Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 10.9s

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: 98.4% 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 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 98.1%

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

   6. pow2 98.1%

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

   7. metadata-eval 98.5%

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

4. Applied egg-rr 98.5%

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

Alternative 2: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0012000000569969416:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0012000000569969416)
   (* 6.28318530718 (* u2 (pow (+ (/ 1.0 u1) -1.0) -0.5)))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0012000000569969416f) {
		tmp = 6.28318530718f * (u2 * powf(((1.0f / u1) + -1.0f), -0.5f));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.0012000000569969416e0) then
        tmp = 6.28318530718e0 * (u2 * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0012000000569969416))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * (Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.0012000000569969416))
		tmp = single(6.28318530718) * (u2 * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00120000006

   1. Initial program 98.5%

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

      1. clear-num 98.5%

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

      2. sqrt-div 98.3%

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

      3. metadata-eval 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. div-sub 98.4%

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

      2. sub-neg 98.4%

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

      3. *-inverses 98.4%

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

      4. metadata-eval 98.4%

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

      5. +-commutative 98.4%

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

      6. remove-double-neg 98.4%

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

      7. unsub-neg 98.4%

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

      8. distribute-neg-frac 98.4%

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

      9. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in u2 around inf 98.5%

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

   8. Taylor expanded in u2 around 0 98.2%

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

      1. associate-*r* 98.3%

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

      2. sub-neg 98.3%

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

      3. metadata-eval 98.3%

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

      4. unpow-1 98.3%

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

      5. metadata-eval 98.3%

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

      6. pow-sqr 98.4%

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

      7. rem-sqrt-square 98.4%

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

      8. rem-square-sqrt 97.7%

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

      9. fabs-sqr 97.7%

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

      10. rem-square-sqrt 98.4%

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

      11. associate-*r* 98.4%

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

   10. Simplified 98.4%

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

   if 0.00120000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 86.9%

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

      1. +-commutative 86.9%

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

   5. Simplified 86.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0012000000569969416:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.02500000037252903:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.02500000037252903)
   (* 6.28318530718 (* u2 (pow (+ (/ 1.0 u1) -1.0) -0.5)))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.02500000037252903f) {
		tmp = 6.28318530718f * (u2 * powf(((1.0f / u1) + -1.0f), -0.5f));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.02500000037252903e0) then
        tmp = 6.28318530718e0 * (u2 * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.02500000037252903))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * (Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.02500000037252903))
		tmp = single(6.28318530718) * (u2 * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0250000004

   1. Initial program 98.5%

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

      1. clear-num 98.5%

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

      2. sqrt-div 98.3%

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

      3. metadata-eval 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. div-sub 98.4%

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

      2. sub-neg 98.4%

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

      3. *-inverses 98.4%

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

      4. metadata-eval 98.4%

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

      5. +-commutative 98.4%

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

      6. remove-double-neg 98.4%

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

      7. unsub-neg 98.4%

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

      8. distribute-neg-frac 98.4%

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

      9. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in u2 around inf 98.5%

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

   8. Taylor expanded in u2 around 0 95.6%

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

      1. associate-*r* 95.6%

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

      2. sub-neg 95.6%

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

      3. metadata-eval 95.6%

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

      4. unpow-1 95.6%

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

      5. metadata-eval 95.6%

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

      6. pow-sqr 95.8%

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

      7. rem-sqrt-square 95.8%

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

      8. rem-square-sqrt 95.2%

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

      9. fabs-sqr 95.2%

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

      10. rem-square-sqrt 95.8%

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

      11. associate-*r* 95.8%

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

   10. Simplified 95.8%

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

   if 0.0250000004 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

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

   3. Taylor expanded in u1 around 0 75.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.02500000037252903:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. expm1-log1p-u 98.1%

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

4. Applied egg-rr 98.1%

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

   1. expm1-undefine 98.1%

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

   2. sub-neg 98.1%

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

   3. log1p-undefine 98.1%

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

   4. rem-exp-log 98.1%

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

   5. associate-+r- 98.2%

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

   6. metadata-eval 98.2%

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

   7. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in u2 around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. *-lft-identity 98.2%

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

   3. associate-*l/ 98.1%

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

   4. associate-/r/ 98.3%

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

   5. div-sub 98.3%

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

   6. sub-neg 98.3%

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

   7. *-inverses 98.3%

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

   8. metadata-eval 98.3%

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

   9. metadata-eval 98.3%

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

   10. sub-neg 98.3%

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

   11. sub-neg 98.3%

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

   12. metadata-eval 98.3%

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

   13. unpow-1 98.3%

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

   14. metadata-eval 98.3%

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

   15. pow-sqr 98.4%

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

   16. rem-sqrt-square 98.4%

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

   17. rem-square-sqrt 97.7%

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

   18. fabs-sqr 97.7%

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

   19. rem-square-sqrt 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

    \[\leadsto \sin \left(u2 \cdot 6.28318530718\right) \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. clear-num 98.3%

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

   2. sqrt-div 98.1%

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

   3. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

   1. div-sub 98.2%

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

   2. sub-neg 98.2%

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

   3. *-inverses 98.2%

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

   4. metadata-eval 98.2%

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

   5. +-commutative 98.2%

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

   6. remove-double-neg 98.2%

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

   7. unsub-neg 98.2%

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

   8. distribute-neg-frac 98.2%

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

   9. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in u2 around inf 98.3%

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

   1. sqrt-div 98.2%

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

   2. metadata-eval 98.2%

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

   3. un-div-inv 98.3%

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

   4. sub-neg 98.3%

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

   5. metadata-eval 98.3%

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

9. Applied egg-rr 98.3%

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

10. Final simplification 98.3%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Final simplification 98.2%

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

Alternative 7: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. clear-num 98.3%

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

   2. sqrt-div 98.1%

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

   3. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

   1. div-sub 98.2%

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

   2. sub-neg 98.2%

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

   3. *-inverses 98.2%

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

   4. metadata-eval 98.2%

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

   5. +-commutative 98.2%

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

   6. remove-double-neg 98.2%

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

   7. unsub-neg 98.2%

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

   8. distribute-neg-frac 98.2%

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

   9. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in u2 around inf 98.3%

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

8. Taylor expanded in u2 around 0 80.1%

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

   1. associate-*r* 80.1%

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

   2. sub-neg 80.1%

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

   3. metadata-eval 80.1%

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

   4. unpow-1 80.1%

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

   5. metadata-eval 80.1%

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

   6. pow-sqr 80.2%

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

   7. rem-sqrt-square 80.2%

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

   8. rem-square-sqrt 79.8%

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

   9. fabs-sqr 79.8%

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

   10. rem-square-sqrt 80.2%

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

   11. associate-*r* 80.2%

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

10. Simplified 80.2%

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

Alternative 8: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 80.2%

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

Alternative 9: 73.0% 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 98.2%

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

3. Taylor expanded in u2 around 0 80.2%

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

4. Taylor expanded in u1 around 0 71.1%

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

   1. +-commutative 85.9%

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

6. Simplified 71.1%

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

7. Final simplification 71.1%

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

Alternative 10: 64.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (sqrtf(u1) * -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) * (-6.28318530718e0)) * -u2
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 80.2%

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

4. Taylor expanded in u1 around 0 63.0%

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

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

   \[\leadsto \color{blue}{-6.28318530718 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 63.0%

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

7. Simplified 63.0%

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

8. Final simplification 63.0%

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

Alternative 11: 64.5% 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 98.2%

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

3. Taylor expanded in u2 around 0 80.2%

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

4. Taylor expanded in u1 around 0 63.0%

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

5. Final simplification 63.0%

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

Reproduce

herbie shell --seed 2024149 
(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))))