Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 15.8s

Alternatives: 13

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

\[\begin{array}{l} \\ {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (pow (+ (/ 1.0 u1) -1.0) -0.5)
  (sin (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * 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 = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   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 98.6%

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

4. Applied egg-rr 98.6%

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

   1. add-cube-cbrt 97.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right) \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}} \]

   2. pow3 97.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right)}^{3}} \]

   3. sqrt-prod 97.0%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{{u2}^{2}} \cdot \sqrt{39.47841760436263}\right)}}\right)}^{3} \]

   4. metadata-eval 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2}} \cdot \color{blue}{6.28318530718}\right)}\right)}^{3} \]

   5. unpow2 97.2%

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

   6. sqrt-prod 96.9%

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

   7. add-sqr-sqrt 97.2%

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

6. Applied egg-rr 97.2%

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

   1. rem-cube-cbrt 98.5%

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

   2. clear-num 98.4%

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

   3. unpow-1 98.4%

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

   4. sqrt-pow1 98.5%

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

   5. div-sub 98.5%

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

   6. *-inverses 98.5%

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

   7. sub-neg 98.5%

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

   8. metadata-eval 98.5%

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

   9. metadata-eval 98.5%

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

   10. *-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   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 98.6%

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

10. Applied egg-rr 98.6%

    \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \]

11. Final simplification 98.6%

    \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \]
12. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (sqrt (* (pow u2 2.0) 39.47841760436263))) (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f))) * sqrtf((u1 / (1.0f - u1)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0))) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263)))) * sqrt((u1 / (single(1.0) - u1)));
end

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   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 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.04399999976158142:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.04399999976158142)
   (* (sqrt (/ u1 (- 1.0 u1))) (* u2 6.28318530718))
   (* (sin (* u2 6.28318530718)) (pow (/ 1.0 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.04399999976158142f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * 6.28318530718f);
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * powf((1.0f / 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) <= 0.04399999976158142e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * 6.28318530718e0)
    else
        tmp = sin((u2 * 6.28318530718e0)) * ((1.0e0 / 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(0.04399999976158142))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(6.28318530718)));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * (Float32(Float32(1.0) / 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(0.04399999976158142))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * single(6.28318530718));
	else
		tmp = sin((u2 * single(6.28318530718))) * ((single(1.0) / u1) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0439999998

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*r* 93.9%

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

   5. Simplified 93.9%

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

   if 0.0439999998 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

      1. add-sqr-sqrt 96.7%

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

      2. sqrt-unprod 97.9%

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

      3. *-commutative 97.9%

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

      4. *-commutative 97.9%

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

      5. swap-sqr 97.5%

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

      6. pow2 97.5%

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

      7. metadata-eval 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. add-cube-cbrt 97.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right) \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}} \]

      2. pow3 97.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right)}^{3}} \]

      3. sqrt-prod 96.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{{u2}^{2}} \cdot \sqrt{39.47841760436263}\right)}}\right)}^{3} \]

      4. metadata-eval 96.8%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2}} \cdot \color{blue}{6.28318530718}\right)}\right)}^{3} \]

      5. unpow2 96.8%

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

      6. sqrt-prod 96.1%

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

      7. add-sqr-sqrt 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. rem-cube-cbrt 97.9%

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

      2. clear-num 97.7%

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

      3. unpow-1 97.7%

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

      4. sqrt-pow1 97.9%

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

      5. div-sub 98.0%

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

      6. *-inverses 98.0%

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

      7. sub-neg 98.0%

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

      8. metadata-eval 98.0%

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

      9. metadata-eval 98.0%

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

      10. *-commutative 98.0%

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

   8. Applied egg-rr 98.0%

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

   9. Taylor expanded in u1 around 0 75.5%

      \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.04399999976158142:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.04399999976158142:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.04399999976158142)
   (* (sqrt (/ u1 (- 1.0 u1))) (* u2 6.28318530718))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.04399999976158142f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * 6.28318530718f);
	} 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) <= 0.04399999976158142e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * 6.28318530718e0)
    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(0.04399999976158142))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(6.28318530718)));
	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(0.04399999976158142))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * single(6.28318530718));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0439999998

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*r* 93.9%

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

   5. Simplified 93.9%

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

   if 0.0439999998 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 75.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.04399999976158142:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
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 98.5%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   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 98.6%

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

4. Applied egg-rr 98.6%

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

   1. add-cube-cbrt 97.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right) \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}} \]

   2. pow3 97.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right)}^{3}} \]

   3. sqrt-prod 97.0%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{{u2}^{2}} \cdot \sqrt{39.47841760436263}\right)}}\right)}^{3} \]

   4. metadata-eval 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2}} \cdot \color{blue}{6.28318530718}\right)}\right)}^{3} \]

   5. unpow2 97.2%

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

   6. sqrt-prod 96.9%

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

   7. add-sqr-sqrt 97.2%

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

6. Applied egg-rr 97.2%

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

   1. rem-cube-cbrt 98.5%

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

   2. clear-num 98.4%

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

   3. unpow-1 98.4%

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

   4. sqrt-pow1 98.5%

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

   5. div-sub 98.5%

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

   6. *-inverses 98.5%

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

   7. sub-neg 98.5%

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

   8. metadata-eval 98.5%

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

   9. metadata-eval 98.5%

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

   10. *-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

   \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
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 98.5%

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

3. Final simplification 98.5%

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

Alternative 7: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 79.2%

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

4. Final simplification 79.2%

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

Alternative 8: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 79.2%

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

   1. *-commutative 79.2%

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

   2. associate-*r* 79.2%

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

5. Simplified 79.2%

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

6. Final simplification 79.2%

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

Alternative 9: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 79.2%

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

4. Taylor expanded in u1 around 0 64.7%

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

5. Final simplification 64.7%

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

Alternative 10: 20.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(3.14159265359)) + (single(6.28318530718) * (u1 * u2));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u1 around 0 87.3%

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

   1. +-commutative 87.3%

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

   2. unpow2 87.3%

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

   3. fma-define 87.3%

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

5. Simplified 87.3%

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

6. Taylor expanded in u2 around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. *-commutative 72.6%

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

   3. +-commutative 72.6%

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

   4. unpow2 72.6%

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

   5. fma-undefine 72.6%

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

8. Simplified 72.6%

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

9. Taylor expanded in u1 around inf 20.2%

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

10. Final simplification 20.2%

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

Alternative 11: 20.6% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(3.14159265359) + (u1 * single(6.28318530718)));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u1 around 0 87.3%

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

   1. +-commutative 87.3%

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

   2. unpow2 87.3%

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

   3. fma-define 87.3%

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

5. Simplified 87.3%

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

6. Taylor expanded in u2 around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. *-commutative 72.6%

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

   3. +-commutative 72.6%

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

   4. unpow2 72.6%

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

   5. fma-undefine 72.6%

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

8. Simplified 72.6%

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

9. Taylor expanded in u1 around inf 20.2%

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

    1. +-commutative 20.2%

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

    2. associate-*r* 20.2%

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

    3. distribute-rgt-out 20.2%

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

11. Simplified 20.2%

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

12. Final simplification 20.2%

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

Alternative 12: 4.8% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u1 around 0 87.3%

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

   1. +-commutative 87.3%

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

   2. unpow2 87.3%

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

   3. fma-define 87.3%

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

5. Simplified 87.3%

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

6. Taylor expanded in u2 around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. *-commutative 72.6%

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

   3. +-commutative 72.6%

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

   4. unpow2 72.6%

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

   5. fma-undefine 72.6%

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

8. Simplified 72.6%

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

9. Taylor expanded in u1 around -inf 4.8%

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

10. Final simplification 4.8%

    \[\leadsto \left(u1 \cdot u2\right) \cdot -6.28318530718 \]
11. Add Preprocessing

Alternative 13: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u1 around 0 87.3%

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

   1. +-commutative 87.3%

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

   2. unpow2 87.3%

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

   3. fma-define 87.3%

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

5. Simplified 87.3%

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

6. Taylor expanded in u2 around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. *-commutative 72.6%

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

   3. +-commutative 72.6%

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

   4. unpow2 72.6%

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

   5. fma-undefine 72.6%

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

8. Simplified 72.6%

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

9. Taylor expanded in u1 around inf 19.2%

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

10. Final simplification 19.2%

    \[\leadsto 6.28318530718 \cdot \left(u1 \cdot u2\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))))