.0) - u1))));
	else
		tmp = sin((single(6.28318530718) * u2)) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0250000004

   1. Initial program 98.6%

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

   2. Taylor expanded in u2 around 0 95.2%

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

      1. associate-*r* 95.3%

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

   4. Simplified 95.3%

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

   if 0.0250000004 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.5%

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

      1. *-commutative 97.5%

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

      2. sqrt-div 97.4%

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

      3. associate-*r/ 97.2%

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

   3. Applied egg-rr 97.2%

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

      1. associate-/l* 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in u1 around 0 71.4%

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

      1. add-sqr-sqrt 71.3%

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

      2. sqrt-unprod 71.4%

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

      3. un-div-inv 71.4%

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

      4. associate-/r* 71.4%

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

      5. add-sqr-sqrt 71.4%

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

   8. Applied egg-rr 71.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02500000037252903:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. clear-num 98.4%

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

   2. associate-/r/ 98.1%

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

3. Applied egg-rr 98.1%

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

   1. associate-/r/ 98.4%

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

5. Applied egg-rr 98.4%

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

6. Final simplification 98.4%

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

Alternative 4: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02500000037252903:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.02500000037252903)
   (* u2 (* 6.28318530718 (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) <= 0.02500000037252903f) {
		tmp = u2 * (6.28318530718f * 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) <= 0.02500000037252903e0) then
        tmp = u2 * (6.28318530718e0 * 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(0.02500000037252903))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * 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(0.02500000037252903))
		tmp = u2 * (single(6.28318530718) * 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) < 0.0250000004

   1. Initial program 98.6%

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

   2. Taylor expanded in u2 around 0 95.2%

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

      1. associate-*r* 95.3%

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

   4. Simplified 95.3%

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

   if 0.0250000004 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.5%

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

   2. Taylor expanded in u1 around 0 71.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02500000037252903:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

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 98.3%

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

2. Final simplification 98.3%

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

Alternative 6: 81.1% 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 98.3%

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

2. Taylor expanded in u2 around 0 78.8%

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

3. Final simplification 78.8%

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

Alternative 7: 81.1% 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 98.3%

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

2. Taylor expanded in u2 around 0 78.8%

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

   1. *-commutative 78.8%

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

   2. associate-*r* 78.8%

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

4. Simplified 78.8%

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

5. Final simplification 78.8%

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

Alternative 8: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 78.8%

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

   1. associate-*r* 78.8%

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

4. Simplified 78.8%

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

5. Final simplification 78.8%

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

Alternative 9: 64.4% 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 98.3%

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

2. Taylor expanded in u2 around 0 78.8%

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

3. Taylor expanded in u1 around 0 62.7%

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

4. Final simplification 62.7%

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

Alternative 10: 64.4% accurate, 2.0× speedup

Math

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

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(Float32(6.28318530718) * u2) * sqrt(u1))
end

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. sqrt-div 98.1%

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

   3. associate-*r/ 98.0%

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

3. Applied egg-rr 98.0%

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

   1. associate-/l* 97.9%

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

5. Simplified 97.9%

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

6. Taylor expanded in u1 around 0 73.3%

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

7. Taylor expanded in u2 around 0 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-*r* 62.7%

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

9. Simplified 62.7%

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

10. Final simplification 62.7%

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

Alternative 11: -0.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 78.8%

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

   1. add-sqr-sqrt 78.3%

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

   2. sqrt-unprod 78.8%

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

   3. *-commutative 78.8%

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

   4. *-commutative 78.8%

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

   5. swap-sqr 78.6%

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

   6. swap-sqr 78.7%

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

   7. add-sqr-sqrt 78.8%

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

   8. pow2 78.8%

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

   9. metadata-eval 79.1%

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

4. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{\sqrt{\left(\frac{u1}{1 - u1} \cdot {u2}^{2}\right) \cdot 39.47841760436263}} \]
5. Step-by-step derivation

   1. associate-*l* 79.0%

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

6. Simplified 79.0%

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

7. Taylor expanded in u1 around -inf -0.0%

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

8. Final simplification -0.0%

   \[\leadsto u2 \cdot \sqrt{-39.47841760436263} \]

Reproduce

herbie shell --seed 2023333 
(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))))