Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 14.9s

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

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

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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. inv-pow 98.4%

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

   3. div-sub 98.5%

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

   4. pow1 98.5%

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

   5. pow1 98.5%

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

   6. pow-div 98.5%

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

   7. metadata-eval 98.5%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{-1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. metadata-eval 98.5%

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

3. Applied egg-rr 98.5%

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

   1. unpow-1 98.5%

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

   2. sub-neg 98.5%

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

   3. metadata-eval 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0008999999845400453:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(u2 \cdot \frac{u1}{1 - u1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0008999999845400453)
   (sqrt (* 39.47841760436263 (* u2 (* u2 (/ u1 (- 1.0 u1))))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0008999999845400453f) {
		tmp = sqrtf((39.47841760436263f * (u2 * (u2 * (u1 / (1.0f - u1))))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (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) <= 0.0008999999845400453e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 * (u2 * (u1 / (1.0e0 - u1))))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (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(0.0008999999845400453))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 * Float32(u2 * Float32(u1 / Float32(Float32(1.0) - u1))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * 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(0.0008999999845400453))
		tmp = sqrt((single(39.47841760436263) * (u2 * (u2 * (u1 / (single(1.0) - u1))))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (single(1.0) + u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 8.99999985e-4

   1. Initial program 98.5%

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

   2. Taylor expanded in u2 around 0 98.5%

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

      1. add-sqr-sqrt 97.5%

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

      2. sqrt-unprod 98.5%

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

      3. swap-sqr 98.3%

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

      4. metadata-eval 98.5%

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

      5. swap-sqr 98.5%

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

      6. add-sqr-sqrt 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 99.0%

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

   6. Simplified 99.0%

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

   if 8.99999985e-4 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.2%

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

      1. expm1-log1p-u 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in u1 around 0 87.1%

      \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   5. Step-by-step derivation

      1. unpow2 87.1%

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

      2. distribute-rgt1-in 87.0%

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

      3. *-commutative 87.0%

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

   6. Simplified 87.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0008999999845400453:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(u2 \cdot \frac{u1}{1 - u1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(u2 \cdot \frac{u1}{1 - u1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.02500000037252903)
   (sqrt (* 39.47841760436263 (* u2 (* u2 (/ 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 = sqrtf((39.47841760436263f * (u2 * (u2 * (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 = sqrt((39.47841760436263e0 * (u2 * (u2 * (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 = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 * Float32(u2 * 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 = sqrt((single(39.47841760436263) * (u2 * (u2 * (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 98.5%

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

   2. Taylor expanded in u2 around 0 95.3%

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

      1. add-sqr-sqrt 94.5%

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

      2. sqrt-unprod 95.3%

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

      3. swap-sqr 95.1%

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

      4. metadata-eval 95.3%

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

      5. swap-sqr 95.3%

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

      6. add-sqr-sqrt 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-*l* 95.7%

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

   6. Simplified 95.7%

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

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

   1. Initial program 98.0%

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

   2. Taylor expanded in u1 around 0 72.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(u2 \cdot \frac{u1}{1 - u1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 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. Final simplification 98.4%

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

Alternative 5: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((single(39.47841760436263) * (u2 * (u2 * (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. Taylor expanded in u2 around 0 82.7%

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

   1. add-sqr-sqrt 82.1%

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

   2. sqrt-unprod 82.7%

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

   3. swap-sqr 82.6%

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

   4. metadata-eval 82.7%

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

   5. swap-sqr 82.7%

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

   6. add-sqr-sqrt 83.0%

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

4. Applied egg-rr 83.0%

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

   1. *-commutative 83.0%

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

   2. associate-*l* 83.0%

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

6. Simplified 83.0%

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

7. Final simplification 83.0%

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

Alternative 6: 81.0% 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. Taylor expanded in u2 around 0 82.7%

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

3. Final simplification 82.7%

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

Alternative 7: 64.2% 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. Taylor expanded in u2 around 0 82.7%

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

3. Taylor expanded in u1 around 0 65.9%

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

4. Final simplification 65.9%

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

Reproduce

herbie shell --seed 2023285 
(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))))