Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.1s

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: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (cbrt (pow (/ u1 (- 1.0 u1)) 1.5)) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf(powf((u1 / (1.0f - u1)), 1.5f)) * sinf((6.28318530718f * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cbrt((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5))) * sin(Float32(Float32(6.28318530718) * u2)))
end

Derivation

1. Initial program 98.4%

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

   1. add-cbrt-cube 98.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. pow1/3 95.9%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. add-sqr-sqrt 96.0%

      \[\leadsto {\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. pow1 96.0%

      \[\leadsto {\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. pow1/2 96.0%

      \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. pow-prod-up 95.9%

      \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. metadata-eval 95.9%

      \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. unpow1/3 98.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Simplified 98.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Add Preprocessing

Alternative 2: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.0013000000035390258:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.0013000000035390258)
     (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))
     (* (* 6.28318530718 u2) (sqrt t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.0013000000035390258f) {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = (6.28318530718f * u2) * sqrtf(t_0);
	}
	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 = u1 / (1.0e0 - u1)
    if (t_0 <= 0.0013000000035390258e0) then
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = (6.28318530718e0 * u2) * sqrt(t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0013000000035390258))
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u1 / (single(1.0) - u1);
	tmp = single(0.0);
	if (t_0 <= single(0.0013000000035390258))
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = (single(6.28318530718) * u2) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.0013

   1. Initial program 98.5%

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

   3. Taylor expanded in u1 around 0 98.3%

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

      1. +-commutative 98.3%

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

   5. Simplified 98.3%

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

   if 0.0013 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   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 82.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0013000000035390258:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.020999999716877937:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.020999999716877937)
   (* (* 6.28318530718 u2) (pow (+ -1.0 (/ 1.0 u1)) -0.5))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.020999999716877937f) {
		tmp = (6.28318530718f * u2) * powf((-1.0f + (1.0f / u1)), -0.5f);
	} 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) <= 0.020999999716877937e0) then
        tmp = (6.28318530718e0 * u2) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
    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(0.020999999716877937))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)));
	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(0.020999999716877937))
		tmp = (single(6.28318530718) * u2) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0209999997

   1. Initial program 98.6%

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

   3. Taylor expanded in u1 around inf 98.5%

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

   4. Taylor expanded in u2 around 0 95.2%

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

      1. associate-*r* 95.2%

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

      2. sub-neg 95.2%

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

      3. metadata-eval 95.2%

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

      4. unpow-1 95.2%

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

      5. metadata-eval 95.2%

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      6. pow-sqr 95.3%

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

      7. rem-sqrt-square 95.3%

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

      8. metadata-eval 95.3%

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \left|{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

      9. pow-sqr 94.7%

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \left|\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.25} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.25}}\right| \]

      10. fabs-sqr 94.7%

          \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{\left({\left(\frac{1}{u1} + -1\right)}^{-0.25} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.25}\right)} \]

      11. pow-sqr 95.3%

          \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(2 \cdot -0.25\right)}} \]

      12. metadata-eval 95.3%

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

   6. Simplified 95.3%

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

   if 0.0209999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   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 u1 around 0 76.3%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.020999999716877937:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 6.28318530718 u2)) (sqrt (/ u1 (+ (- 2.0 u1) -1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf((u1 / ((2.0f - 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 = sin((6.28318530718e0 * u2)) * sqrt((u1 / ((2.0e0 - u1) + (-1.0e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 / ((single(2.0) - u1) + single(-1.0))));
end

Derivation

1. Initial program 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. expm1-undefine 98.3%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. sub-neg 98.3%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. log1p-undefine 98.3%

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

   4. rem-exp-log 98.3%

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

   5. associate-+r- 98.5%

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

   6. metadata-eval 98.5%

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

   7. metadata-eval 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

   \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}} \]
8. 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 6: 81.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(6.28318530718 \cdot u2\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 6.28318530718 u2) (pow (+ -1.0 (/ 1.0 u1)) -0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * powf((-1.0f + (1.0f / 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 = (6.28318530718e0 * u2) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
end

Derivation

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 u1 around inf 98.3%

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

4. Taylor expanded in u2 around 0 81.0%

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

   1. associate-*r* 81.0%

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

   2. sub-neg 81.0%

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

   3. metadata-eval 81.0%

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

   4. unpow-1 81.0%

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

   5. metadata-eval 81.0%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   6. pow-sqr 81.0%

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

   7. rem-sqrt-square 81.0%

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

   8. metadata-eval 81.0%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \left|{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

   9. pow-sqr 80.6%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \left|\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.25} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.25}}\right| \]

   10. fabs-sqr 80.6%

       \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{\left({\left(\frac{1}{u1} + -1\right)}^{-0.25} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.25}\right)} \]

   11. pow-sqr 81.0%

       \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(2 \cdot -0.25\right)}} \]

   12. metadata-eval 81.0%

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

6. Simplified 81.0%

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

7. Final simplification 81.0%

   \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \]
8. 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 98.4%

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

3. Taylor expanded in u2 around 0 81.0%

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

4. Final simplification 81.0%

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

Alternative 8: 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 98.4%

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

3. Taylor expanded in u2 around 0 80.9%

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

4. Final simplification 80.9%

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

Alternative 9: 73.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 80.9%

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

4. Taylor expanded in u1 around 0 74.4%

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

   1. +-commutative 88.2%

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

6. Simplified 74.4%

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

7. Final simplification 74.4%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\right) \]
8. Add Preprocessing

Alternative 10: 64.7% 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 98.4%

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

3. Taylor expanded in u2 around 0 80.9%

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

4. Taylor expanded in u1 around 0 66.6%

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

5. Final simplification 66.6%

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

Reproduce

herbie shell --seed 2024139 
(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))))