Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 14.0s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

Alternative 2: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999995231628418:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9999995231628418)
   (* (+ 1.0 (* (* u2 u2) -19.739208802181317)) (sqrt (* u1 (+ u1 1.0))))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9999995231628418f) {
		tmp = (1.0f + ((u2 * u2) * -19.739208802181317f)) * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = 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 (cos((6.28318530718e0 * u2)) <= 0.9999995231628418e0) then
        tmp = (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))) * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9999995231628418))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = 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 (cos((single(6.28318530718) * u2)) <= single(0.9999995231628418))
		tmp = (single(1.0) + ((u2 * u2) * single(-19.739208802181317))) * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999999523

   1. Initial program 98.9%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      2. clear-num N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      3. sqrt-div N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

      4. metadata-eval N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      7. cos-lowering-cos.f32 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      10. pow-lowering-pow.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      11. div-sub N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\frac{1}{2}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\frac{1}{2}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\frac{1}{2}}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\frac{1}{2}}} \]

      15. +-lowering-+.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{\frac{1}{2}}} \]

      16. /-lowering-/.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      7. unpow2 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      9. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      10. /-lowering-/.f32 N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} - 1}}} \]

      11. sub-neg N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      13. +-commutative N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      14. +-lowering-+.f32 N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      15. /-lowering-/.f32 72.4

          \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{-1 + \color{blue}{\frac{1}{u1}}}} \]

   7. Simplified 72.4%

      \[\leadsto \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \]

      3. +-lowering-+.f32 67.2

         \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \]

   10. Simplified 67.2%

       \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \]

   if 0.999999523 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. sqrt-lowering-sqrt.f32 N/A

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

      13. *-rgt-identity N/A

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

      14. /-lowering-/.f32 N/A

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

      15. associate-*r* N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   5. Simplified 99.1%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999995231628418:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999961853027344:\\ \;\;\;\;\frac{1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317}{\sqrt{\frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9999961853027344)
   (/ (+ 1.0 (* (* u2 u2) -19.739208802181317)) (sqrt (/ 1.0 u1)))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9999961853027344f) {
		tmp = (1.0f + ((u2 * u2) * -19.739208802181317f)) / sqrtf((1.0f / u1));
	} else {
		tmp = 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 (cos((6.28318530718e0 * u2)) <= 0.9999961853027344e0) then
        tmp = (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))) / sqrt((1.0e0 / u1))
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9999961853027344))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))) / sqrt(Float32(Float32(1.0) / u1)));
	else
		tmp = 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 (cos((single(6.28318530718) * u2)) <= single(0.9999961853027344))
		tmp = (single(1.0) + ((u2 * u2) * single(-19.739208802181317))) / sqrt((single(1.0) / u1));
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99999619

   1. Initial program 98.8%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      2. clear-num N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      3. sqrt-div N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

      4. metadata-eval N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      7. cos-lowering-cos.f32 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      10. pow-lowering-pow.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      11. div-sub N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\frac{1}{2}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\frac{1}{2}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\frac{1}{2}}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\frac{1}{2}}} \]

      15. +-lowering-+.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{\frac{1}{2}}} \]

      16. /-lowering-/.f32 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{\sqrt{\frac{1}{u1}}}} \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{\sqrt{\frac{1}{u1}}}} \]

      2. /-lowering-/.f32 78.2

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

   7. Simplified 78.2%

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

   8. Taylor expanded in u2 around 0

      \[\leadsto \frac{\color{blue}{1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}}}{\sqrt{\frac{1}{u1}}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}}}{\sqrt{\frac{1}{u1}}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \frac{1 + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}}}{\sqrt{\frac{1}{u1}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)}}{\sqrt{\frac{1}{u1}}} \]

      4. *-lowering-*.f32 58.5

         \[\leadsto \frac{1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}}{\sqrt{\frac{1}{u1}}} \]

   10. Simplified 58.5%

       \[\leadsto \frac{\color{blue}{1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)}}{\sqrt{\frac{1}{u1}}} \]

   if 0.99999619 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. sqrt-lowering-sqrt.f32 N/A

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

      13. *-rgt-identity N/A

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

      14. /-lowering-/.f32 N/A

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

      15. associate-*r* N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   5. Simplified 98.2%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999961853027344:\\ \;\;\;\;\frac{1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317}{\sqrt{\frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999961853027344:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9999961853027344)
   (* (+ 1.0 (* (* u2 u2) -19.739208802181317)) (sqrt u1))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9999961853027344f) {
		tmp = (1.0f + ((u2 * u2) * -19.739208802181317f)) * sqrtf(u1);
	} else {
		tmp = 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 (cos((6.28318530718e0 * u2)) <= 0.9999961853027344e0) then
        tmp = (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))) * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9999961853027344))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))) * sqrt(u1));
	else
		tmp = 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 (cos((single(6.28318530718) * u2)) <= single(0.9999961853027344))
		tmp = (single(1.0) + ((u2 * u2) * single(-19.739208802181317))) * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99999619

   1. Initial program 98.8%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      2. clear-num N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      3. sqrt-div N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

      4. metadata-eval N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      7. cos-lowering-cos.f32 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      10. pow-lowering-pow.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      11. div-sub N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\frac{1}{2}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\frac{1}{2}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\frac{1}{2}}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\frac{1}{2}}} \]

      15. +-lowering-+.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{\frac{1}{2}}} \]

      16. /-lowering-/.f32 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      7. unpow2 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      9. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      10. /-lowering-/.f32 N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} - 1}}} \]

      11. sub-neg N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      13. +-commutative N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      14. +-lowering-+.f32 N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      15. /-lowering-/.f32 68.5

          \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{-1 + \color{blue}{\frac{1}{u1}}}} \]

   7. Simplified 68.5%

      \[\leadsto \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{u1}} \]
   9. Step-by-step derivation

   10. Simplified 58.5%

       \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{u1}} \]

   if 0.99999619 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. sqrt-lowering-sqrt.f32 N/A

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

      13. *-rgt-identity N/A

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

      14. /-lowering-/.f32 N/A

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

      15. associate-*r* N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   5. Simplified 98.2%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999961853027344:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999961853027344:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9999961853027344)
   (* (+ 1.0 (* u2 (* u2 -19.739208802181317))) (sqrt u1))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9999961853027344f) {
		tmp = (1.0f + (u2 * (u2 * -19.739208802181317f))) * sqrtf(u1);
	} else {
		tmp = 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 (cos((6.28318530718e0 * u2)) <= 0.9999961853027344e0) then
        tmp = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9999961853027344))
		tmp = Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))) * sqrt(u1));
	else
		tmp = 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 (cos((single(6.28318530718) * u2)) <= single(0.9999961853027344))
		tmp = (single(1.0) + (u2 * (u2 * single(-19.739208802181317)))) * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99999619

   1. Initial program 98.8%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      2. clear-num N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      3. sqrt-div N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

      4. metadata-eval N/A

         \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      7. cos-lowering-cos.f32 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      10. pow-lowering-pow.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

      11. div-sub N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\frac{1}{2}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\frac{1}{2}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\frac{1}{2}}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\frac{1}{2}}} \]

      15. +-lowering-+.f32 N/A

          \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{\frac{1}{2}}} \]

      16. /-lowering-/.f32 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      7. unpow2 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      9. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      10. /-lowering-/.f32 N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} - 1}}} \]

      11. sub-neg N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      13. +-commutative N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      14. +-lowering-+.f32 N/A

          \[\leadsto \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

      15. /-lowering-/.f32 68.5

          \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{-1 + \color{blue}{\frac{1}{u1}}}} \]

   7. Simplified 68.5%

      \[\leadsto \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{-1 + \frac{1}{u1}}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{-1 + \frac{1}{u1}}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{-1 + \frac{1}{u1}}\right)}^{\frac{1}{2}}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \]

      4. inv-pow N/A

         \[\leadsto {\color{blue}{\left({\left(-1 + \frac{1}{u1}\right)}^{-1}\right)}}^{\frac{1}{2}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \]

      5. pow-pow N/A

         \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \]

      6. pow-lowering-pow.f32 N/A

         \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto {\color{blue}{\left(-1 + \frac{1}{u1}\right)}}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto {\left(-1 + \color{blue}{\frac{1}{u1}}\right)}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \left(1 + \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right) \cdot u2}\right) \]

      12. *-commutative N/A

          \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \left(1 + \color{blue}{u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)}\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \left(1 + \color{blue}{u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \left(1 + u2 \cdot \color{blue}{\left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right) \]

      15. *-lowering-*.f32 68.4

          \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + u2 \cdot \color{blue}{\left(u2 \cdot -19.739208802181317\right)}\right) \]

   9. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)} \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \]
   11. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \]

       2. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \]

       3. +-lowering-+.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \]

       4. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000}}\right) \]

       5. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right) \]

       6. associate-*l* N/A

          \[\leadsto \sqrt{u1} \cdot \left(1 + \color{blue}{u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(1 + u2 \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)}\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(1 + \color{blue}{u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(1 + u2 \cdot \color{blue}{\left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right) \]

       10. *-lowering-*.f32 58.5

           \[\leadsto \sqrt{u1} \cdot \left(1 + u2 \cdot \color{blue}{\left(u2 \cdot -19.739208802181317\right)}\right) \]

   12. Simplified 58.5%

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)} \]

   if 0.99999619 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. sqrt-lowering-sqrt.f32 N/A

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

      13. *-rgt-identity N/A

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

      14. /-lowering-/.f32 N/A

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

      15. associate-*r* N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   5. Simplified 98.2%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999961853027344:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ (* u1 (- -1.0 u1)) (+ -1.0 (* u1 u1))))
  (+
   1.0
   (*
    (* u2 u2)
    (+
     -19.739208802181317
     (* u2 (* u2 (+ 64.93939402268539 (* (* u2 u2) -85.45681720672748)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 * (-1.0f - u1)) / (-1.0f + (u1 * u1)))) * (1.0f + ((u2 * u2) * (-19.739208802181317f + (u2 * (u2 * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f)))))));
}

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)) / ((-1.0e0) + (u1 * u1)))) * (1.0e0 + ((u2 * u2) * ((-19.739208802181317e0) + (u2 * (u2 * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0))))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(u1 * Float32(Float32(-1.0) - u1)) / Float32(Float32(-1.0) + Float32(u1 * u1)))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(u2 * Float32(u2 * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748)))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 * (single(-1.0) - u1)) / (single(-1.0) + (u1 * u1)))) * (single(1.0) + ((u2 * u2) * (single(-19.739208802181317) + (u2 * (u2 * (single(64.93939402268539) + ((u2 * u2) * single(-85.45681720672748))))))));
end

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 99.1%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\frac{\color{blue}{u1 \cdot \left(-1 \cdot u1 - 1\right)}}{-1 + u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{u1 \cdot \left(-1 \cdot u1 - 1\right)}}{-1 + u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \color{blue}{\left(-1 \cdot u1 + \left(\mathsf{neg}\left(1\right)\right)\right)}}{-1 + u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 \cdot u1 + \color{blue}{-1}\right)}{-1 + u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \color{blue}{\left(-1 + -1 \cdot u1\right)}}{-1 + u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)}{-1 + u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. unsub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \color{blue}{\left(-1 - u1\right)}}{-1 + u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. --lowering--.f32 99.1

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

7. Simplified 99.1%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right) \]

   17. *-lowering-*.f32 94.5

       \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \color{blue}{\left(u2 \cdot u2\right)} \cdot -85.45681720672748\right)\right)\right)\right) \]

10. Simplified 94.5%

    \[\leadsto \sqrt{\frac{u1 \cdot \left(-1 - u1\right)}{-1 + u1 \cdot u1}} \cdot \color{blue}{\left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\right)} \]
11. Add Preprocessing

Alternative 7: 92.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (pow (+ -1.0 (/ 1.0 u1)) -0.5)
  (+
   1.0
   (*
    (* u2 u2)
    (+
     -19.739208802181317
     (* (* u2 u2) (+ 64.93939402268539 (* (* u2 u2) -85.45681720672748))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((-1.0f + (1.0f / u1)), -0.5f) * (1.0f + ((u2 * u2) * (-19.739208802181317f + ((u2 * u2) * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f))))));
}

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) + (1.0e0 / u1)) ** (-0.5e0)) * (1.0e0 + ((u2 * u2) * ((-19.739208802181317e0) + ((u2 * u2) * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0)))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5)) * (single(1.0) + ((u2 * u2) * (single(-19.739208802181317) + ((u2 * u2) * (single(64.93939402268539) + ((u2 * u2) * single(-85.45681720672748)))))));
end

Derivation

1. Initial program 99.2%

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   2. clear-num N/A

      \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   3. sqrt-div N/A

      \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

   4. metadata-eval N/A

      \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   7. cos-lowering-cos.f32 N/A

      \[\leadsto \frac{\color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   9. pow1/2 N/A

      \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

   10. pow-lowering-pow.f32 N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

   11. div-sub N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\frac{1}{2}}} \]

   12. sub-neg N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\frac{1}{2}}} \]

   13. *-inverses N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\frac{1}{2}}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\frac{1}{2}}} \]

   15. +-lowering-+.f32 N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{\frac{1}{2}}} \]

   16. /-lowering-/.f32 98.7

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\color{blue}{1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]
6. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \frac{\color{blue}{1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \frac{1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \frac{1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   5. sub-neg N/A

      \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   9. unpow2 N/A

      \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   14. *-commutative N/A

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   16. unpow2 N/A

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   17. *-lowering-*.f32 94.3

       \[\leadsto \frac{1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \color{blue}{\left(u2 \cdot u2\right)} \cdot -85.45681720672748\right)\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{0.5}} \]

7. Simplified 94.3%

   \[\leadsto \frac{\color{blue}{1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{0.5}} \]

9. Applied egg-rr 94.4%

   \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)} \]
10. Add Preprocessing

Alternative 8: 91.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot 64.93939402268539\right) + \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (+
   (* (* u2 u2) (* (* u2 u2) 64.93939402268539))
   (+ 1.0 (* (* u2 u2) -19.739208802181317)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (((u2 * u2) * ((u2 * u2) * 64.93939402268539f)) + (1.0f + ((u2 * u2) * -19.739208802181317f)));
}

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 * u2) * ((u2 * u2) * 64.93939402268539e0)) + (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(Float32(u2 * u2) * Float32(Float32(u2 * u2) * Float32(64.93939402268539))) + Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (((u2 * u2) * ((u2 * u2) * single(64.93939402268539))) + (single(1.0) + ((u2 * u2) * single(-19.739208802181317))));
end

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)\right)} \]

5. Simplified 91.7%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot 64.93939402268539\right) + \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right)\right)} \]

6. Final simplification 91.7%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot 64.93939402268539\right) + \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\right) \]
7. Add Preprocessing

Alternative 9: 90.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ 1.0 (/ (- 1.0 u1) u1)))
  (+
   1.0
   (* u2 (* u2 (+ -19.739208802181317 (* (* u2 u2) 64.93939402268539)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((1.0f / ((1.0f - u1) / u1))) * (1.0f + (u2 * (u2 * (-19.739208802181317f + ((u2 * u2) * 64.93939402268539f)))));
}

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((1.0e0 / ((1.0e0 - u1) / u1))) * (1.0e0 + (u2 * (u2 * ((-19.739208802181317e0) + ((u2 * u2) * 64.93939402268539e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - u1) / u1))) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(64.93939402268539)))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((single(1.0) / ((single(1.0) - u1) / u1))) * (single(1.0) + (u2 * (u2 * (single(-19.739208802181317) + ((u2 * u2) * single(64.93939402268539))))));
end

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 99.1%

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

   1. frac-2neg N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(u1 \cdot \left(u1 + 1\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. clear-num N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(u1 \cdot \left(u1 + 1\right)\right)\right)\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}{\color{blue}{u1 \cdot \left(u1 + 1\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. associate-/l/ N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}{u1 + 1}}{u1}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. distribute-neg-in N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{1} + \left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. sub-neg N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{1 - u1 \cdot u1}}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{1 \cdot 1} - u1 \cdot u1}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{1 \cdot 1 - u1 \cdot u1}{\color{blue}{1 + u1}}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. flip-- N/A

       \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{1 - u1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. --lowering--.f32 99.0

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
8. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}}\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}}\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right) \]

   13. *-lowering-*.f32 91.6

       \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + \color{blue}{\left(u2 \cdot u2\right)} \cdot 64.93939402268539\right)\right)\right) \]

9. Simplified 91.6%

   \[\leadsto \sqrt{\frac{1}{\frac{1 - u1}{u1}}} \cdot \color{blue}{\left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right)\right)} \]
10. Add Preprocessing

Alternative 10: 90.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right)}{{\left(-1 + \frac{1}{u1}\right)}^{0.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/
  (+ 1.0 (* u2 (* u2 (+ -19.739208802181317 (* (* u2 u2) 64.93939402268539)))))
  (pow (+ -1.0 (/ 1.0 u1)) 0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (1.0f + (u2 * (u2 * (-19.739208802181317f + ((u2 * u2) * 64.93939402268539f))))) / 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 = (1.0e0 + (u2 * (u2 * ((-19.739208802181317e0) + ((u2 * u2) * 64.93939402268539e0))))) / (((-1.0e0) + (1.0e0 / u1)) ** 0.5e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(64.93939402268539)))))) / (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(0.5)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(1.0) + (u2 * (u2 * (single(-19.739208802181317) + ((u2 * u2) * single(64.93939402268539)))))) / ((single(-1.0) + (single(1.0) / u1)) ^ single(0.5));
end

Derivation

1. Initial program 99.2%

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   2. clear-num N/A

      \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   3. sqrt-div N/A

      \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

   4. metadata-eval N/A

      \[\leadsto \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   7. cos-lowering-cos.f32 N/A

      \[\leadsto \frac{\color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   9. pow1/2 N/A

      \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

   10. pow-lowering-pow.f32 N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}}} \]

   11. div-sub N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\frac{1}{2}}} \]

   12. sub-neg N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\frac{1}{2}}} \]

   13. *-inverses N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\frac{1}{2}}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\frac{1}{2}}} \]

   15. +-lowering-+.f32 N/A

       \[\leadsto \frac{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}{{\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{\frac{1}{2}}} \]

   16. /-lowering-/.f32 98.7

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\color{blue}{1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]
6. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \frac{\color{blue}{1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1 + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \frac{1 + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{1 + u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   8. +-commutative N/A

      \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}}\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}}\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   12. unpow2 N/A

       \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{\frac{1}{2}}} \]

   13. *-lowering-*.f32 91.5

       \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + \color{blue}{\left(u2 \cdot u2\right)} \cdot 64.93939402268539\right)\right)}{{\left(\frac{1}{u1} + -1\right)}^{0.5}} \]

7. Simplified 91.5%

   \[\leadsto \frac{\color{blue}{1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right)}}{{\left(\frac{1}{u1} + -1\right)}^{0.5}} \]

8. Final simplification 91.5%

   \[\leadsto \frac{1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right)}{{\left(-1 + \frac{1}{u1}\right)}^{0.5}} \]
9. Add Preprocessing

Alternative 11: 87.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
}

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))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
end

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{{u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000}} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left({u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \color{blue}{\left({u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000} + 1\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   7. *-commutative N/A

      \[\leadsto \left(\color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \left(\color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   11. *-rgt-identity N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

   12. sub-neg N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   13. rgt-mult-inverse N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   14. mul-1-neg N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   16. mul-1-neg N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

   17. *-rgt-identity N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

   18. distribute-lft-in N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   19. +-commutative N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   20. sub-neg N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

   21. associate-*r* N/A

       \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

5. Simplified 88.2%

   \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

6. Final simplification 88.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \]
7. Add Preprocessing

Alternative 12: 87.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* u2 (* u2 -19.739208802181317)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
}

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))) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
end

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 99.1%

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

   1. frac-2neg N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(u1 \cdot \left(u1 + 1\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. clear-num N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(u1 \cdot \left(u1 + 1\right)\right)\right)\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}{\color{blue}{u1 \cdot \left(u1 + 1\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. associate-/l/ N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)}{u1 + 1}}{u1}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. distribute-neg-in N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{1} + \left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. sub-neg N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{1 - u1 \cdot u1}}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{\color{blue}{1 \cdot 1} - u1 \cdot u1}{u1 + 1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\frac{\frac{1 \cdot 1 - u1 \cdot u1}{\color{blue}{1 + u1}}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. flip-- N/A

       \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{1 - u1}}{u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. --lowering--.f32 99.0

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   7. *-commutative N/A

      \[\leadsto \left(1 + \color{blue}{{u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000}}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   8. unpow2 N/A

      \[\leadsto \left(1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   9. associate-*l* N/A

      \[\leadsto \left(1 + \color{blue}{u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   10. *-commutative N/A

       \[\leadsto \left(1 + u2 \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \left(1 + \color{blue}{u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   12. *-commutative N/A

       \[\leadsto \left(1 + u2 \cdot \color{blue}{\left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \left(1 + u2 \cdot \color{blue}{\left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   14. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \left(1 + u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   15. sub-neg N/A

       \[\leadsto \left(1 + u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right) \cdot \sqrt{\frac{u1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   16. mul-1-neg N/A

       \[\leadsto \left(1 + u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right) \cdot \sqrt{\frac{u1}{1 + \color{blue}{-1 \cdot u1}}} \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \left(1 + u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 + -1 \cdot u1}}} \]

   18. mul-1-neg N/A

       \[\leadsto \left(1 + u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right) \cdot \sqrt{\frac{u1}{1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}} \]

   19. sub-neg N/A

       \[\leadsto \left(1 + u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right) \cdot \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \]

   20. --lowering--.f32 88.2

       \[\leadsto \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \]

9. Simplified 88.2%

   \[\leadsto \color{blue}{\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

10. Final simplification 88.2%

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \]
11. Add Preprocessing

Alternative 13: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. sqrt-lowering-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. /-lowering-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Simplified 78.5%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
6. Add Preprocessing

Alternative 14: 71.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. sqrt-lowering-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. /-lowering-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Simplified 78.5%

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

6. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. +-lowering-+.f32 70.5

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

8. Simplified 70.5%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \]
9. Add Preprocessing

Alternative 15: 63.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. sqrt-lowering-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. /-lowering-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Simplified 78.5%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 62.9

      \[\leadsto \color{blue}{\sqrt{u1}} \]

8. Simplified 62.9%

   \[\leadsto \color{blue}{\sqrt{u1}} \]
9. Add Preprocessing

Alternative 16: 19.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (+
  1.0
  (*
   (* u2 u2)
   (+
    -19.739208802181317
    (* u2 (* u2 (+ 64.93939402268539 (* (* u2 u2) -85.45681720672748))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f + ((u2 * u2) * (-19.739208802181317f + (u2 * (u2 * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f))))));
}

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 + ((u2 * u2) * ((-19.739208802181317e0) + (u2 * (u2 * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0)))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(u2 * Float32(u2 * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) + ((u2 * u2) * (single(-19.739208802181317) + (u2 * (u2 * (single(64.93939402268539) + ((u2 * u2) * single(-85.45681720672748)))))));
end

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 91.2%

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

5. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
6. Step-by-step derivation

   1. cos-lowering-cos.f32 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-lowering-*.f32 19.9

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

7. Simplified 19.9%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \color{blue}{1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto 1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)} \]

   3. unpow2 N/A

      \[\leadsto 1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto 1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \]

   5. sub-neg N/A

      \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)} \]

   6. metadata-eval N/A

      \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right) \]

   7. +-commutative N/A

      \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

   8. +-lowering-+.f32 N/A

      \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

   9. unpow2 N/A

      \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} + u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right) \]

   17. *-lowering-*.f32 19.8

       \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \color{blue}{\left(u2 \cdot u2\right)} \cdot -85.45681720672748\right)\right)\right) \]

10. Simplified 19.8%

    \[\leadsto \color{blue}{1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)} \]
11. Add Preprocessing

Alternative 17: 19.6% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ 1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (+ 1.0 (* u2 (* u2 -19.739208802181317))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f + (u2 * (u2 * -19.739208802181317f));
}

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 + (u2 * (u2 * (-19.739208802181317e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) + (u2 * (u2 * single(-19.739208802181317)));
end

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 91.2%

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

5. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
6. Step-by-step derivation

   1. cos-lowering-cos.f32 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-lowering-*.f32 19.9

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

7. Simplified 19.9%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}} \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \color{blue}{1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}} \]

   2. *-commutative N/A

      \[\leadsto 1 + \color{blue}{{u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000}} \]

   3. unpow2 N/A

      \[\leadsto 1 + \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-98696044010906577398881}{5000000000000000000000} \]

   4. associate-*l* N/A

      \[\leadsto 1 + \color{blue}{u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)} \]

   5. *-commutative N/A

      \[\leadsto 1 + u2 \cdot \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)} \]

   6. *-lowering-*.f32 N/A

      \[\leadsto 1 + \color{blue}{u2 \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right)} \]

   7. *-commutative N/A

      \[\leadsto 1 + u2 \cdot \color{blue}{\left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)} \]

   8. *-lowering-*.f32 19.7

      \[\leadsto 1 + u2 \cdot \color{blue}{\left(u2 \cdot -19.739208802181317\right)} \]

10. Simplified 19.7%

    \[\leadsto \color{blue}{1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)} \]
11. Add Preprocessing

Alternative 18: 19.1% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 1.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f;
}

Fortran

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 91.2%

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

5. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
6. Step-by-step derivation

   1. cos-lowering-cos.f32 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-lowering-*.f32 19.9

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

7. Simplified 19.9%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 18.8%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))