Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 13.7s

Alternatives: 17

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 97.5%

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

   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 98.2%

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

   6. pow2 98.2%

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

   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. Final simplification 98.4%

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

Alternative 2: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.004000000189989805:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.004000000189989805)
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))
   (* 6.28318530718 (/ u2 (sqrt (+ (/ 1.0 u1) -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.004000000189989805f) {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = 6.28318530718f * (u2 / sqrtf(((1.0f / 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 ((u1 / (1.0e0 - u1)) <= 0.004000000189989805e0) then
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = 6.28318530718e0 * (u2 / sqrt(((1.0e0 / u1) + (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.004000000189989805))
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = Float32(Float32(6.28318530718) * Float32(u2 / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u1 / (single(1.0) - u1)) <= single(0.004000000189989805))
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = single(6.28318530718) * (u2 / sqrt(((single(1.0) / u1) + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00400000019

   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 u1 around 0 96.7%

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

      1. +-commutative 78.1%

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

   5. Simplified 96.7%

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

   if 0.00400000019 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. sqrt-div 81.9%

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

      3. clear-num 81.8%

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

      4. un-div-inv 81.9%

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

      5. sqrt-undiv 82.1%

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

   5. Applied egg-rr 82.1%

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

      1. div-sub 82.3%

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

      2. sub-neg 82.3%

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

      3. *-inverses 82.3%

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

      4. metadata-eval 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.004000000189989805:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.032099999487400055:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.032099999487400055)
   (* u2 (* 6.28318530718 (pow (+ (/ 1.0 u1) -1.0) -0.5)))
   (/ (sin (* u2 6.28318530718)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.032099999487400055f) {
		tmp = u2 * (6.28318530718f * powf(((1.0f / u1) + -1.0f), -0.5f));
	} 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) <= 0.032099999487400055e0) then
        tmp = u2 * (6.28318530718e0 * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)))
    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(0.032099999487400055))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * (Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5))));
	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(0.032099999487400055))
		tmp = u2 * (single(6.28318530718) * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)));
	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) < 0.0320999995

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 94.9%

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

      1. add-exp-log 90.6%

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

      2. *-commutative 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-*l* 90.6%

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

   5. Applied egg-rr 90.6%

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

      1. rem-exp-log 94.9%

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

      2. *-commutative 94.9%

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

      3. *-commutative 94.9%

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

      4. clear-num 95.0%

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

      5. sqrt-div 94.7%

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

      6. metadata-eval 94.7%

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

      7. pow1/2 94.7%

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

      8. pow-flip 95.0%

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

      9. div-sub 95.0%

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

      10. *-inverses 95.0%

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

      11. sub-neg 95.0%

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

      12. metadata-eval 95.0%

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

      13. metadata-eval 95.0%

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

   7. Applied egg-rr 95.0%

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

   if 0.0320999995 < (*.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. add-sqr-sqrt 96.4%

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

      2. sqrt-unprod 97.7%

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

      3. *-commutative 97.7%

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

      4. *-commutative 97.7%

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

      5. swap-sqr 97.7%

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

      6. pow2 97.7%

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

      7. metadata-eval 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. sqrt-div 97.8%

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

      3. clear-num 97.5%

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

      4. un-div-inv 97.6%

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

      5. *-commutative 97.6%

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

      6. sqrt-prod 96.9%

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

      7. metadata-eval 97.4%

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

      8. sqrt-pow1 97.4%

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

      9. metadata-eval 97.4%

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

      10. pow1 97.4%

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

      11. *-commutative 97.4%

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

      12. sqrt-undiv 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u1 around 0 77.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.032099999487400055:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.032099999487400055:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.032099999487400055)
   (* u2 (* 6.28318530718 (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) <= 0.032099999487400055f) {
		tmp = u2 * (6.28318530718f * 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) <= 0.032099999487400055e0) then
        tmp = u2 * (6.28318530718e0 * (((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(0.032099999487400055))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * (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(0.032099999487400055))
		tmp = u2 * (single(6.28318530718) * (((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) < 0.0320999995

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 94.9%

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

      1. add-exp-log 90.6%

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

      2. *-commutative 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-*l* 90.6%

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

   5. Applied egg-rr 90.6%

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

      1. rem-exp-log 94.9%

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

      2. *-commutative 94.9%

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

      3. *-commutative 94.9%

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

      4. clear-num 95.0%

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

      5. sqrt-div 94.7%

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

      6. metadata-eval 94.7%

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

      7. pow1/2 94.7%

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

      8. pow-flip 95.0%

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

      9. div-sub 95.0%

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

      10. *-inverses 95.0%

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

      11. sub-neg 95.0%

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

      12. metadata-eval 95.0%

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

      13. metadata-eval 95.0%

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

   7. Applied egg-rr 95.0%

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

   if 0.0320999995 < (*.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 77.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.032099999487400055:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * 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. 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 98.2%

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

   6. pow2 98.2%

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

   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. Step-by-step derivation

   1. *-commutative 98.4%

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

   2. sqrt-div 98.2%

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

   3. clear-num 98.1%

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

   4. un-div-inv 98.1%

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

   5. *-commutative 98.1%

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

   6. sqrt-prod 97.8%

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

   7. metadata-eval 97.9%

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

   8. sqrt-pow1 97.9%

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

   9. metadata-eval 97.9%

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

   10. pow1 97.9%

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

   11. *-commutative 97.9%

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

   12. sqrt-undiv 98.3%

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

6. Applied egg-rr 98.3%

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

   1. div-inv 98.2%

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

   2. pow1/2 98.2%

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

   3. pow-flip 98.4%

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

   4. div-sub 98.3%

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

   5. *-inverses 98.3%

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

   6. sub-neg 98.3%

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

   7. metadata-eval 98.3%

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

   8. metadata-eval 98.3%

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

8. Applied egg-rr 98.3%

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

9. Final simplification 98.3%

   \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
10. 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: 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. 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 98.2%

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

   6. pow2 98.2%

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

   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. Step-by-step derivation

   1. *-commutative 98.4%

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

   2. sqrt-div 98.2%

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

   3. clear-num 98.1%

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

   4. un-div-inv 98.1%

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

   5. *-commutative 98.1%

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

   6. sqrt-prod 97.8%

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

   7. metadata-eval 97.9%

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

   8. sqrt-pow1 97.9%

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

   9. metadata-eval 97.9%

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

   10. pow1 97.9%

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

   11. *-commutative 97.9%

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

   12. sqrt-undiv 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 8: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(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 = 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. Taylor expanded in u2 around 0 79.7%

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

   1. add-exp-log 76.5%

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

   2. *-commutative 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l* 76.5%

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

5. Applied egg-rr 76.5%

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

   1. rem-exp-log 79.7%

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

   2. *-commutative 79.7%

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

   3. *-commutative 79.7%

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

   4. clear-num 79.7%

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

   5. sqrt-div 79.6%

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

   6. metadata-eval 79.6%

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

   7. pow1/2 79.6%

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

   8. pow-flip 79.7%

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

   9. div-sub 79.8%

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

   10. *-inverses 79.8%

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

   11. sub-neg 79.8%

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

   12. metadata-eval 79.8%

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

   13. metadata-eval 79.8%

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

7. Applied egg-rr 79.8%

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

8. Final simplification 79.8%

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

Alternative 9: 73.6% 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 79.7%

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

4. Taylor expanded in u1 around 0 72.0%

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

   1. +-commutative 72.0%

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

6. Simplified 72.0%

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

7. Final simplification 72.0%

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

Alternative 10: 81.8% 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 79.7%

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

4. Final simplification 79.7%

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

Alternative 11: 81.8% 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. Taylor expanded in u2 around 0 79.7%

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

   1. *-commutative 79.7%

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

   2. sqrt-div 79.5%

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

   3. clear-num 79.5%

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

   4. un-div-inv 79.5%

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

   5. sqrt-undiv 79.8%

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

5. Applied egg-rr 79.8%

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

   1. div-sub 79.7%

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

   2. sub-neg 79.7%

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

   3. *-inverses 79.7%

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

   4. metadata-eval 79.7%

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

7. Simplified 79.7%

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

8. Final simplification 79.7%

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

Alternative 12: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 / 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. Taylor expanded in u2 around 0 79.7%

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

   1. *-commutative 79.7%

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

   2. sqrt-div 79.5%

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

   3. clear-num 79.5%

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

   4. un-div-inv 79.5%

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

   5. sqrt-undiv 79.8%

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

5. Applied egg-rr 79.8%

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

6. Final simplification 79.8%

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

Alternative 13: 65.3% 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 79.7%

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

4. Taylor expanded in u1 around 0 63.1%

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

5. Final simplification 63.1%

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

Alternative 14: 65.3% 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 79.7%

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

4. Taylor expanded in u1 around 0 63.1%

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

   1. *-commutative 63.1%

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

   2. associate-*l* 63.1%

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

6. Simplified 63.1%

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

7. Final simplification 63.1%

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

Alternative 15: 18.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \left(u2 \cdot 3.14159265359 + 6.28318530718 \cdot \left(u1 \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u1 (+ (* u2 3.14159265359) (* 6.28318530718 (* u1 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * ((u2 * 3.14159265359f) + (6.28318530718f * (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 = u1 * ((u2 * 3.14159265359e0) + (6.28318530718e0 * (u1 * u2)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 * Float32(Float32(u2 * Float32(3.14159265359)) + Float32(Float32(6.28318530718) * Float32(u1 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 * ((u2 * single(3.14159265359)) + (single(6.28318530718) * (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 79.7%

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

4. Taylor expanded in u1 around 0 76.0%

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

5. Taylor expanded in u1 around inf 18.7%

   \[\leadsto \color{blue}{{u1}^{2} \cdot \left(3.14159265359 \cdot \frac{u2}{u1} + 6.28318530718 \cdot u2\right)} \]

6. Taylor expanded in u1 around 0 18.7%

   \[\leadsto \color{blue}{u1 \cdot \left(3.14159265359 \cdot u2 + 6.28318530718 \cdot \left(u1 \cdot u2\right)\right)} \]

7. Final simplification 18.7%

   \[\leadsto u1 \cdot \left(u2 \cdot 3.14159265359 + 6.28318530718 \cdot \left(u1 \cdot u2\right)\right) \]
8. Add Preprocessing

Alternative 16: 18.7% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \left(u2 \cdot \left(3.14159265359 + u1 \cdot 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u1 (* u2 (+ 3.14159265359 (* u1 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * (u2 * (3.14159265359f + (u1 * 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 = u1 * (u2 * (3.14159265359e0 + (u1 * 6.28318530718e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 * Float32(u2 * Float32(Float32(3.14159265359) + Float32(u1 * Float32(6.28318530718)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 * (u2 * (single(3.14159265359) + (u1 * 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. Taylor expanded in u2 around 0 79.7%

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

4. Taylor expanded in u1 around 0 76.0%

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

5. Taylor expanded in u1 around inf 18.7%

   \[\leadsto \color{blue}{{u1}^{2} \cdot \left(3.14159265359 \cdot \frac{u2}{u1} + 6.28318530718 \cdot u2\right)} \]

6. Taylor expanded in u1 around 0 18.7%

   \[\leadsto \color{blue}{u1 \cdot \left(3.14159265359 \cdot u2 + 6.28318530718 \cdot \left(u1 \cdot u2\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 18.7%

      \[\leadsto u1 \cdot \color{blue}{\left(6.28318530718 \cdot \left(u1 \cdot u2\right) + 3.14159265359 \cdot u2\right)} \]

   2. associate-*r* 18.7%

      \[\leadsto u1 \cdot \left(\color{blue}{\left(6.28318530718 \cdot u1\right) \cdot u2} + 3.14159265359 \cdot u2\right) \]

   3. distribute-rgt-out 18.7%

      \[\leadsto u1 \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 \cdot u1 + 3.14159265359\right)\right)} \]

8. Simplified 18.7%

   \[\leadsto \color{blue}{u1 \cdot \left(u2 \cdot \left(6.28318530718 \cdot u1 + 3.14159265359\right)\right)} \]

9. Final simplification 18.7%

   \[\leadsto u1 \cdot \left(u2 \cdot \left(3.14159265359 + u1 \cdot 6.28318530718\right)\right) \]
10. Add Preprocessing

Alternative 17: 18.5% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ 3.14159265359 \cdot \left(u1 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 3.14159265359 (* u1 u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 3.14159265359f * (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 = 3.14159265359e0 * (u1 * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(3.14159265359) * Float32(u1 * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(3.14159265359) * (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 79.7%

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

4. Taylor expanded in u1 around 0 76.0%

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

5. Taylor expanded in u1 around inf 18.7%

   \[\leadsto \color{blue}{{u1}^{2} \cdot \left(3.14159265359 \cdot \frac{u2}{u1} + 6.28318530718 \cdot u2\right)} \]

6. Taylor expanded in u1 around 0 18.5%

   \[\leadsto \color{blue}{3.14159265359 \cdot \left(u1 \cdot u2\right)} \]
7. Step-by-step derivation

   1. *-commutative 18.5%

      \[\leadsto 3.14159265359 \cdot \color{blue}{\left(u2 \cdot u1\right)} \]

8. Simplified 18.5%

   \[\leadsto \color{blue}{3.14159265359 \cdot \left(u2 \cdot u1\right)} \]

9. Final simplification 18.5%

   \[\leadsto 3.14159265359 \cdot \left(u1 \cdot u2\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(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))))