Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 15.7s

Alternatives: 20

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. flip3-- N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. associate-/r/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. associate-*l/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}{{1}^{3} - {u1}^{3}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. frac-2neg N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. lower-neg.f32 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. +-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(\left(u1 \cdot u1 + 1 \cdot u1\right) + 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. distribute-lft-in N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + u1 \cdot 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + \color{blue}{u1}\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1, u1 \cdot u1 + 1 \cdot u1, u1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. *-lft-identity N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   14. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{1} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. sub-neg N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left({u1}^{3}\right)\right)\right)}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   16. cube-neg N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(1 + \color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

4. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 98.7

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

6. Applied rewrites 98.7%

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. flip3-- N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. associate-/r/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. associate-*l/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}{{1}^{3} - {u1}^{3}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. frac-2neg N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. lower-neg.f32 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. +-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(\left(u1 \cdot u1 + 1 \cdot u1\right) + 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. distribute-lft-in N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + u1 \cdot 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + \color{blue}{u1}\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1, u1 \cdot u1 + 1 \cdot u1, u1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. *-lft-identity N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   14. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{1} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. sub-neg N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left({u1}^{3}\right)\right)\right)}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   16. cube-neg N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(1 + \color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

4. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 98.7

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. sqr-neg N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-neg.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 4: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.4000000059604645:\\ \;\;\;\;\sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.4000000059604645)
   (*
    (sqrt (/ u1 (/ (+ (* u1 u1) -1.0) (- -1.0 u1))))
    (*
     u2
     (fma
      (* u2 u2)
      (fma
       (* u2 u2)
       (fma (* u2 u2) -76.70585975309672 81.6052492761019)
       -41.341702240407926)
      6.28318530718)))
   (* (sin (* 6.28318530718 u2)) (sqrt (fma u1 (fma u1 u1 u1) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.4000000059604645f) {
		tmp = sqrtf((u1 / (((u1 * u1) + -1.0f) / (-1.0f - u1)))) * (u2 * fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f), -41.341702240407926f), 6.28318530718f));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.4000000059604645))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(Float32(u1 * u1) + Float32(-1.0)) / Float32(Float32(-1.0) - u1)))) * Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.400000006

   1. Initial program 98.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. sqr-neg N/A

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-neg.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      15. lower-*.f32 98.9

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 98.9%

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)} \]

   if 0.400000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. lower-fma.f32 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.4000000059604645:\\ \;\;\;\;\sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.5)
   (*
    (sqrt (/ u1 (/ (+ (* u1 u1) -1.0) (- -1.0 u1))))
    (*
     u2
     (fma
      (* u2 u2)
      (fma
       (* u2 u2)
       (fma (* u2 u2) -76.70585975309672 81.6052492761019)
       -41.341702240407926)
      6.28318530718)))
   (* (sin (* 6.28318530718 u2)) (sqrt (fma u1 u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.5f) {
		tmp = sqrtf((u1 / (((u1 * u1) + -1.0f) / (-1.0f - u1)))) * (u2 * fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f), -41.341702240407926f), 6.28318530718f));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(fmaf(u1, u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.5))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(Float32(u1 * u1) + Float32(-1.0)) / Float32(Float32(-1.0) - u1)))) * Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.5

   1. Initial program 98.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. sqr-neg N/A

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-neg.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      15. lower-*.f32 98.8

          \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 98.8%

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)} \]

   if 0.5 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 87.3

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

   5. Applied rewrites 87.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Final simplification 98.7%

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

Alternative 7: 93.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (/ (+ (* u1 u1) -1.0) (- -1.0 u1))))
  (*
   u2
   (fma
    (* u2 u2)
    (fma
     (* u2 u2)
     (fma (* u2 u2) -76.70585975309672 81.6052492761019)
     -41.341702240407926)
    6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (((u1 * u1) + -1.0f) / (-1.0f - u1)))) * (u2 * fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f), -41.341702240407926f), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(Float32(u1 * u1) + Float32(-1.0)) / Float32(Float32(-1.0) - u1)))) * Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.7%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. sqr-neg N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-neg.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   13. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   15. lower-*.f32 92.8

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

7. Applied rewrites 92.8%

   \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)} \]

8. Final simplification 92.8%

   \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]
9. Add Preprocessing

Alternative 8: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.07000000029802322:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.07000000029802322)
   (*
    u2
    (*
     (sqrt (/ u1 (- 1.0 u1)))
     (fma u2 (* u2 -41.341702240407926) 6.28318530718)))
   (*
    (*
     u2
     (fma
      (* u2 u2)
      (fma
       (* u2 u2)
       (fma (* u2 u2) -76.70585975309672 81.6052492761019)
       -41.341702240407926)
      6.28318530718))
    (sqrt (fma u1 (fma u1 u1 u1) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.07000000029802322f) {
		tmp = u2 * (sqrtf((u1 / (1.0f - u1))) * fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f));
	} else {
		tmp = (u2 * fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f), -41.341702240407926f), 6.28318530718f)) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.07000000029802322))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0700000003

   1. Initial program 98.8%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-commutative N/A

         \[\leadsto u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \color{blue}{\left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto u2 \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)\right)} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)} \]

   if 0.0700000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.4%

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

      1. flip3-- N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. associate-/r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}{{1}^{3} - {u1}^{3}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. frac-2neg N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. lower-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(\left(u1 \cdot u1 + 1 \cdot u1\right) + 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + u1 \cdot 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + \color{blue}{u1}\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1, u1 \cdot u1 + 1 \cdot u1, u1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{1} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      15. sub-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left({u1}^{3}\right)\right)\right)}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      16. cube-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(1 + \color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      15. lower-*.f32 73.4

          \[\leadsto \sqrt{\frac{-\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{-\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(1 + u1\right) + 1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + u1 \cdot 1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + \color{blue}{u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(1 + u1\right), u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{\left(1 + u1\right) \cdot u1}, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{\left(u1 + 1\right)} \cdot u1, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot u1 + u1}, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      8. lower-fma.f32 71.0

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

   10. Applied rewrites 71.0%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.07000000029802322:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (/ (+ (* u1 u1) -1.0) (- -1.0 u1))))
  (*
   u2
   (fma
    (* u2 u2)
    (fma u2 (* u2 81.6052492761019) -41.341702240407926)
    6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (((u1 * u1) + -1.0f) / (-1.0f - u1)))) * (u2 * fmaf((u2 * u2), fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(Float32(u1 * u1) + Float32(-1.0)) / Float32(Float32(-1.0) - u1)))) * Float32(u2 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.7%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. sqr-neg N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-neg.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{u2 \cdot \left(u2 \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, u2 \cdot \left(u2 \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   12. lower-*.f32 89.7

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, \color{blue}{u2 \cdot 81.6052492761019}, -41.341702240407926\right), 6.28318530718\right)\right) \]

7. Applied rewrites 89.7%

   \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\left(-u1\right) - 1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

8. Final simplification 89.7%

   \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 + -1}{-1 - u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]
9. Add Preprocessing

Alternative 10: 92.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17000000178813934:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.17000000178813934)
   (*
    u2
    (*
     (sqrt (/ u1 (- 1.0 u1)))
     (fma u2 (* u2 -41.341702240407926) 6.28318530718)))
   (*
    (*
     u2
     (fma
      (* u2 u2)
      (fma
       (* u2 u2)
       (fma (* u2 u2) -76.70585975309672 81.6052492761019)
       -41.341702240407926)
      6.28318530718))
    (sqrt (fma u1 u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.17000000178813934f) {
		tmp = u2 * (sqrtf((u1 / (1.0f - u1))) * fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f));
	} else {
		tmp = (u2 * fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f), -41.341702240407926f), 6.28318530718f)) * sqrtf(fmaf(u1, u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.17000000178813934))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.170000002

   1. Initial program 98.8%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-commutative N/A

         \[\leadsto u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \color{blue}{\left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto u2 \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)} \]

   if 0.170000002 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.2%

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

      1. flip3-- N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. associate-/r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}{{1}^{3} - {u1}^{3}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. frac-2neg N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. lower-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{neg}\left(u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(\left(u1 \cdot u1 + 1 \cdot u1\right) + 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + u1 \cdot 1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\left(u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + \color{blue}{u1}\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1, u1 \cdot u1 + 1 \cdot u1, u1\right)}\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)\right)}{\mathsf{neg}\left(\left({1}^{3} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{1} - {u1}^{3}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      15. sub-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left({u1}^{3}\right)\right)\right)}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      16. cube-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{\mathsf{neg}\left(\left(1 + \color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      15. lower-*.f32 70.0

          \[\leadsto \sqrt{\frac{-\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 70.0%

      \[\leadsto \sqrt{\frac{-\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{-1 + u1 \cdot \left(u1 \cdot u1\right)}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot u1 + u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      4. lower-fma.f32 67.3

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

   10. Applied rewrites 67.3%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.17000000178813934:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, u2 \cdot \left(u2 \cdot 81.6052492761019\right), \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (sqrt (/ u1 (- 1.0 u1)))
   (fma
    (* u2 u2)
    (* u2 (* u2 81.6052492761019))
    (fma u2 (* u2 -41.341702240407926) 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf((u1 / (1.0f - u1))) * fmaf((u2 * u2), (u2 * (u2 * 81.6052492761019f)), fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(u2 * u2), Float32(u2 * Float32(u2 * Float32(81.6052492761019))), fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)\right)\right)} \]

5. Applied rewrites 89.6%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, u2 \cdot \left(u2 \cdot 81.6052492761019\right), \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\right)} \]
6. Add Preprocessing

Alternative 12: 87.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.01600000075995922:\\ \;\;\;\;u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.01600000075995922)
     (*
      u2
      (*
       (sqrt (fma u1 (fma u1 u1 u1) u1))
       (fma -41.341702240407926 (* u2 u2) 6.28318530718)))
     (* (* 6.28318530718 u2) (sqrt t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.01600000075995922f) {
		tmp = u2 * (sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1)) * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	} else {
		tmp = (6.28318530718f * u2) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.01600000075995922))
		tmp = Float32(u2 * Float32(sqrt(fma(u1, fma(u1, u1, u1), u1)) * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.0160000008

   1. Initial program 98.8%

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

   3. Taylor expanded in u1 around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left({u1}^{2} + 1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      17. unpow2 N/A

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

      18. lower-fma.f32 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right)} \]

   8. Applied rewrites 87.1%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right) \cdot \mathsf{fma}\left(u1, u1, 1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(1 + u1\right) + 1\right)}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       2. distribute-lft-in N/A

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + u1 \cdot 1}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       3. *-rgt-identity N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + \color{blue}{u1}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(1 + u1\right), u1\right)}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \color{blue}{\left(1 + u1\right) \cdot u1}, u1\right)} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \color{blue}{\left(u1 + 1\right)} \cdot u1, u1\right)} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       7. distribute-lft1-in N/A

          \[\leadsto u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot u1 + u1}, u1\right)} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       8. lower-fma.f32 87.0

          \[\leadsto u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \]

   11. Applied rewrites 87.0%

       \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \]

   if 0.0160000008 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 76.5%

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

Alternative 13: 86.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.0011599999852478504:\\ \;\;\;\;u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.0011599999852478504)
     (*
      u2
      (*
       (sqrt (fma u1 u1 u1))
       (fma -41.341702240407926 (* u2 u2) 6.28318530718)))
     (* (* 6.28318530718 u2) (sqrt t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.0011599999852478504f) {
		tmp = u2 * (sqrtf(fmaf(u1, u1, u1)) * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	} else {
		tmp = (6.28318530718f * u2) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0011599999852478504))
		tmp = Float32(u2 * Float32(sqrt(fma(u1, u1, u1)) * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00115999999

   1. Initial program 98.7%

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

   3. Taylor expanded in u1 around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left({u1}^{2} + 1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      17. unpow2 N/A

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

      18. lower-fma.f32 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right)} \]

   8. Applied rewrites 86.9%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right) \cdot \mathsf{fma}\left(u1, u1, 1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       2. +-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       3. distribute-lft1-in N/A

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot u1 + u1}} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

       4. lower-fma.f32 86.6

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \]

   11. Applied rewrites 86.6%

       \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \]

   if 0.00115999999 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 78.1%

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

Alternative 14: 84.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006500000134110451)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (* u2 (* (fma -41.341702240407926 (* u2 u2) 6.28318530718) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006500000134110451f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = u2 * (fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.006500000134110451))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(u2 * Float32(fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00650000013

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 96.9%

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

   if 0.00650000013 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.7%

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

   3. Taylor expanded in u1 around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left({u1}^{2} + 1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      17. unpow2 N/A

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

      18. lower-fma.f32 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right)} \]

   8. Applied rewrites 61.7%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right) \cdot \mathsf{fma}\left(u1, u1, 1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

       2. lower-sqrt.f32 N/A

          \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1}} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

       6. lower-*.f32 55.5

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   11. Applied rewrites 55.5%

       \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (sqrt (/ u1 (- 1.0 u1)))
   (fma u2 (* u2 -41.341702240407926) 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf((u1 / (1.0f - u1))) * fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. *-commutative N/A

      \[\leadsto u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \color{blue}{\left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   3. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)\right)} \]

5. Applied rewrites 87.1%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)} \]
6. Add Preprocessing

Alternative 16: 79.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006500000134110451)
   (* (* 6.28318530718 u2) (sqrt (fma u1 (fma u1 u1 u1) u1)))
   (* u2 (* (fma -41.341702240407926 (* u2 u2) 6.28318530718) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006500000134110451f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = u2 * (fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.006500000134110451))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = Float32(u2 * Float32(fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00650000013

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f32 90.9

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

   8. Applied rewrites 90.9%

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

   if 0.00650000013 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.7%

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

   3. Taylor expanded in u1 around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left({u1}^{2} + 1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      17. unpow2 N/A

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

      18. lower-fma.f32 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right)} \]

   8. Applied rewrites 61.7%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right) \cdot \mathsf{fma}\left(u1, u1, 1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

       2. lower-sqrt.f32 N/A

          \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1}} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

       6. lower-*.f32 55.5

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   11. Applied rewrites 55.5%

       \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006500000134110451)
   (* (* 6.28318530718 u2) (sqrt (fma u1 u1 u1)))
   (* u2 (* (fma -41.341702240407926 (* u2 u2) 6.28318530718) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006500000134110451f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, u1, u1));
	} else {
		tmp = u2 * (fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.006500000134110451))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, u1, u1)));
	else
		tmp = Float32(u2 * Float32(fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00650000013

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 87.0

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

   8. Applied rewrites 87.0%

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

   if 0.00650000013 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.7%

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

   3. Taylor expanded in u1 around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left({u1}^{2} + 1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      17. unpow2 N/A

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

      18. lower-fma.f32 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\left(1 + {u1}^{2}\right) \cdot \left(u1 + {u1}^{2}\right)}\right)} \]

   8. Applied rewrites 61.7%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right) \cdot \mathsf{fma}\left(u1, u1, 1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

       2. lower-sqrt.f32 N/A

          \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1}} \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

       6. lower-*.f32 55.5

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   11. Applied rewrites 55.5%

       \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-rgt-identity N/A

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

   18. lower-/.f32 N/A

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Applied rewrites 78.2%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f32 71.1

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

8. Applied rewrites 71.1%

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

Alternative 19: 64.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-rgt-identity N/A

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

   18. lower-/.f32 N/A

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Applied rewrites 78.2%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 62.1

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

8. Applied rewrites 62.1%

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

   1. lift-sqrt.f32 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 62.1

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

10. Applied rewrites 62.1%

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

11. Final simplification 62.1%

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

Alternative 20: 64.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-rgt-identity N/A

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

   18. lower-/.f32 N/A

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Applied rewrites 78.2%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 62.1

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

8. Applied rewrites 62.1%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))