Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 98.9% → 98.9%

Time: 13.4s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

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

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

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

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

   9. *-lft-identity N/A

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

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

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. distribute-lft-neg-out N/A

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

   17. sqr-neg N/A

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

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

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

   19. *-lowering-*.f32 98.9

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.0949999988079071:\\ \;\;\;\;\mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\sqrt{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (cos (* 6.28318530718 u2)) (sqrt (/ u1 (- 1.0 u1))))
      0.0949999988079071)
   (*
    (fma -19.739208802181317 (* u2 u2) 1.0)
    (sqrt (fma u1 (fma u1 u1 u1) u1)))
   (/ (sqrt u1) (sqrt (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((cosf((6.28318530718f * u2)) * sqrtf((u1 / (1.0f - u1)))) <= 0.0949999988079071f) {
		tmp = fmaf(-19.739208802181317f, (u2 * u2), 1.0f) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = sqrtf(u1) / sqrtf((1.0f - u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))) <= Float32(0.0949999988079071))
		tmp = Float32(fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = Float32(sqrt(u1) / sqrt(Float32(Float32(1.0) - u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 0.0949999988

   1. Initial program 98.9%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f32 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lft-identity N/A

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

      10. *-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. unpow2 N/A

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

      13. distribute-lft-in N/A

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

      14. distribute-rgt-in N/A

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

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

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

   8. Simplified 89.4%

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

   if 0.0949999988 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 98.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

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

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

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

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

      13. *-rgt-identity N/A

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

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

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

      15. associate-*r* N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   5. Simplified 86.0%

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

      1. sqrt-div N/A

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

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

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

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

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

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

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

      5. --lowering--.f32 86.1

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

   7. Applied egg-rr 86.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.0949999988079071:\\ \;\;\;\;\mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\sqrt{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Final simplification 98.9%

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

Alternative 4: 93.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathsf{fma}\left(u2 \cdot u2, t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (fma
    (* u2 u2)
    (*
     t_0
     (fma
      (* u2 u2)
      (* (* u2 u2) -85.45681720672748)
      (fma u2 (* u2 64.93939402268539) -19.739208802181317)))
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return fmaf((u2 * u2), (t_0 * fmaf((u2 * u2), ((u2 * u2) * -85.45681720672748f), fmaf(u2, (u2 * 64.93939402268539f), -19.739208802181317f))), t_0);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return fma(Float32(u2 * u2), Float32(t_0 * fma(Float32(u2 * u2), Float32(Float32(u2 * u2) * Float32(-85.45681720672748)), fma(u2, Float32(u2 * Float32(64.93939402268539)), Float32(-19.739208802181317)))), t_0)
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 95.5%

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

Alternative 5: 92.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. clear-num N/A

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

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

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

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

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

   8. /-lowering-/.f32 98.7

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f32 N/A

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 95.3

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

7. Simplified 95.3%

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

8. Taylor expanded in u1 around 0

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

   1. lft-mult-inverse N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-in N/A

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

   9. lft-mult-inverse N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. --lowering--.f32 95.3

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

10. Simplified 95.3%

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

Alternative 6: 92.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

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

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

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

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

   9. *-lft-identity N/A

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

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

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. distribute-lft-neg-out N/A

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

   17. sqr-neg N/A

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

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

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

   19. *-lowering-*.f32 98.9

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

4. Applied egg-rr 98.9%

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

   1. frac-2neg N/A

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

   2. clear-num N/A

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

   3. remove-double-neg N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-/l/ N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. flip-- N/A

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

   13. associate-/r/ N/A

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

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

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f32 N/A

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 95.3

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

9. Simplified 95.3%

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

10. Final simplification 95.3%

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

Alternative 7: 91.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.15000000596046448)
   (* (sqrt (/ u1 (- 1.0 u1))) (fma u2 (* u2 -19.739208802181317) 1.0))
   (*
    (fma
     (* u2 u2)
     (fma
      (* u2 u2)
      (fma (* u2 u2) -85.45681720672748 64.93939402268539)
      -19.739208802181317)
     1.0)
    (sqrt (fma u1 u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.15000000596046448f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * fmaf(u2, (u2 * -19.739208802181317f), 1.0f);
	} else {
		tmp = fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f), -19.739208802181317f), 1.0f) * 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(0.15000000596046448))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)));
	else
		tmp = Float32(fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)), Float32(-19.739208802181317)), Float32(1.0)) * 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) < 0.150000006

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-rgt-identity N/A

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

      11. sub-neg N/A

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

      12. rgt-mult-inverse N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-in N/A

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

      18. +-commutative N/A

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

      19. sub-neg N/A

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

      20. associate-*r* N/A

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

   5. Simplified 98.6%

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

   if 0.150000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.4%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

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

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

      8. /-lowering-/.f32 96.3

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 72.0

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

   7. Simplified 72.0%

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

   8. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f32 63.9

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

   10. Simplified 63.9%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

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

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

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

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

   9. *-lft-identity N/A

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

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

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. distribute-lft-neg-out N/A

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

   17. sqr-neg N/A

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

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

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

   19. *-lowering-*.f32 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 93.3

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

7. Simplified 93.3%

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

8. Final simplification 93.3%

   \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right)}{\left(--1\right) - u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right), 1\right) \]
9. Add Preprocessing

Alternative 9: 91.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 93.2%

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

Alternative 10: 90.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

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

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

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

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

   9. *-lft-identity N/A

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

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

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. distribute-lft-neg-out N/A

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

   17. sqr-neg N/A

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

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

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

   19. *-lowering-*.f32 98.9

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

4. Applied egg-rr 98.9%

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

   1. frac-2neg N/A

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

   2. clear-num N/A

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

   3. remove-double-neg N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-/l/ N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. flip-- N/A

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

   13. associate-/r/ N/A

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

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

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 93.1

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

9. Simplified 93.1%

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

10. Final simplification 93.1%

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

Alternative 11: 87.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. *-rgt-identity N/A

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

   11. sub-neg N/A

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

   12. rgt-mult-inverse N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-frac2 N/A

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

   15. mul-1-neg N/A

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

   16. *-rgt-identity N/A

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

   17. distribute-lft-in N/A

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

   18. +-commutative N/A

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

   19. sub-neg N/A

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

   20. associate-*r* N/A

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

5. Simplified 90.4%

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

6. Final simplification 90.4%

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

Alternative 12: 79.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

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

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

   13. *-rgt-identity N/A

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

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

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Simplified 82.9%

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

Alternative 13: 73.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

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

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

   13. *-rgt-identity N/A

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

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

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Simplified 82.9%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. accelerator-lowering-fma.f32 76.6

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

8. Simplified 76.6%

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

Alternative 14: 70.7% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

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

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

   13. *-rgt-identity N/A

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

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

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Simplified 82.9%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. accelerator-lowering-fma.f32 73.5

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

8. Simplified 73.5%

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

Alternative 15: 62.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

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

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

   13. *-rgt-identity N/A

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

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

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Simplified 82.9%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 65.4

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

8. Simplified 65.4%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))