Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 11.4s

Alternatives: 18

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

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* u1 (/ 1.0 (- 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 / (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 / (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) / 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) / (single(1.0) - u1)))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.2%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 98.0%

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

   6. pow2 98.0%

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

   7. metadata-eval 98.3%

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

4. Applied egg-rr 98.3%

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

   1. clear-num 98.3%

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

   2. associate-/r/ 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

   \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \]
8. 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 98.2%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 98.0%

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

   6. pow2 98.0%

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

   7. metadata-eval 98.3%

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.010999999940395355)
   (* (* u2 6.28318530718) (pow (+ -1.0 (/ 1.0 u1)) -0.5))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.010999999940395355f) {
		tmp = (u2 * 6.28318530718f) * powf((-1.0f + (1.0f / u1)), -0.5f);
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 * (1.0f + u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.010999999940395355e0) then
        tmp = (u2 * 6.28318530718e0) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (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(0.010999999940395355))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.010999999940395355))
		tmp = (u2 * single(6.28318530718)) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 * (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) < 0.0109999999

   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 98.0%

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

      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 98.3%

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

      6. pow2 98.3%

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

      7. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in u2 around 0 96.6%

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

      1. pow1 96.6%

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

      2. sqrt-prod 96.7%

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

      3. sqrt-div 96.3%

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

      4. metadata-eval 96.3%

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

      5. associate-/r/ 96.7%

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

      6. sqrt-div 96.8%

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

      7. inv-pow 96.8%

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

      8. sqrt-pow2 96.8%

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

      9. div-sub 96.8%

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

      10. *-inverses 96.8%

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

      11. sub-neg 96.8%

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

      12. metadata-eval 96.8%

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

      13. metadata-eval 96.8%

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

      14. metadata-eval 96.6%

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

      15. unpow1/2 96.6%

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

      16. *-commutative 96.6%

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

      17. unpow1/2 96.6%

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

      18. metadata-eval 96.8%

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

   9. Applied egg-rr 96.8%

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

       1. unpow1 96.8%

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

       2. *-commutative 96.8%

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

       3. *-commutative 96.8%

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

   11. Simplified 96.8%

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

   if 0.0109999999 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0 87.7%

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

      1. +-commutative 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.010999999940395355)
   (* (* u2 6.28318530718) (pow (+ -1.0 (/ 1.0 u1)) -0.5))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.010999999940395355f) {
		tmp = (u2 * 6.28318530718f) * powf((-1.0f + (1.0f / u1)), -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) <= 0.010999999940395355e0) then
        tmp = (u2 * 6.28318530718e0) * (((-1.0e0) + (1.0e0 / u1)) ** (-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(0.010999999940395355))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ 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(0.010999999940395355))
		tmp = (u2 * single(6.28318530718)) * ((single(-1.0) + (single(1.0) / u1)) ^ 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) < 0.0109999999

   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 98.0%

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

      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 98.3%

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

      6. pow2 98.3%

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

      7. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in u2 around 0 96.6%

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

      1. pow1 96.6%

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

      2. sqrt-prod 96.7%

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

      3. sqrt-div 96.3%

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

      4. metadata-eval 96.3%

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

      5. associate-/r/ 96.7%

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

      6. sqrt-div 96.8%

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

      7. inv-pow 96.8%

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

      8. sqrt-pow2 96.8%

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

      9. div-sub 96.8%

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

      10. *-inverses 96.8%

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

      11. sub-neg 96.8%

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

      12. metadata-eval 96.8%

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

      13. metadata-eval 96.8%

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

      14. metadata-eval 96.6%

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

      15. unpow1/2 96.6%

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

      16. *-commutative 96.6%

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

      17. unpow1/2 96.6%

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

      18. metadata-eval 96.8%

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

   9. Applied egg-rr 96.8%

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

       1. unpow1 96.8%

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

       2. *-commutative 96.8%

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

       3. *-commutative 96.8%

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

   11. Simplified 96.8%

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

   if 0.0109999999 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0 75.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.010999999940395355:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. clear-num 98.1%

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

   2. sqrt-div 98.3%

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

   3. metadata-eval 98.3%

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) * powf((-1.0f - (-1.0f / u1)), -0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((u2 * 6.28318530718e0)) * (((-1.0e0) - ((-1.0e0) / u1)) ** (-0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. clear-num 98.1%

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

   2. sqrt-div 98.3%

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

   3. metadata-eval 98.3%

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

4. Applied egg-rr 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. inv-pow 98.3%

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

   3. sqrt-pow2 98.2%

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

   4. div-sub 98.2%

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

   5. *-inverses 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. metadata-eval 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-lft-identity 98.2%

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

   2. metadata-eval 98.2%

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

   3. sub-neg 98.2%

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

   4. *-inverses 98.2%

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

   5. div-sub 98.2%

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

   6. remove-double-neg 98.2%

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

   7. distribute-neg-frac2 98.2%

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

   8. distribute-neg-frac 98.2%

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

   9. sub-neg 98.2%

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

   10. +-commutative 98.2%

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

   11. distribute-neg-in 98.2%

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

   12. remove-double-neg 98.2%

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

   13. sub-neg 98.2%

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

   14. div-sub 98.2%

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

   15. distribute-frac-neg2 98.2%

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

   16. *-inverses 98.2%

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

   17. metadata-eval 98.2%

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

   18. distribute-frac-neg2 98.2%

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

   19. distribute-neg-frac 98.2%

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

   20. metadata-eval 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 7: 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 8: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 98.0%

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

   6. pow2 98.0%

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

   7. metadata-eval 98.3%

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

4. Applied egg-rr 98.3%

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

   1. clear-num 98.3%

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

   2. associate-/r/ 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in u2 around 0 79.9%

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

   1. pow1 79.9%

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

   2. sqrt-prod 80.0%

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

   3. sqrt-div 79.7%

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

   4. metadata-eval 79.7%

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

   5. associate-/r/ 80.0%

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

   6. sqrt-div 80.0%

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

   7. inv-pow 80.0%

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

   8. sqrt-pow2 80.0%

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

   9. div-sub 80.1%

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

   10. *-inverses 80.1%

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

   11. sub-neg 80.1%

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

   12. metadata-eval 80.1%

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

   13. metadata-eval 80.1%

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

   14. metadata-eval 79.9%

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

   15. unpow1/2 79.9%

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

   16. *-commutative 79.9%

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

   17. unpow1/2 79.9%

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

   18. metadata-eval 80.1%

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

9. Applied egg-rr 80.1%

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

    1. unpow1 80.1%

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

    2. *-commutative 80.1%

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

    3. *-commutative 80.1%

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

11. Simplified 80.1%

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

12. Final simplification 80.1%

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

Alternative 9: 81.8% 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 98.2%

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

3. Taylor expanded in u2 around 0 80.0%

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

4. Final simplification 80.0%

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

Alternative 10: 81.8% 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 98.2%

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

3. Taylor expanded in u2 around 0 79.9%

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

4. Final simplification 79.9%

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

Alternative 11: 73.0% accurate, 1.9× speedup

Math

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

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 98.2%

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

3. Taylor expanded in u2 around 0 79.9%

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

4. Taylor expanded in u1 around 0 70.5%

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

   1. +-commutative 85.4%

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

6. Simplified 70.5%

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

7. Final simplification 70.5%

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

Alternative 12: 64.6% accurate, 2.0× speedup

Math

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

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(u2 * Float32(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 u1 around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. *-commutative 85.4%

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

   3. distribute-lft1-in 85.4%

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

   4. fma-define 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 70.5%

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

   1. +-commutative 70.5%

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

   2. unpow2 70.5%

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

   3. fma-undefine 70.5%

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

   4. rem-square-sqrt 70.5%

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

   5. fma-define 70.5%

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

   6. hypot-undefine 70.5%

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

8. Simplified 70.5%

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

9. Taylor expanded in u1 around 0 62.6%

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

    1. associate-*r* 62.7%

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

    2. *-commutative 62.7%

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

    3. *-commutative 62.7%

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

11. Simplified 62.7%

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

12. Final simplification 62.7%

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

Alternative 13: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 79.9%

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

4. Taylor expanded in u1 around 0 62.6%

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

5. Final simplification 62.6%

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

Alternative 14: 20.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 * ((u2 * single(6.28318530718)) + (single(3.14159265359) * (u2 / 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 u1 around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. *-commutative 85.4%

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

   3. distribute-lft1-in 85.4%

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

   4. fma-define 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 70.5%

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

   1. +-commutative 70.5%

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

   2. unpow2 70.5%

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

   3. fma-undefine 70.5%

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

   4. rem-square-sqrt 70.5%

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

   5. fma-define 70.5%

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

   6. hypot-undefine 70.5%

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

8. Simplified 70.5%

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

9. Taylor expanded in u1 around inf 20.6%

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

10. Final simplification 20.6%

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

Alternative 15: 20.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * (u1 * (single(1.0) + (single(0.5) / 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 u1 around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. *-commutative 85.4%

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

   3. distribute-lft1-in 85.4%

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

   4. fma-define 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 70.5%

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

   1. +-commutative 70.5%

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

   2. unpow2 70.5%

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

   3. fma-undefine 70.5%

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

   4. rem-square-sqrt 70.5%

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

   5. fma-define 70.5%

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

   6. hypot-undefine 70.5%

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

8. Simplified 70.5%

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

9. Taylor expanded in u1 around inf 20.6%

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

    1. associate-*r/ 20.6%

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

    2. metadata-eval 20.6%

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

11. Simplified 20.6%

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

Alternative 16: 20.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u1 * (u2 + ((u2 * single(0.5)) / 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 u1 around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. *-commutative 85.4%

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

   3. distribute-lft1-in 85.4%

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

   4. fma-define 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 70.5%

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

   1. +-commutative 70.5%

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

   2. unpow2 70.5%

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

   3. fma-undefine 70.5%

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

   4. rem-square-sqrt 70.5%

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

   5. fma-define 70.5%

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

   6. hypot-undefine 70.5%

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

8. Simplified 70.5%

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

9. Taylor expanded in u1 around inf 20.6%

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

    1. associate-*r/ 20.6%

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

11. Simplified 20.6%

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

12. Final simplification 20.6%

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

Alternative 17: 19.5% 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 98.2%

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

3. Taylor expanded in u1 around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. *-commutative 85.4%

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

   3. distribute-lft1-in 85.4%

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

   4. fma-define 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 70.5%

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

   1. +-commutative 70.5%

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

   2. unpow2 70.5%

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

   3. fma-undefine 70.5%

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

   4. rem-square-sqrt 70.5%

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

   5. fma-define 70.5%

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

   6. hypot-undefine 70.5%

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

8. Simplified 70.5%

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

9. Taylor expanded in u1 around inf 19.2%

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

Alternative 18: 4.7% 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 98.2%

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

3. Taylor expanded in u1 around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. *-commutative 85.4%

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

   3. distribute-lft1-in 85.4%

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

   4. fma-define 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 70.5%

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

   1. +-commutative 70.5%

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

   2. unpow2 70.5%

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

   3. fma-undefine 70.5%

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

   4. rem-square-sqrt 70.5%

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

   5. fma-define 70.5%

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

   6. hypot-undefine 70.5%

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

8. Simplified 70.5%

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

9. Taylor expanded in u1 around -inf 5.3%

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

10. Final simplification 5.3%

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

Reproduce

herbie shell --seed 2024130 
(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))))