Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5

Percentage Accurate: 60.4% → 98.1%

Time: 2.3min

Alternatives: 16

Speedup: 1.3×

Specification

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (/
 (- (log (- 1.0 u0)))
 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Initial Program: 60.4% accurate, 1.0× speedup

Math

\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (/
 (- (log (- 1.0 u0)))
 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Alternative 1: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, t\_0, sin2phi\right)}{\left(-alphay\right) \cdot t\_1}} \cdot alphay\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0 \cdot u0, u0\right)}{t\_0 + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (/ cos2phi (* alphax alphax))) (t_1 (log (- 1.0 u0))))
  (if (<= t_1 -0.03999999910593033)
    (*
     (/
      1.0
      (/ (fma (* alphay alphay) t_0 sin2phi) (* (- alphay) t_1)))
     alphay)
    (/
     (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) (* u0 u0) u0)
     (+ t_0 (/ 1.0 (/ (* alphay alphay) sin2phi)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (t_1 <= -0.03999999910593033f) {
		tmp = (1.0f / (fmaf((alphay * alphay), t_0, sin2phi) / (-alphay * t_1))) * alphay;
	} else {
		tmp = fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), (u0 * u0), u0) / (t_0 + (1.0f / ((alphay * alphay) / sin2phi)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(Float32(1.0) / Float32(fma(Float32(alphay * alphay), t_0, sin2phi) / Float32(Float32(-alphay) * t_1))) * alphay);
	else
		tmp = Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), Float32(u0 * u0), u0) / Float32(t_0 + Float32(Float32(1.0) / Float32(Float32(alphay * alphay) / sin2phi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0399999991

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \color{blue}{\left(\left(-alphay\right) \cdot alphay\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\left(-alphay\right) \cdot \log \left(1 - u0\right)}} \cdot alphay} \]

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{sin2phi \cdot \frac{1}{alphay \cdot alphay}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot alphay} \cdot sin2phi}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\color{blue}{1 \cdot 1}}{alphay \cdot alphay} \cdot sin2phi} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1 \cdot 1}{\color{blue}{alphay \cdot alphay}} \cdot sin2phi} \]

      6. times-frac N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\left(\frac{1}{alphay} \cdot \frac{1}{alphay}\right)} \cdot sin2phi} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\color{blue}{\frac{1}{alphay}} \cdot \frac{1}{alphay}\right) \cdot sin2phi} \]

      8. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\frac{1}{alphay} \cdot \color{blue}{\frac{1}{alphay}}\right) \cdot sin2phi} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      10. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay}} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{alphay} \cdot \color{blue}{\left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      13. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      14. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay}{\color{blue}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      16. associate-/r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      17. div-flip-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{\frac{alphay}{\frac{1}{alphay}}}}} \]

      18. div-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      19. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\frac{alphay}{\color{blue}{\frac{1}{alphay}}}}{sin2phi}}} \]

      20. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{\frac{alphay}{1} \cdot alphay}}{sin2phi}}} \]

      21. /-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay} \cdot alphay}{sin2phi}}} \]

      22. unpow1 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{1}} \cdot alphay}{sin2phi}}} \]

      23. pow-plus N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{\left(1 + 1\right)}}}{sin2phi}}} \]

      24. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{{alphay}^{\color{blue}{2}}}{sin2phi}}} \]

      25. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      26. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      27. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(sin2phi\right)\right)\right)}}}} \]

      28. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{sin2phi}}}} \]

   8. Applied rewrites 93.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq -0.029999999329447746:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, t\_0, sin2phi\right)}{\left(-alphay\right) \cdot t\_1}} \cdot alphay\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0 \cdot u0, u0\right)}{t\_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (/ cos2phi (* alphax alphax))) (t_1 (log (- 1.0 u0))))
  (if (<= t_1 -0.029999999329447746)
    (*
     (/
      1.0
      (/ (fma (* alphay alphay) t_0 sin2phi) (* (- alphay) t_1)))
     alphay)
    (/
     (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) (* u0 u0) u0)
     (+ t_0 (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (t_1 <= -0.029999999329447746f) {
		tmp = (1.0f / (fmaf((alphay * alphay), t_0, sin2phi) / (-alphay * t_1))) * alphay;
	} else {
		tmp = fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), (u0 * u0), u0) / (t_0 + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.029999999329447746))
		tmp = Float32(Float32(Float32(1.0) / Float32(fma(Float32(alphay * alphay), t_0, sin2phi) / Float32(Float32(-alphay) * t_1))) * alphay);
	else
		tmp = Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), Float32(u0 * u0), u0) / Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0299999993

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \color{blue}{\left(\left(-alphay\right) \cdot alphay\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\left(-alphay\right) \cdot \log \left(1 - u0\right)}} \cdot alphay} \]

   if -0.0299999993 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, t\_0, sin2phi\right)}{\left(-alphay\right) \cdot t\_1}} \cdot alphay\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (/ cos2phi (* alphax alphax))) (t_1 (log (- 1.0 u0))))
  (if (<= t_1 -0.03999999910593033)
    (*
     (/
      1.0
      (/ (fma (* alphay alphay) t_0 sin2phi) (* (- alphay) t_1)))
     alphay)
    (/
     (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
     (+ (/ sin2phi (* alphay alphay)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (t_1 <= -0.03999999910593033f) {
		tmp = (1.0f / (fmaf((alphay * alphay), t_0, sin2phi) / (-alphay * t_1))) * alphay;
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / ((sin2phi / (alphay * alphay)) + t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(Float32(1.0) / Float32(fma(Float32(alphay * alphay), t_0, sin2phi) / Float32(Float32(-alphay) * t_1))) * alphay);
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0399999991

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \color{blue}{\left(\left(-alphay\right) \cdot alphay\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\left(-alphay\right) \cdot \log \left(1 - u0\right)}} \cdot alphay} \]

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot 1 + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}\right) + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{u0 \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0 \cdot \frac{1}{2}\right) + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \frac{1}{2} \cdot u0\right) + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. associate-+l+ N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \left(\frac{1}{2} \cdot \color{blue}{u0} + 1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{u0 \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \left(\color{blue}{\frac{1}{2} \cdot u0} + 1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{u0 \cdot \left(\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot \left(u0 \cdot u0\right) + \left(\color{blue}{\frac{1}{2} \cdot u0} + 1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. +-commutative N/A

          \[\leadsto \frac{u0 \cdot \left(\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot \left(u0 \cdot u0\right) + \left(1 + \color{blue}{\frac{1}{2} \cdot u0}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      15. lower-fma.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, \color{blue}{u0 \cdot u0}, 1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      16. lift-+.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, \color{blue}{u0} \cdot u0, 1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      17. +-commutative N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\frac{1}{4} \cdot u0 + \frac{1}{3}, \color{blue}{u0} \cdot u0, 1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      18. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\frac{1}{4} \cdot u0 + \frac{1}{3}, u0 \cdot u0, 1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      19. lower-fma.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), \color{blue}{u0} \cdot u0, 1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      20. lower-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0 \cdot \color{blue}{u0}, 1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      21. +-commutative N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0 \cdot u0, \frac{1}{2} \cdot u0 + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      22. lower-fma.f32 93.1%

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0 \cdot u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.1%

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), \color{blue}{u0 \cdot u0}, \mathsf{fma}\left(0.5, u0, 1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0 \cdot u0, \mathsf{fma}\left(\frac{1}{2}, u0, 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0 \cdot u0, \mathsf{fma}\left(\frac{1}{2}, u0, 1\right)\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 93.1%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0 \cdot u0, \mathsf{fma}\left(0.5, u0, 1\right)\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lift-fma.f32 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right) \cdot \left(u0 \cdot u0\right) + \mathsf{fma}\left(\frac{1}{2}, u0, 1\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lift-fma.f32 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right) \cdot \left(u0 \cdot u0\right) + \left(\frac{1}{2} \cdot u0 + 1\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. associate-+r+ N/A

         \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right) \cdot \left(u0 \cdot u0\right) + \frac{1}{2} \cdot u0\right) + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. add-flip N/A

         \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right) \cdot \left(u0 \cdot u0\right) + \frac{1}{2} \cdot u0\right) - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right) \cdot \left(u0 \cdot u0\right) + \frac{1}{2} \cdot u0\right) - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right) \cdot u0\right) \cdot u0 + \frac{1}{2} \cdot u0\right) - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. distribute-rgt-out N/A

          \[\leadsto \frac{\left(u0 \cdot \left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right) \cdot u0 + \frac{1}{2}\right) - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-fma.f32 N/A

          \[\leadsto \frac{\left(u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0 - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. add-flip-rev N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 93.1%

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq -0.014800000004470348:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, t\_0, sin2phi\right)}{\left(-alphay\right) \cdot t\_1}} \cdot alphay\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{t\_0 + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (/ cos2phi (* alphax alphax))) (t_1 (log (- 1.0 u0))))
  (if (<= t_1 -0.014800000004470348)
    (*
     (/
      1.0
      (/ (fma (* alphay alphay) t_0 sin2phi) (* (- alphay) t_1)))
     alphay)
    (/
     (fma (fma 0.3333333333333333 u0 0.5) (* u0 u0) u0)
     (+ t_0 (/ 1.0 (/ (* alphay alphay) sin2phi)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (t_1 <= -0.014800000004470348f) {
		tmp = (1.0f / (fmaf((alphay * alphay), t_0, sin2phi) / (-alphay * t_1))) * alphay;
	} else {
		tmp = fmaf(fmaf(0.3333333333333333f, u0, 0.5f), (u0 * u0), u0) / (t_0 + (1.0f / ((alphay * alphay) / sin2phi)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.014800000004470348))
		tmp = Float32(Float32(Float32(1.0) / Float32(fma(Float32(alphay * alphay), t_0, sin2phi) / Float32(Float32(-alphay) * t_1))) * alphay);
	else
		tmp = Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), Float32(u0 * u0), u0) / Float32(t_0 + Float32(Float32(1.0) / Float32(Float32(alphay * alphay) / sin2phi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0148

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \color{blue}{\left(\left(-alphay\right) \cdot alphay\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(-alphay\right)\right) \cdot alphay} \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\left(-alphay\right) \cdot \log \left(1 - u0\right)}} \cdot alphay} \]

   if -0.0148 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{sin2phi \cdot \frac{1}{alphay \cdot alphay}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot alphay} \cdot sin2phi}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\color{blue}{1 \cdot 1}}{alphay \cdot alphay} \cdot sin2phi} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1 \cdot 1}{\color{blue}{alphay \cdot alphay}} \cdot sin2phi} \]

      6. times-frac N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\left(\frac{1}{alphay} \cdot \frac{1}{alphay}\right)} \cdot sin2phi} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\color{blue}{\frac{1}{alphay}} \cdot \frac{1}{alphay}\right) \cdot sin2phi} \]

      8. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\frac{1}{alphay} \cdot \color{blue}{\frac{1}{alphay}}\right) \cdot sin2phi} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      10. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay}} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{alphay} \cdot \color{blue}{\left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      13. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      14. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay}{\color{blue}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      16. associate-/r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      17. div-flip-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{\frac{alphay}{\frac{1}{alphay}}}}} \]

      18. div-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      19. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\frac{alphay}{\color{blue}{\frac{1}{alphay}}}}{sin2phi}}} \]

      20. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{\frac{alphay}{1} \cdot alphay}}{sin2phi}}} \]

      21. /-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay} \cdot alphay}{sin2phi}}} \]

      22. unpow1 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{1}} \cdot alphay}{sin2phi}}} \]

      23. pow-plus N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{\left(1 + 1\right)}}}{sin2phi}}} \]

      24. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{{alphay}^{\color{blue}{2}}}{sin2phi}}} \]

      25. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      26. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      27. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(sin2phi\right)\right)\right)}}}} \]

      28. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{sin2phi}}}} \]

   8. Applied rewrites 93.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq -0.014800000004470348:\\ \;\;\;\;\frac{1}{\frac{\frac{sin2phi}{alphay \cdot alphay} + t\_0}{-t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{t\_0 + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (/ cos2phi (* alphax alphax))) (t_1 (log (- 1.0 u0))))
  (if (<= t_1 -0.014800000004470348)
    (/ 1.0 (/ (+ (/ sin2phi (* alphay alphay)) t_0) (- t_1)))
    (/
     (fma (fma 0.3333333333333333 u0 0.5) (* u0 u0) u0)
     (+ t_0 (/ 1.0 (/ (* alphay alphay) sin2phi)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (t_1 <= -0.014800000004470348f) {
		tmp = 1.0f / (((sin2phi / (alphay * alphay)) + t_0) / -t_1);
	} else {
		tmp = fmaf(fmaf(0.3333333333333333f, u0, 0.5f), (u0 * u0), u0) / (t_0 + (1.0f / ((alphay * alphay) / sin2phi)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.014800000004470348))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(sin2phi / Float32(alphay * alphay)) + t_0) / Float32(-t_1)));
	else
		tmp = Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), Float32(u0 * u0), u0) / Float32(t_0 + Float32(Float32(1.0) / Float32(Float32(alphay * alphay) / sin2phi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0148

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(-\log \left(1 - u0\right)\right) \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(-\log \left(1 - u0\right)\right) \cdot 1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      4. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{\left(-\log \left(1 - u0\right)\right) \cdot 1}}} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{\color{blue}{-\log \left(1 - u0\right)}}} \]

      6. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{-\log \left(1 - u0\right)}}} \]

      7. lower-/.f32 59.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{-\log \left(1 - u0\right)}}} \]

      8. lift-+.f32 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}}{-\log \left(1 - u0\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}{-\log \left(1 - u0\right)}} \]

      10. lower-+.f32 59.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}{-\log \left(1 - u0\right)}} \]

   3. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}{-\log \left(1 - u0\right)}}} \]

   if -0.0148 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{sin2phi \cdot \frac{1}{alphay \cdot alphay}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot alphay} \cdot sin2phi}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\color{blue}{1 \cdot 1}}{alphay \cdot alphay} \cdot sin2phi} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1 \cdot 1}{\color{blue}{alphay \cdot alphay}} \cdot sin2phi} \]

      6. times-frac N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\left(\frac{1}{alphay} \cdot \frac{1}{alphay}\right)} \cdot sin2phi} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\color{blue}{\frac{1}{alphay}} \cdot \frac{1}{alphay}\right) \cdot sin2phi} \]

      8. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\frac{1}{alphay} \cdot \color{blue}{\frac{1}{alphay}}\right) \cdot sin2phi} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      10. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay}} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{alphay} \cdot \color{blue}{\left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      13. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      14. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay}{\color{blue}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      16. associate-/r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      17. div-flip-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{\frac{alphay}{\frac{1}{alphay}}}}} \]

      18. div-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      19. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\frac{alphay}{\color{blue}{\frac{1}{alphay}}}}{sin2phi}}} \]

      20. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{\frac{alphay}{1} \cdot alphay}}{sin2phi}}} \]

      21. /-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay} \cdot alphay}{sin2phi}}} \]

      22. unpow1 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{1}} \cdot alphay}{sin2phi}}} \]

      23. pow-plus N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{\left(1 + 1\right)}}}{sin2phi}}} \]

      24. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{{alphay}^{\color{blue}{2}}}{sin2phi}}} \]

      25. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      26. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      27. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(sin2phi\right)\right)\right)}}}} \]

      28. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{sin2phi}}}} \]

   8. Applied rewrites 93.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.014600000344216824:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_1 + t\_0}{-\log \left(1 - u0\right)}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (/ cos2phi (* alphax alphax)))
       (t_1 (/ sin2phi (* alphay alphay))))
  (if (<= u0 0.014600000344216824)
    (/ (fma (fma 0.3333333333333333 u0 0.5) (* u0 u0) u0) (+ t_0 t_1))
    (/ 1.0 (/ (+ t_1 t_0) (- (log (- 1.0 u0))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = sin2phi / (alphay * alphay);
	float tmp;
	if (u0 <= 0.014600000344216824f) {
		tmp = fmaf(fmaf(0.3333333333333333f, u0, 0.5f), (u0 * u0), u0) / (t_0 + t_1);
	} else {
		tmp = 1.0f / ((t_1 + t_0) / -logf((1.0f - u0)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.014600000344216824))
		tmp = Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), Float32(u0 * u0), u0) / Float32(t_0 + t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(t_1 + t_0) / Float32(-log(Float32(Float32(1.0) - u0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0146000003

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.0146000003 < u0

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(-\log \left(1 - u0\right)\right) \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(-\log \left(1 - u0\right)\right) \cdot 1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      4. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{\left(-\log \left(1 - u0\right)\right) \cdot 1}}} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{\color{blue}{-\log \left(1 - u0\right)}}} \]

      6. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{-\log \left(1 - u0\right)}}} \]

      7. lower-/.f32 59.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}{-\log \left(1 - u0\right)}}} \]

      8. lift-+.f32 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}}{-\log \left(1 - u0\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}{-\log \left(1 - u0\right)}} \]

      10. lower-+.f32 59.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}{-\log \left(1 - u0\right)}} \]

   3. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}{-\log \left(1 - u0\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq -0.014800000004470348:\\ \;\;\;\;\frac{-t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{t\_1}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (log (- 1.0 u0)))
       (t_1
        (+
         (/ cos2phi (* alphax alphax))
         (/ sin2phi (* alphay alphay)))))
  (if (<= t_0 -0.014800000004470348)
    (/ (- t_0) t_1)
    (/ (fma (fma 0.3333333333333333 u0 0.5) (* u0 u0) u0) t_1))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float t_1 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	float tmp;
	if (t_0 <= -0.014800000004470348f) {
		tmp = -t_0 / t_1;
	} else {
		tmp = fmaf(fmaf(0.3333333333333333f, u0, 0.5f), (u0 * u0), u0) / t_1;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.014800000004470348))
		tmp = Float32(Float32(-t_0) / t_1);
	else
		tmp = Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), Float32(u0 * u0), u0) / t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0148

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if -0.0148 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;t\_0 \leq -0.003599999938160181:\\ \;\;\;\;\frac{-t\_0}{t\_1 + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0 \cdot u0, u0\right)}{t\_1 + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (log (- 1.0 u0))) (t_1 (/ cos2phi (* alphax alphax))))
  (if (<= t_0 -0.003599999938160181)
    (/ (- t_0) (+ t_1 (/ sin2phi (* alphay alphay))))
    (/
     (fma 0.5 (* u0 u0) u0)
     (+ t_1 (/ 1.0 (/ (* alphay alphay) sin2phi)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float t_1 = cos2phi / (alphax * alphax);
	float tmp;
	if (t_0 <= -0.003599999938160181f) {
		tmp = -t_0 / (t_1 + (sin2phi / (alphay * alphay)));
	} else {
		tmp = fmaf(0.5f, (u0 * u0), u0) / (t_1 + (1.0f / ((alphay * alphay) / sin2phi)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003599999938160181))
		tmp = Float32(Float32(-t_0) / Float32(t_1 + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(fma(Float32(0.5), Float32(u0 * u0), u0) / Float32(t_1 + Float32(Float32(1.0) / Float32(Float32(alphay * alphay) / sin2phi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.00359999994

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if -0.00359999994 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{sin2phi \cdot \frac{1}{alphay \cdot alphay}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot alphay} \cdot sin2phi}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\color{blue}{1 \cdot 1}}{alphay \cdot alphay} \cdot sin2phi} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1 \cdot 1}{\color{blue}{alphay \cdot alphay}} \cdot sin2phi} \]

      6. times-frac N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\left(\frac{1}{alphay} \cdot \frac{1}{alphay}\right)} \cdot sin2phi} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\color{blue}{\frac{1}{alphay}} \cdot \frac{1}{alphay}\right) \cdot sin2phi} \]

      8. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \left(\frac{1}{alphay} \cdot \color{blue}{\frac{1}{alphay}}\right) \cdot sin2phi} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      10. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay}} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{alphay} \cdot \color{blue}{\left(\frac{1}{alphay} \cdot sin2phi\right)}} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      13. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      14. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay}{\color{blue}{\frac{1}{alphay} \cdot sin2phi}}}} \]

      16. associate-/r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      17. div-flip-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{\frac{alphay}{\frac{1}{alphay}}}}} \]

      18. div-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}}} \]

      19. lift-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\frac{alphay}{\color{blue}{\frac{1}{alphay}}}}{sin2phi}}} \]

      20. associate-/r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{\frac{alphay}{1} \cdot alphay}}{sin2phi}}} \]

      21. /-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay} \cdot alphay}{sin2phi}}} \]

      22. unpow1 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{1}} \cdot alphay}{sin2phi}}} \]

      23. pow-plus N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{{alphay}^{\left(1 + 1\right)}}}{sin2phi}}} \]

      24. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{{alphay}^{\color{blue}{2}}}{sin2phi}}} \]

      25. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      26. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}}} \]

      27. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(sin2phi\right)\right)\right)}}}} \]

      28. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{\color{blue}{sin2phi}}}} \]

   8. Applied rewrites 93.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{u0} \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 87.6%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{u0} \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.003000000026077032:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0 \cdot u0, u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0
        (+
         (/ cos2phi (* alphax alphax))
         (/ sin2phi (* alphay alphay)))))
  (if (<= u0 0.003000000026077032)
    (/ (fma 0.5 (* u0 u0) u0) t_0)
    (/ (- (log (- 1.0 u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	float tmp;
	if (u0 <= 0.003000000026077032f) {
		tmp = fmaf(0.5f, (u0 * u0), u0) / t_0;
	} else {
		tmp = -logf((1.0f - u0)) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.003000000026077032))
		tmp = Float32(fma(Float32(0.5), Float32(u0 * u0), u0) / t_0);
	else
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00300000003

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{u0} \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.7%

      \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{u0} \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.00300000003 < u0

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.014999999664723873:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{sin2phi}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (if (<= u0 0.014999999664723873)
  (/
   (fma 0.5 (* u0 u0) u0)
   (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
  (* (/ 1.0 (/ sin2phi (log (- 1.0 u0)))) (* (- alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (u0 <= 0.014999999664723873f) {
		tmp = fmaf(0.5f, (u0 * u0), u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (1.0f / (sin2phi / logf((1.0f - u0)))) * (-alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.014999999664723873))
		tmp = Float32(fma(Float32(0.5), Float32(u0 * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(sin2phi / log(Float32(Float32(1.0) - u0)))) * Float32(Float32(-alphay) * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0149999997

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{1}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0 + \color{blue}{1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{1} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \frac{u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + u0}{\color{blue}{1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. /-rgt-identity N/A

         \[\leadsto \frac{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right) \cdot 1 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot \left(u0 \cdot u0\right) + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. lower-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{u0} \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.7%

      \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{u0} \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.0149999997 < u0

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

   4. Taylor expanded in alphax around inf

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 48.5%

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{\frac{sin2phi}{t\_0}} \cdot \left(\left(-alphay\right) \cdot alphay\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{1}{\frac{alphay \cdot alphay}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (log (- 1.0 u0))))
  (if (<= t_0 -0.0020000000949949026)
    (* (/ 1.0 (/ sin2phi t_0)) (* (- alphay) alphay))
    (/
     (- (- u0))
     (+
      (/ 1.0 (/ (* alphay alphay) sin2phi))
      (/ cos2phi (* alphax alphax)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float tmp;
	if (t_0 <= -0.0020000000949949026f) {
		tmp = (1.0f / (sin2phi / t_0)) * (-alphay * alphay);
	} else {
		tmp = -(-u0) / ((1.0f / ((alphay * alphay) / sin2phi)) + (cos2phi / (alphax * alphax)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = log((1.0e0 - u0))
    if (t_0 <= (-0.0020000000949949026e0)) then
        tmp = (1.0e0 / (sin2phi / t_0)) * (-alphay * alphay)
    else
        tmp = -(-u0) / ((1.0e0 / ((alphay * alphay) / sin2phi)) + (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0020000000949949026))
		tmp = Float32(Float32(Float32(1.0) / Float32(sin2phi / t_0)) * Float32(Float32(-alphay) * alphay));
	else
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(Float32(1.0) / Float32(Float32(alphay * alphay) / sin2phi)) + Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log((single(1.0) - u0));
	tmp = single(0.0);
	if (t_0 <= single(-0.0020000000949949026))
		tmp = (single(1.0) / (sin2phi / t_0)) * (-alphay * alphay);
	else
		tmp = -(-u0) / ((single(1.0) / ((alphay * alphay) / sin2phi)) + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.00200000009

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

   4. Taylor expanded in alphax around inf

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 48.5%

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   if -0.00200000009 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 76.0%

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 76.0%

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-neg.f32 76.0%

         \[\leadsto \frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      5. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

      6. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lower-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      2. mult-flip N/A

         \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{sin2phi \cdot \frac{1}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{1}{alphay \cdot alphay} \cdot sin2phi} + \frac{cos2phi}{alphax \cdot alphax}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-\left(-u0\right)}{\frac{\color{blue}{1 \cdot 1}}{alphay \cdot alphay} \cdot sin2phi + \frac{cos2phi}{alphax \cdot alphax}} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\frac{1 \cdot 1}{\color{blue}{alphay \cdot alphay}} \cdot sin2phi + \frac{cos2phi}{alphax \cdot alphax}} \]

      6. times-frac N/A

         \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\left(\frac{1}{alphay} \cdot \frac{1}{alphay}\right)} \cdot sin2phi + \frac{cos2phi}{alphax \cdot alphax}} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\left(\color{blue}{\frac{1}{alphay}} \cdot \frac{1}{alphay}\right) \cdot sin2phi + \frac{cos2phi}{alphax \cdot alphax}} \]

      8. lift-/.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\left(\frac{1}{alphay} \cdot \color{blue}{\frac{1}{alphay}}\right) \cdot sin2phi + \frac{cos2phi}{alphax \cdot alphax}} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{1}{alphay} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right)} + \frac{cos2phi}{alphax \cdot alphax}} \]

      10. lift-/.f32 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{1}{alphay}} \cdot \left(\frac{1}{alphay} \cdot sin2phi\right) + \frac{cos2phi}{alphax \cdot alphax}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{alphay} \cdot \color{blue}{\left(\frac{1}{alphay} \cdot sin2phi\right)} + \frac{cos2phi}{alphax \cdot alphax}} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{1}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      13. lift-/.f32 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      14. lift-/.f32 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\color{blue}{\frac{alphay}{\frac{1}{alphay} \cdot sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{alphay}{\color{blue}{\frac{1}{alphay} \cdot sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      16. associate-/r* N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\color{blue}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      17. div-flip-rev N/A

          \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{sin2phi}{\frac{alphay}{\frac{1}{alphay}}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      18. div-flip N/A

          \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{1}{\frac{\frac{alphay}{\frac{1}{alphay}}}{sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      19. lift-/.f32 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{\frac{alphay}{\color{blue}{\frac{1}{alphay}}}}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      20. associate-/r/ N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{\color{blue}{\frac{alphay}{1} \cdot alphay}}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      21. /-rgt-identity N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{\color{blue}{alphay} \cdot alphay}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      22. unpow1 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{\color{blue}{{alphay}^{1}} \cdot alphay}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      23. pow-plus N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{\color{blue}{{alphay}^{\left(1 + 1\right)}}}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      24. metadata-eval N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{{alphay}^{\color{blue}{2}}}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      25. pow2 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      26. lift-*.f32 N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{\color{blue}{alphay \cdot alphay}}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      27. remove-double-neg N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{alphay \cdot alphay}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(sin2phi\right)\right)\right)}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      28. remove-double-neg N/A

          \[\leadsto \frac{-\left(-u0\right)}{\frac{1}{\frac{alphay \cdot alphay}{\color{blue}{sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   8. Applied rewrites 75.9%

      \[\leadsto \frac{-\left(-u0\right)}{\color{blue}{\frac{1}{\frac{alphay \cdot alphay}{sin2phi}}} + \frac{cos2phi}{alphax \cdot alphax}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{\frac{sin2phi}{t\_0}} \cdot \left(\left(-alphay\right) \cdot alphay\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (let* ((t_0 (log (- 1.0 u0))))
  (if (<= t_0 -0.0020000000949949026)
    (* (/ 1.0 (/ sin2phi t_0)) (* (- alphay) alphay))
    (/
     (- (- u0))
     (+
      (/ sin2phi (* alphay alphay))
      (/ cos2phi (* alphax alphax)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float tmp;
	if (t_0 <= -0.0020000000949949026f) {
		tmp = (1.0f / (sin2phi / t_0)) * (-alphay * alphay);
	} else {
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = log((1.0e0 - u0))
    if (t_0 <= (-0.0020000000949949026e0)) then
        tmp = (1.0e0 / (sin2phi / t_0)) * (-alphay * alphay)
    else
        tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0020000000949949026))
		tmp = Float32(Float32(Float32(1.0) / Float32(sin2phi / t_0)) * Float32(Float32(-alphay) * alphay));
	else
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log((single(1.0) - u0));
	tmp = single(0.0);
	if (t_0 <= single(-0.0020000000949949026))
		tmp = (single(1.0) / (sin2phi / t_0)) * (-alphay * alphay);
	else
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.00200000009

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

   4. Taylor expanded in alphax around inf

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 48.5%

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   if -0.00200000009 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 76.0%

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 76.0%

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-neg.f32 76.0%

         \[\leadsto \frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      5. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

      6. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lower-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.8399999579532533 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{sin2phi}{\left(-alphay\right) \cdot alphay}} \cdot \left(\mathsf{fma}\left(-0.5, u0, -1\right) \cdot u0\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (if (<= sin2phi 1.8399999579532533e-10)
  (/
   (- (- u0))
   (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
  (*
   (/ 1.0 (/ sin2phi (* (- alphay) alphay)))
   (* (fma -0.5 u0 -1.0) u0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.8399999579532533e-10f) {
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (1.0f / (sin2phi / (-alphay * alphay))) * (fmaf(-0.5f, u0, -1.0f) * u0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.8399999579532533e-10))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(sin2phi / Float32(Float32(-alphay) * alphay))) * Float32(fma(Float32(-0.5), u0, Float32(-1.0)) * u0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.83999996e-10

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 76.0%

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 76.0%

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-neg.f32 76.0%

         \[\leadsto \frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      5. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

      6. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lower-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   if 1.83999996e-10 < sin2phi

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

   4. Taylor expanded in u0 around 0

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

      2. lower--.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - \color{blue}{1}\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

      3. lower-*.f32 87.2%

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   6. Applied rewrites 87.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   7. Taylor expanded in alphax around inf

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(-alphay\right) \cdot alphay\right) \cdot \frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}} \]

       3. lift-/.f32 N/A

          \[\leadsto \left(\left(-alphay\right) \cdot alphay\right) \cdot \color{blue}{\frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}} \]

       4. mult-flip-rev N/A

          \[\leadsto \color{blue}{\frac{\left(-alphay\right) \cdot alphay}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}} \]

       5. lift-/.f32 N/A

          \[\leadsto \frac{\left(-alphay\right) \cdot alphay}{\color{blue}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}} \]

       6. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{\left(-alphay\right) \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)} \]

       7. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\left(-alphay\right) \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)} \]

   11. Applied rewrites 66.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{sin2phi}{\left(-alphay\right) \cdot alphay}} \cdot \left(\mathsf{fma}\left(-0.5, u0, -1\right) \cdot u0\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.8399999579532533 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{sin2phi}{\mathsf{fma}\left(-0.5, u0, -1\right) \cdot u0}} \cdot alphay\right) \cdot \left(-alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (if (<= sin2phi 1.8399999579532533e-10)
  (/
   (- (- u0))
   (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
  (*
   (* (/ 1.0 (/ sin2phi (* (fma -0.5 u0 -1.0) u0))) alphay)
   (- alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.8399999579532533e-10f) {
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((1.0f / (sin2phi / (fmaf(-0.5f, u0, -1.0f) * u0))) * alphay) * -alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.8399999579532533e-10))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(sin2phi / Float32(fma(Float32(-0.5), u0, Float32(-1.0)) * u0))) * alphay) * Float32(-alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.83999996e-10

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 76.0%

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 76.0%

      \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-neg.f32 76.0%

         \[\leadsto \frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      5. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

      6. lower-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lower-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   if 1.83999996e-10 < sin2phi

   1. Initial program 60.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   3. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

   4. Taylor expanded in u0 around 0

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

      2. lower--.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - \color{blue}{1}\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

      3. lower-*.f32 87.2%

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   6. Applied rewrites 87.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   7. Taylor expanded in alphax around inf

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

       2. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}} \cdot \color{blue}{\left(\left(-alphay\right) \cdot alphay\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}} \cdot \color{blue}{\left(alphay \cdot \left(-alphay\right)\right)} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}} \cdot alphay\right) \cdot \left(-alphay\right)} \]

       5. lower-*.f32 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\frac{sin2phi}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}} \cdot alphay\right) \cdot \left(-alphay\right)} \]

   11. Applied rewrites 66.4%

       \[\leadsto \color{blue}{\left(\frac{1}{\frac{sin2phi}{\mathsf{fma}\left(-0.5, u0, -1\right) \cdot u0}} \cdot alphay\right) \cdot \left(-alphay\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.0% accurate, 1.3× speedup

Math

\[\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (/
 (- (- u0))
 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 60.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

2. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Step-by-step derivation

   1. lower-*.f32 76.0%

      \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 76.0%

   \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{--1 \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{-\left(\mathsf{neg}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower-neg.f32 76.0%

      \[\leadsto \frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-neg.f32 N/A

      \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

   5. lower-neg.f32 N/A

      \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. lower-neg.f32 N/A

      \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lower-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

6. Applied rewrites 76.0%

   \[\leadsto \color{blue}{\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
7. Add Preprocessing

Alternative 16: 58.7% accurate, 1.6× speedup

Math

\[\frac{1}{\frac{sin2phi}{u0 \cdot -1}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :precision binary32
  (* (/ 1.0 (/ sin2phi (* u0 -1.0))) (* (- alphay) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (1.0f / (sin2phi / (u0 * -1.0f))) * (-alphay * alphay);
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (1.0e0 / (sin2phi / (u0 * (-1.0e0)))) * (-alphay * alphay)
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(1.0) / Float32(sin2phi / Float32(u0 * Float32(-1.0)))) * Float32(Float32(-alphay) * alphay))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (single(1.0) / (sin2phi / (u0 * single(-1.0)))) * (-alphay * alphay);
end

Derivation

1. Initial program 60.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. lift-neg.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. distribute-frac-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   5. lift-+.f32 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

   7. add-to-fraction N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\mathsf{neg}\left(\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{alphay \cdot alphay}}\right)} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi}{\mathsf{neg}\left(alphay \cdot alphay\right)}}} \]

   9. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(alphay \cdot alphay\right) + sin2phi} \cdot \left(\mathsf{neg}\left(alphay \cdot alphay\right)\right)} \]

3. Applied rewrites 60.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\log \left(1 - u0\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right)} \]

4. Taylor expanded in u0 around 0

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
5. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   2. lower--.f32 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - \color{blue}{1}\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

   3. lower-*.f32 87.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

6. Applied rewrites 87.2%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

7. Taylor expanded in alphax around inf

   \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
8. Step-by-step derivation

9. Applied rewrites 66.4%

   \[\leadsto \frac{1}{\frac{\color{blue}{sin2phi}}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]

10. Taylor expanded in u0 around 0

    \[\leadsto \frac{1}{\frac{sin2phi}{u0 \cdot -1}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
11. Step-by-step derivation

12. Applied rewrites 58.7%

    \[\leadsto \frac{1}{\frac{sin2phi}{u0 \cdot -1}} \cdot \left(\left(-alphay\right) \cdot alphay\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))