Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.8%

Time: 13.1s

Alternatives: 22

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{1}{u1 + 1} - \left(u1 + -1\right) \cdot \frac{u1}{-1 + u1 \cdot u1}}{\left(u1 + -1\right) \cdot \frac{1}{u1 + 1}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

6. Applied egg-rr 98.9%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f32 N/A

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

   7. flip3-- N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f32 N/A

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

   10. div-inv N/A

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

   11. associate-*l* N/A

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

   12. inv-pow N/A

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

   13. pow-plus N/A

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

   14. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1 \cdot {\left(1 + u1\right)}^{\color{blue}{0}}}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. *-lft-identity N/A

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

8. Applied egg-rr 99.0%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.054999999701976776:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -85.45681720672748, u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))) (t_1 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 t_1) 0.054999999701976776)
     (* t_0 (sqrt (fma u1 (fma u1 u1 u1) u1)))
     (fma
      (* t_1 (* u2 u2))
      (fma
       (* (* u2 u2) -85.45681720672748)
       (* u2 u2)
       (fma u2 (* u2 64.93939402268539) -19.739208802181317))
      t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((t_0 * t_1) <= 0.054999999701976776f) {
		tmp = t_0 * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = fmaf((t_1 * (u2 * u2)), fmaf(((u2 * u2) * -85.45681720672748f), (u2 * u2), fmaf(u2, (u2 * 64.93939402268539f), -19.739208802181317f)), t_1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.054999999701976776))
		tmp = Float32(t_0 * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = fma(Float32(t_1 * Float32(u2 * u2)), fma(Float32(Float32(u2 * u2) * Float32(-85.45681720672748)), Float32(u2 * u2), fma(u2, Float32(u2 * Float32(64.93939402268539)), Float32(-19.739208802181317))), t_1);
	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))) < 0.0549999997

   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. lower-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. lower-fma.f32 98.7

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

   5. Simplified 98.7%

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

   if 0.0549999997 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.1%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.054999999701976776:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -85.45681720672748, u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.027000000700354576:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{\mathsf{fma}\left(-u1, u1, 1\right)} \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (cos (* 6.28318530718 u2)) (sqrt (/ u1 (- 1.0 u1))))
      0.027000000700354576)
   (* (sqrt (fma u1 u1 u1)) (fma -19.739208802181317 (* u2 u2) 1.0))
   (sqrt (* (/ u1 (fma (- u1) u1 1.0)) (+ u1 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((cosf((6.28318530718f * u2)) * sqrtf((u1 / (1.0f - u1)))) <= 0.027000000700354576f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * fmaf(-19.739208802181317f, (u2 * u2), 1.0f);
	} else {
		tmp = sqrtf(((u1 / fmaf(-u1, u1, 1.0f)) * (u1 + 1.0f)));
	}
	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(0.027000000700354576))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)));
	else
		tmp = sqrt(Float32(Float32(u1 / fma(Float32(-u1), u1, Float32(1.0))) * Float32(u1 + Float32(1.0))));
	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))) < 0.0270000007

   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. lower-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. lower-fma.f32 98.7

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

   5. Simplified 98.7%

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

   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. lower-*.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. lower-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. lower-*.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. *-lft-identity N/A

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

      12. unpow2 N/A

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

      13. distribute-rgt-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. lower-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 89.9%

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

   9. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft1-in N/A

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

       4. lower-fma.f32 89.7

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

   11. Simplified 89.7%

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

   if 0.0270000007 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.0%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 87.8

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

   5. Simplified 87.8%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. lift-+.f32 N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. lift-+.f32 N/A

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

      11. lower-*.f32 N/A

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

   7. Applied egg-rr 87.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.027000000700354576:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{\mathsf{fma}\left(-u1, u1, 1\right)} \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.027000000700354576:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \frac{-1}{u1 + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (cos (* 6.28318530718 u2)) (sqrt (/ u1 (- 1.0 u1))))
      0.027000000700354576)
   (* (sqrt (fma u1 u1 u1)) (fma -19.739208802181317 (* u2 u2) 1.0))
   (sqrt (* u1 (/ -1.0 (+ u1 -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((cosf((6.28318530718f * u2)) * sqrtf((u1 / (1.0f - u1)))) <= 0.027000000700354576f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * fmaf(-19.739208802181317f, (u2 * u2), 1.0f);
	} else {
		tmp = sqrtf((u1 * (-1.0f / (u1 + -1.0f))));
	}
	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(0.027000000700354576))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)));
	else
		tmp = sqrt(Float32(u1 * Float32(Float32(-1.0) / Float32(u1 + Float32(-1.0)))));
	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))) < 0.0270000007

   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. lower-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. lower-fma.f32 98.7

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

   5. Simplified 98.7%

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

   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. lower-*.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. lower-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. lower-*.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. *-lft-identity N/A

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

      12. unpow2 N/A

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

      13. distribute-rgt-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. lower-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 89.9%

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

   9. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft1-in N/A

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

       4. lower-fma.f32 89.7

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

   11. Simplified 89.7%

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

   if 0.0270000007 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.0%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 87.8

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

   5. Simplified 87.8%

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

      1. lift--.f32 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. metadata-eval N/A

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

      6. lift--.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. lift-+.f32 N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f32 87.8

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

   7. Applied egg-rr 87.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.027000000700354576:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \frac{-1}{u1 + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9997000098228455:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9997000098228455)
   (*
    (sqrt (fma u1 (fma u1 u1 u1) u1))
    (fma
     (* u2 u2)
     (fma
      (* u2 u2)
      (fma (* u2 u2) -85.45681720672748 64.93939402268539)
      -19.739208802181317)
     1.0))
   (* (sqrt (/ u1 (- 1.0 u1))) (fma u2 (* u2 -19.739208802181317) 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9997000098228455f) {
		tmp = sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1)) * fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f), -19.739208802181317f), 1.0f);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * fmaf(u2, (u2 * -19.739208802181317f), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9997000098228455))
		tmp = Float32(sqrt(fma(u1, fma(u1, u1, 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)));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99970001

   1. Initial program 97.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. lower-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. lower-fma.f32 93.9

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

   5. Simplified 93.9%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 76.6

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

   8. Simplified 76.6%

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

   if 0.99970001 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)} \]
   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. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      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. lower-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. lower-*.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower--.f32 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9997000098228455:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6000000238418579:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1 \cdot \left(u1 \cdot u1\right)} \cdot \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.6000000238418579)
   (*
    (sqrt (* (/ u1 (- 1.0 (* u1 (* u1 u1)))) (+ 1.0 (fma u1 u1 u1))))
    (fma
     (* u2 u2)
     (fma
      u2
      (* u2 (fma (* u2 u2) -85.45681720672748 64.93939402268539))
      -19.739208802181317)
     1.0))
   (* (cos (* 6.28318530718 u2)) (sqrt (fma u1 u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.6000000238418579f) {
		tmp = sqrtf(((u1 / (1.0f - (u1 * (u1 * u1)))) * (1.0f + fmaf(u1, u1, u1)))) * fmaf((u2 * u2), fmaf(u2, (u2 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f)), -19.739208802181317f), 1.0f);
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(fmaf(u1, u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.6000000238418579))
		tmp = Float32(sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - Float32(u1 * Float32(u1 * u1)))) * Float32(Float32(1.0) + fma(u1, u1, u1)))) * fma(Float32(u2 * u2), fma(u2, Float32(u2 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539))), Float32(-19.739208802181317)), Float32(1.0)));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.600000024

   1. Initial program 99.3%

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

   4. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{1}{u1 + 1} - \left(u1 + -1\right) \cdot \frac{u1}{-1 + u1 \cdot u1}}{\left(u1 + -1\right) \cdot \frac{1}{u1 + 1}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   6. Applied egg-rr 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f32 N/A

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

      7. flip3-- N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f32 N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. inv-pow N/A

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

      13. pow-plus N/A

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

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{u1 \cdot {\left(1 + u1\right)}^{\color{blue}{0}}}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lft-identity N/A

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in u2 around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f32 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f32 N/A

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

       5. sub-neg N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f32 N/A

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

       10. lower-*.f32 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f32 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f32 99.2

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

   11. Simplified 99.2%

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

   if 0.600000024 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.7%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 88.8

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

   5. Simplified 88.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6000000238418579:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1 \cdot \left(u1 \cdot u1\right)} \cdot \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% 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 8: 93.3% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (/ u1 (- 1.0 (* u1 (* u1 u1)))) (+ 1.0 (fma u1 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(((u1 / (1.0f - (u1 * (u1 * u1)))) * (1.0f + fmaf(u1, 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(u1 / Float32(Float32(1.0) - Float32(u1 * Float32(u1 * u1)))) * Float32(Float32(1.0) + fma(u1, u1, u1)))) * fma(Float32(u2 * u2), fma(u2, Float32(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

4. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{1}{u1 + 1} - \left(u1 + -1\right) \cdot \frac{u1}{-1 + u1 \cdot u1}}{\left(u1 + -1\right) \cdot \frac{1}{u1 + 1}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

6. Applied egg-rr 98.9%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f32 N/A

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

   7. flip3-- N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f32 N/A

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

   10. div-inv N/A

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

   11. associate-*l* N/A

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

   12. inv-pow N/A

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

   13. pow-plus N/A

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

   14. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1 \cdot {\left(1 + u1\right)}^{\color{blue}{0}}}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. *-lft-identity N/A

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

8. Applied egg-rr 99.0%

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

9. Taylor expanded in u2 around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f32 N/A

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

    5. sub-neg N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. metadata-eval N/A

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

    9. lower-fma.f32 N/A

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

    10. lower-*.f32 N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. lower-fma.f32 N/A

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

    14. unpow2 N/A

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

    15. lower-*.f32 95.0

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

11. Simplified 95.0%

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

Alternative 9: 91.4% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (/ u1 (- 1.0 (* u1 (* u1 u1)))) (+ 1.0 (fma u1 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(((u1 / (1.0f - (u1 * (u1 * u1)))) * (1.0f + fmaf(u1, 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(Float32(u1 / Float32(Float32(1.0) - Float32(u1 * Float32(u1 * u1)))) * Float32(Float32(1.0) + fma(u1, 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

4. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{1}{u1 + 1} - \left(u1 + -1\right) \cdot \frac{u1}{-1 + u1 \cdot u1}}{\left(u1 + -1\right) \cdot \frac{1}{u1 + 1}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

6. Applied egg-rr 98.9%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f32 N/A

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

   7. flip3-- N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f32 N/A

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

   10. div-inv N/A

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

   11. associate-*l* N/A

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

   12. inv-pow N/A

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

   13. pow-plus N/A

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

   14. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1 \cdot {\left(1 + u1\right)}^{\color{blue}{0}}}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. *-lft-identity N/A

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

8. Applied egg-rr 99.0%

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

9. Taylor expanded in u2 around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f32 N/A

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

    5. sub-neg N/A

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

    6. *-commutative N/A

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

    7. metadata-eval N/A

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

    8. lower-fma.f32 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f32 93.5

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

11. Simplified 93.5%

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

Alternative 10: 91.5% 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

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. associate-+l+ N/A

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

5. Simplified 93.5%

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

Alternative 11: 86.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.007000000216066837:\\ \;\;\;\;\mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{u1 \cdot \left(u1 + \mathsf{fma}\left(u1, u1, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{\mathsf{fma}\left(-u1, u1, 1\right)} \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.007000000216066837)
   (*
    (fma -19.739208802181317 (* u2 u2) 1.0)
    (sqrt (* u1 (+ u1 (fma u1 u1 1.0)))))
   (sqrt (* (/ u1 (fma (- u1) u1 1.0)) (+ u1 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.007000000216066837f) {
		tmp = fmaf(-19.739208802181317f, (u2 * u2), 1.0f) * sqrtf((u1 * (u1 + fmaf(u1, u1, 1.0f))));
	} else {
		tmp = sqrtf(((u1 / fmaf(-u1, u1, 1.0f)) * (u1 + 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.007000000216066837))
		tmp = Float32(fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)) * sqrt(Float32(u1 * Float32(u1 + fma(u1, u1, Float32(1.0))))));
	else
		tmp = sqrt(Float32(Float32(u1 / fma(Float32(-u1), u1, Float32(1.0))) * Float32(u1 + Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00700000022

   1. Initial program 99.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. lower-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. lower-fma.f32 99.0

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

   5. Simplified 99.0%

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

   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. lower-*.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. lower-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. lower-*.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. *-lft-identity N/A

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

      12. unpow2 N/A

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

      13. distribute-rgt-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. lower-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 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}} \]
   9. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. *-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. lift-+.f32 N/A

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

      6. lower-*.f32 90.4

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

      7. lift-+.f32 N/A

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

      8. lift-fma.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. lower-fma.f32 90.4

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

   10. Applied egg-rr 90.4%

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

   if 0.00700000022 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 87.4

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

   5. Simplified 87.4%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. lift-+.f32 N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. lift-+.f32 N/A

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

      11. lower-*.f32 N/A

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

   7. Applied egg-rr 87.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.007000000216066837:\\ \;\;\;\;\mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{u1 \cdot \left(u1 + \mathsf{fma}\left(u1, u1, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{\mathsf{fma}\left(-u1, u1, 1\right)} \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.007000000216066837:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{\mathsf{fma}\left(-u1, u1, 1\right)} \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.007000000216066837)
   (*
    (sqrt (fma u1 (fma u1 u1 u1) u1))
    (fma -19.739208802181317 (* u2 u2) 1.0))
   (sqrt (* (/ u1 (fma (- u1) u1 1.0)) (+ u1 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.007000000216066837f) {
		tmp = sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1)) * fmaf(-19.739208802181317f, (u2 * u2), 1.0f);
	} else {
		tmp = sqrtf(((u1 / fmaf(-u1, u1, 1.0f)) * (u1 + 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.007000000216066837))
		tmp = Float32(sqrt(fma(u1, fma(u1, u1, u1), u1)) * fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)));
	else
		tmp = sqrt(Float32(Float32(u1 / fma(Float32(-u1), u1, Float32(1.0))) * Float32(u1 + Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00700000022

   1. Initial program 99.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. lower-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. lower-fma.f32 99.0

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

   5. Simplified 99.0%

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

   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. lower-*.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. lower-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. lower-*.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. *-lft-identity N/A

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

      12. unpow2 N/A

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

      13. distribute-rgt-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. lower-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 90.4%

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

   if 0.00700000022 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 87.4

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

   5. Simplified 87.4%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. lift-+.f32 N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. lift-+.f32 N/A

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

      11. lower-*.f32 N/A

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

   7. Applied egg-rr 87.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.007000000216066837:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{\mathsf{fma}\left(-u1, u1, 1\right)} \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.4% 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. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   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. lower-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. lower-*.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower--.f32 90.9

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

5. Simplified 90.9%

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

6. Final simplification 90.9%

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

Alternative 14: 83.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (fma u2 (* u2 -19.739208802181317) 1.0) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = fmaf(u2, (u2 * -19.739208802181317f), 1.0f) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009999999776482582))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 96.8

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

   5. Simplified 96.8%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. lower-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. lower-fma.f32 93.5

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

   5. Simplified 93.5%

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

   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. lower-*.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. lower-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. lower-*.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. *-lft-identity N/A

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

      12. unpow2 N/A

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

      13. distribute-rgt-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. lower-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 65.5%

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

   9. Taylor expanded in u1 around 0

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

       1. lower-*.f32 N/A

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

       2. lower-sqrt.f32 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-fma.f32 N/A

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

       8. lower-*.f32 58.5

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

   11. Simplified 58.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.9% 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. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

Alternative 16: 74.0% 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. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

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. lower-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. lower-fma.f32 77.8

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

8. Simplified 77.8%

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

Alternative 17: 71.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \mathsf{fma}\left(u1, 0.5, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (sqrt u1) (fma u1 0.5 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * fmaf(u1, 0.5f, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * fma(u1, Float32(0.5), 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

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

   1. lift--.f32 N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. pow1/2 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow1/2 N/A

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

   9. sqrt-div N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-sqrt.f32 82.3

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

7. Applied egg-rr 82.3%

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

8. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot u1 + 1\right)} \cdot \sqrt{u1} \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{u1 \cdot \frac{1}{2}} + 1\right) \cdot \sqrt{u1} \]

   3. lower-fma.f32 75.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \sqrt{u1} \]

10. Simplified 75.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \sqrt{u1} \]

11. Final simplification 75.0%

    \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(u1, 0.5, 1\right) \]
12. Add Preprocessing

Alternative 18: 71.1% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

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. lower-fma.f32 74.8

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

8. Simplified 74.8%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]
9. Step-by-step derivation

   1. distribute-lft1-in N/A

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

   2. lift-+.f32 N/A

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

   3. lower-*.f32 74.8

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

10. Applied egg-rr 74.8%

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

11. Final simplification 74.8%

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

Alternative 19: 71.2% 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. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

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. lower-fma.f32 74.8

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

8. Simplified 74.8%

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

Alternative 20: 63.0% 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. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 66.3

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

8. Simplified 66.3%

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

Alternative 21: 20.2% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ u1 + 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (+ u1 0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 + 0.5f;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u1 + 0.5e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 + single(0.5);
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. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

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. lower-fma.f32 74.8

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

8. Simplified 74.8%

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

9. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{u1}\right)} \]
10. Step-by-step derivation

    1. distribute-rgt-in N/A

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

    2. *-lft-identity N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

       \[\leadsto u1 + \frac{1}{2} \cdot \color{blue}{1} \]

    5. metadata-eval N/A

       \[\leadsto u1 + \color{blue}{\frac{1}{2}} \]

    6. lower-+.f32 19.9

       \[\leadsto \color{blue}{u1 + 0.5} \]

11. Simplified 19.9%

    \[\leadsto \color{blue}{u1 + 0.5} \]
12. Add Preprocessing

Alternative 22: 4.4% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = -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. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 82.8

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

5. Simplified 82.8%

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

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. lower-fma.f32 74.8

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

8. Simplified 74.8%

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

9. Taylor expanded in u1 around -inf

   \[\leadsto \color{blue}{-1 \cdot u1} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(u1\right)} \]

    2. lower-neg.f32 4.2

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

11. Simplified 4.2%

    \[\leadsto \color{blue}{-u1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))