Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 10.6s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.5× speedup

Math

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

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 97.5%

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

   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 98.0%

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

   6. pow2 98.0%

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

   7. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

Alternative 2: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004999999888241291:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.004999999888241291)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (* (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) <= 0.004999999888241291f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} 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) <= 0.004999999888241291e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    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(0.004999999888241291))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	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(0.004999999888241291))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	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) < 0.00499999989

   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. 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 98.3%

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

      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 98.3%

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

      2. sub-neg 98.3%

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

      3. *-inverses 98.3%

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

      4. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in u2 around 0 97.4%

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

      1. associate-*r* 97.4%

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

      2. *-commutative 97.4%

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

      3. associate-*r* 97.4%

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

      4. sub-neg 97.4%

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

      5. metadata-eval 97.4%

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

   9. Simplified 97.4%

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

       1. add-sqr-sqrt 96.9%

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

       2. sqrt-unprod 97.4%

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

       3. *-commutative 97.4%

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

       4. *-commutative 97.4%

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

       5. swap-sqr 97.3%

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

       6. add-sqr-sqrt 97.5%

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

       7. metadata-eval 97.5%

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

       8. sub-neg 97.5%

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

       9. *-inverses 97.5%

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

       10. div-sub 97.7%

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

       11. clear-num 97.7%

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

       12. metadata-eval 98.2%

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

   11. Applied egg-rr 98.2%

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

   if 0.00499999989 < (*.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 82.4%

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

      1. +-commutative 82.4%

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

   5. Simplified 82.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004999999888241291:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0170499999076128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0170499999076128)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (/ (sin (* u2 6.28318530718)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0170499999076128f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) / sqrtf((1.0f / u1));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.0170499999076128e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    else
        tmp = sin((u2 * 6.28318530718e0)) / sqrt((1.0e0 / 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(0.0170499999076128))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.0170499999076128))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	else
		tmp = sin((u2 * single(6.28318530718))) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0170499999

   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. 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 98.3%

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

      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 98.3%

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

      2. sub-neg 98.3%

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

      3. *-inverses 98.3%

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

      4. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in u2 around 0 96.2%

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

      1. associate-*r* 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*r* 96.1%

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

      4. sub-neg 96.1%

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

      5. metadata-eval 96.1%

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

   9. Simplified 96.1%

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

       1. add-sqr-sqrt 95.7%

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

       2. sqrt-unprod 96.1%

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

       3. *-commutative 96.1%

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

       4. *-commutative 96.1%

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

       5. swap-sqr 96.1%

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

       6. add-sqr-sqrt 96.3%

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

       7. metadata-eval 96.3%

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

       8. sub-neg 96.3%

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

       9. *-inverses 96.3%

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

       10. div-sub 96.5%

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

       11. clear-num 96.4%

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

       12. metadata-eval 96.9%

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

   11. Applied egg-rr 96.9%

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

   if 0.0170499999 < (*.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. Step-by-step derivation

      1. clear-num 97.6%

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

      2. sqrt-div 97.3%

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

      3. metadata-eval 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. div-sub 97.2%

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

      2. sub-neg 97.2%

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

      3. *-inverses 97.2%

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

      4. metadata-eval 97.2%

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

   6. Simplified 97.2%

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

      1. associate-*l/ 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-commutative 97.5%

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

   8. Applied egg-rr 97.5%

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

   9. Taylor expanded in u1 around 0 73.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0170499999076128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0170499999076128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{{u1}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0170499999076128)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (/ (sin (* u2 6.28318530718)) (pow u1 -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0170499999076128f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) / powf(u1, -0.5f);
	}
	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) <= 0.0170499999076128e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    else
        tmp = sin((u2 * 6.28318530718e0)) / (u1 ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0170499999076128))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) / (u1 ^ Float32(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.0170499999076128))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	else
		tmp = sin((u2 * single(6.28318530718))) / (u1 ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0170499999

   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. 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 98.3%

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

      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 98.3%

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

      2. sub-neg 98.3%

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

      3. *-inverses 98.3%

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

      4. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in u2 around 0 96.2%

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

      1. associate-*r* 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*r* 96.1%

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

      4. sub-neg 96.1%

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

      5. metadata-eval 96.1%

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

   9. Simplified 96.1%

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

       1. add-sqr-sqrt 95.7%

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

       2. sqrt-unprod 96.1%

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

       3. *-commutative 96.1%

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

       4. *-commutative 96.1%

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

       5. swap-sqr 96.1%

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

       6. add-sqr-sqrt 96.3%

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

       7. metadata-eval 96.3%

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

       8. sub-neg 96.3%

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

       9. *-inverses 96.3%

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

       10. div-sub 96.5%

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

       11. clear-num 96.4%

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

       12. metadata-eval 96.9%

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

   11. Applied egg-rr 96.9%

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

   if 0.0170499999 < (*.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. Step-by-step derivation

      1. clear-num 97.6%

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

      2. sqrt-div 97.3%

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

      3. metadata-eval 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. div-sub 97.2%

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

      2. sub-neg 97.2%

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

      3. *-inverses 97.2%

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

      4. metadata-eval 97.2%

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

   6. Simplified 97.2%

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

      1. associate-*l/ 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-commutative 97.5%

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

   8. Applied egg-rr 97.5%

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

   9. Taylor expanded in u1 around 0 73.4%

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

       1. *-un-lft-identity 73.4%

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

       2. inv-pow 73.4%

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

       3. sqrt-pow1 73.4%

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

       4. metadata-eval 73.4%

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

   11. Applied egg-rr 73.4%

       \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\color{blue}{1 \cdot {u1}^{-0.5}}} \]
   12. Step-by-step derivation

       1. *-lft-identity 73.4%

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

   13. Simplified 73.4%

       \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\color{blue}{{u1}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0170499999076128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{{u1}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0170499999076128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0170499999076128)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0170499999076128f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} 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) <= 0.0170499999076128e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    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(0.0170499999076128))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	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(0.0170499999076128))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	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) < 0.0170499999

   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. 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 98.3%

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

      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 98.3%

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

      2. sub-neg 98.3%

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

      3. *-inverses 98.3%

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

      4. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in u2 around 0 96.2%

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

      1. associate-*r* 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*r* 96.1%

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

      4. sub-neg 96.1%

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

      5. metadata-eval 96.1%

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

   9. Simplified 96.1%

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

       1. add-sqr-sqrt 95.7%

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

       2. sqrt-unprod 96.1%

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

       3. *-commutative 96.1%

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

       4. *-commutative 96.1%

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

       5. swap-sqr 96.1%

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

       6. add-sqr-sqrt 96.3%

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

       7. metadata-eval 96.3%

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

       8. sub-neg 96.3%

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

       9. *-inverses 96.3%

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

       10. div-sub 96.5%

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

       11. clear-num 96.4%

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

       12. metadata-eval 96.9%

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

   11. Applied egg-rr 96.9%

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

   if 0.0170499999 < (*.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 73.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0170499999076128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) / sqrt((single(1.0) / (u1 / (single(1.0) - 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. Step-by-step derivation

   1. clear-num 98.1%

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

   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 98.0%

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

   2. sub-neg 98.0%

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

   3. *-inverses 98.0%

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

   4. metadata-eval 98.0%

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

6. Simplified 98.0%

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

   1. associate-*l/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. *-commutative 98.1%

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

8. Applied egg-rr 98.1%

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

9. Taylor expanded in u1 around 0 98.2%

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

    1. neg-mul-1 98.2%

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

    2. sub-neg 98.2%

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

11. Simplified 98.2%

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

    1. clear-num 98.2%

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

    2. inv-pow 98.2%

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

13. Applied egg-rr 98.2%

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

    1. unpow-1 98.2%

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

15. Simplified 98.2%

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

Alternative 7: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) / sqrt(((single(1.0) - u1) / 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. Step-by-step derivation

   1. clear-num 98.1%

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

   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 98.0%

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

   2. sub-neg 98.0%

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

   3. *-inverses 98.0%

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

   4. metadata-eval 98.0%

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

6. Simplified 98.0%

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

   1. associate-*l/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. *-commutative 98.1%

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

8. Applied egg-rr 98.1%

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

9. Taylor expanded in u1 around 0 98.2%

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

    1. neg-mul-1 98.2%

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

    2. sub-neg 98.2%

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

11. Simplified 98.2%

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

Alternative 8: 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 9: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * 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. clear-num 98.1%

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

   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 98.0%

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

   2. sub-neg 98.0%

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

   3. *-inverses 98.0%

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

   4. metadata-eval 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in u2 around 0 81.8%

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

   1. associate-*r* 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-*r* 81.8%

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

   4. sub-neg 81.8%

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

   5. metadata-eval 81.8%

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

9. Simplified 81.8%

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

    1. add-sqr-sqrt 81.5%

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

    2. sqrt-unprod 81.8%

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

    3. *-commutative 81.8%

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

    4. *-commutative 81.8%

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

    5. swap-sqr 81.8%

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

    6. add-sqr-sqrt 81.9%

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

    7. metadata-eval 81.9%

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

    8. sub-neg 81.9%

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

    9. *-inverses 81.9%

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

    10. div-sub 82.0%

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

    11. clear-num 82.0%

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

    12. metadata-eval 82.3%

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

11. Applied egg-rr 82.3%

    \[\leadsto u2 \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}} \]
12. Add Preprocessing

Alternative 10: 81.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 / 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 98.1%

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

   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 98.0%

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

   2. sub-neg 98.0%

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

   3. *-inverses 98.0%

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

   4. metadata-eval 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in u2 around 0 81.8%

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

   1. associate-*r* 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-*r* 81.8%

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

   4. sub-neg 81.8%

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

   5. metadata-eval 81.8%

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

9. Simplified 81.8%

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

    1. associate-*r* 81.8%

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

    2. *-commutative 81.8%

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

    3. sqrt-div 81.9%

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

    4. metadata-eval 81.9%

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

    5. un-div-inv 81.9%

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

    6. *-commutative 81.9%

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

11. Applied egg-rr 81.9%

    \[\leadsto \color{blue}{\frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1} + -1}}} \]
12. Step-by-step derivation

    1. *-commutative 81.9%

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

    2. *-lft-identity 81.9%

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

    3. times-frac 81.9%

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

    4. metadata-eval 81.9%

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

13. Simplified 81.9%

    \[\leadsto \color{blue}{6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}}} \]
14. Add Preprocessing

Alternative 11: 81.5% 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 81.9%

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

Alternative 12: 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 81.9%

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

4. Taylor expanded in u1 around 0 72.7%

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

   1. +-commutative 84.8%

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

6. Simplified 72.7%

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

7. Final simplification 72.7%

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

Alternative 13: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) / sqrt((single(1.0) / 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. Step-by-step derivation

   1. clear-num 98.1%

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

   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 98.0%

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

   2. sub-neg 98.0%

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

   3. *-inverses 98.0%

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

   4. metadata-eval 98.0%

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

6. Simplified 98.0%

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

   1. associate-*l/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. *-commutative 98.1%

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

8. Applied egg-rr 98.1%

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

9. Taylor expanded in u1 around 0 73.8%

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

10. Taylor expanded in u2 around 0 64.1%

    \[\leadsto \frac{\color{blue}{6.28318530718 \cdot u2}}{\sqrt{\frac{1}{u1}}} \]

11. Final simplification 64.1%

    \[\leadsto \frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1}}} \]
12. Add Preprocessing

Alternative 14: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) * 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 81.9%

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

4. Taylor expanded in u1 around 0 64.1%

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

   1. *-commutative 64.1%

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

   2. associate-*l* 64.1%

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

   3. *-commutative 64.1%

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

6. Simplified 64.1%

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

7. Final simplification 64.1%

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

Alternative 15: 64.6% 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 81.9%

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

4. Taylor expanded in u1 around 0 64.1%

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

5. Final simplification 64.1%

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

Reproduce

herbie shell --seed 2024173 
(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))))