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

Percentage Accurate: 60.2% → 97.3%

Time: 13.7s

Alternatives: 17

Speedup: 3.5×

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

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

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)
    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: 60.2% accurate, 1.0× speedup

Math

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

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)
    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: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.984000027179718:\\ \;\;\;\;\frac{-0.5 \cdot \log \left({\left(1 - u0\right)}^{2}\right)}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot u0 + \left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- 1.0 u0) 0.984000027179718)
     (/
      (* -0.5 (log (pow (- 1.0 u0) 2.0)))
      (+ (* (/ cos2phi alphax) (/ 1.0 alphax)) t_0))
     (/
      (+
       (* u0 u0)
       (*
        (* (- (+ (/ 1.0 (* u0 u0)) 0.3333333333333333) (/ 0.5 u0)) (* u0 u0))
        u0))
      (+ (/ cos2phi (* alphax alphax)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if ((1.0f - u0) <= 0.984000027179718f) {
		tmp = (-0.5f * logf(powf((1.0f - u0), 2.0f))) / (((cos2phi / alphax) * (1.0f / alphax)) + t_0);
	} else {
		tmp = ((u0 * u0) + (((((1.0f / (u0 * u0)) + 0.3333333333333333f) - (0.5f / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 = sin2phi / (alphay * alphay)
    if ((1.0e0 - u0) <= 0.984000027179718e0) then
        tmp = ((-0.5e0) * log(((1.0e0 - u0) ** 2.0e0))) / (((cos2phi / alphax) * (1.0e0 / alphax)) + t_0)
    else
        tmp = ((u0 * u0) + (((((1.0e0 / (u0 * u0)) + 0.3333333333333333e0) - (0.5e0 / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + t_0)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.984000027179718))
		tmp = Float32(Float32(Float32(-0.5) * log((Float32(Float32(1.0) - u0) ^ Float32(2.0)))) / Float32(Float32(Float32(cos2phi / alphax) * Float32(Float32(1.0) / alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(u0 * u0) + Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(u0 * u0)) + Float32(0.3333333333333333)) - Float32(Float32(0.5) / u0)) * Float32(u0 * u0)) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.984000027179718))
		tmp = (single(-0.5) * log(((single(1.0) - u0) ^ single(2.0)))) / (((cos2phi / alphax) * (single(1.0) / alphax)) + t_0);
	else
		tmp = ((u0 * u0) + (((((single(1.0) / (u0 * u0)) + single(0.3333333333333333)) - (single(0.5) / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.984000027

   1. Initial program 94.4%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. log-pow N/A

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

      9. lower-*.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f32 94.4

          \[\leadsto \frac{-0.5 \cdot \log \color{blue}{\left({\left(1 - u0\right)}^{2}\right)}}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied rewrites 94.4%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \log \left({\left(1 - u0\right)}^{2}\right)}}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.984000027 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 51.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lift--.f32 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. log-prod N/A

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

      8. lower-+.f32 N/A

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

   4. Applied rewrites 69.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{-1}{-1 + u0 \cdot u0}\right) + \mathsf{log1p}\left(u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{{u0}^{2}} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 83.2

         \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 83.2%

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 83.2

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

   10. Applied rewrites 82.9%

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

   11. Taylor expanded in u0 around inf

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

   13. Applied rewrites 98.2%

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

Alternative 2: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.984000027179718:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{\frac{alphax}{cos2phi} \cdot alphax} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot u0 + \left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- 1.0 u0) 0.984000027179718)
     (/ (log (- 1.0 u0)) (- (/ -1.0 (* (/ alphax cos2phi) alphax)) t_0))
     (/
      (+
       (* u0 u0)
       (*
        (* (- (+ (/ 1.0 (* u0 u0)) 0.3333333333333333) (/ 0.5 u0)) (* u0 u0))
        u0))
      (+ (/ cos2phi (* alphax alphax)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if ((1.0f - u0) <= 0.984000027179718f) {
		tmp = logf((1.0f - u0)) / ((-1.0f / ((alphax / cos2phi) * alphax)) - t_0);
	} else {
		tmp = ((u0 * u0) + (((((1.0f / (u0 * u0)) + 0.3333333333333333f) - (0.5f / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 = sin2phi / (alphay * alphay)
    if ((1.0e0 - u0) <= 0.984000027179718e0) then
        tmp = log((1.0e0 - u0)) / (((-1.0e0) / ((alphax / cos2phi) * alphax)) - t_0)
    else
        tmp = ((u0 * u0) + (((((1.0e0 / (u0 * u0)) + 0.3333333333333333e0) - (0.5e0 / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + t_0)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.984000027179718))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(Float32(-1.0) / Float32(Float32(alphax / cos2phi) * alphax)) - t_0));
	else
		tmp = Float32(Float32(Float32(u0 * u0) + Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(u0 * u0)) + Float32(0.3333333333333333)) - Float32(Float32(0.5) / u0)) * Float32(u0 * u0)) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.984000027179718))
		tmp = log((single(1.0) - u0)) / ((single(-1.0) / ((alphax / cos2phi) * alphax)) - t_0);
	else
		tmp = ((u0 * u0) + (((((single(1.0) / (u0 * u0)) + single(0.3333333333333333)) - (single(0.5) / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.984000027

   1. Initial program 94.4%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. clear-num N/A

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

      4. lift-/.f32 N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-/.f32 94.4

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

   6. Applied rewrites 94.4%

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

   if 0.984000027 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 51.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lift--.f32 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. log-prod N/A

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

      8. lower-+.f32 N/A

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

   4. Applied rewrites 69.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{-1}{-1 + u0 \cdot u0}\right) + \mathsf{log1p}\left(u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{{u0}^{2}} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 83.2

         \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 83.2%

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 83.2

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

   10. Applied rewrites 82.9%

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

   11. Taylor expanded in u0 around inf

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

   13. Applied rewrites 98.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.984000027179718:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{\frac{alphax}{cos2phi} \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot u0 + \left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.984000027179718:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot u0 + \left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (if (<= (- 1.0 u0) 0.984000027179718)
     (/ (- (log (- 1.0 u0))) t_0)
     (/
      (+
       (* u0 u0)
       (*
        (* (- (+ (/ 1.0 (* u0 u0)) 0.3333333333333333) (/ 0.5 u0)) (* u0 u0))
        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 ((1.0f - u0) <= 0.984000027179718f) {
		tmp = -logf((1.0f - u0)) / t_0;
	} else {
		tmp = ((u0 * u0) + (((((1.0f / (u0 * u0)) + 0.3333333333333333f) - (0.5f / u0)) * (u0 * u0)) * u0)) / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))
    if ((1.0e0 - u0) <= 0.984000027179718e0) then
        tmp = -log((1.0e0 - u0)) / t_0
    else
        tmp = ((u0 * u0) + (((((1.0e0 / (u0 * u0)) + 0.3333333333333333e0) - (0.5e0 / u0)) * (u0 * u0)) * u0)) / t_0
    end if
    code = tmp
end function

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 (Float32(Float32(1.0) - u0) <= Float32(0.984000027179718))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / t_0);
	else
		tmp = Float32(Float32(Float32(u0 * u0) + Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(u0 * u0)) + Float32(0.3333333333333333)) - Float32(Float32(0.5) / u0)) * Float32(u0 * u0)) * u0)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.984000027179718))
		tmp = -log((single(1.0) - u0)) / t_0;
	else
		tmp = ((u0 * u0) + (((((single(1.0) / (u0 * u0)) + single(0.3333333333333333)) - (single(0.5) / u0)) * (u0 * u0)) * u0)) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.984000027

   1. Initial program 94.4%

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

   if 0.984000027 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 51.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lift--.f32 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. log-prod N/A

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

      8. lower-+.f32 N/A

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

   4. Applied rewrites 69.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{-1}{-1 + u0 \cdot u0}\right) + \mathsf{log1p}\left(u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{{u0}^{2}} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 83.2

         \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 83.2%

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 83.2

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

   10. Applied rewrites 82.9%

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

   11. Taylor expanded in u0 around inf

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

   13. Applied rewrites 98.2%

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

Alternative 4: 91.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot u0 + \left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (+
   (* u0 u0)
   (*
    (* (- (+ (/ 1.0 (* u0 u0)) 0.3333333333333333) (/ 0.5 u0)) (* u0 u0))
    u0))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((u0 * u0) + (((((1.0f / (u0 * u0)) + 0.3333333333333333f) - (0.5f / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 * u0) + (((((1.0e0 / (u0 * u0)) + 0.3333333333333333e0) - (0.5e0 / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(u0 * u0) + Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(u0 * u0)) + Float32(0.3333333333333333)) - Float32(Float32(0.5) / u0)) * Float32(u0 * u0)) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = ((u0 * u0) + (((((single(1.0) / (u0 * u0)) + single(0.3333333333333333)) - (single(0.5) / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 59.6%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lift--.f32 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

4. Applied rewrites 60.5%

   \[\leadsto \frac{\color{blue}{\log \left(\frac{-1}{-1 + u0 \cdot u0}\right) + \mathsf{log1p}\left(u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{{u0}^{2}} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 75.2

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 75.2%

   \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 75.2

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

10. Applied rewrites 75.2%

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

11. Taylor expanded in u0 around inf

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

13. Applied rewrites 90.5%

    \[\leadsto \frac{u0 \cdot u0 + \left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
14. Add Preprocessing

Alternative 5: 91.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot u0 + \left(\left(0.3333333333333333 - \frac{0.5 - \frac{1}{u0}}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (+
   (* u0 u0)
   (* (* (- 0.3333333333333333 (/ (- 0.5 (/ 1.0 u0)) u0)) (* u0 u0)) u0))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((u0 * u0) + (((0.3333333333333333f - ((0.5f - (1.0f / u0)) / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 * u0) + (((0.3333333333333333e0 - ((0.5e0 - (1.0e0 / u0)) / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(u0 * u0) + Float32(Float32(Float32(Float32(0.3333333333333333) - Float32(Float32(Float32(0.5) - Float32(Float32(1.0) / u0)) / u0)) * Float32(u0 * u0)) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = ((u0 * u0) + (((single(0.3333333333333333) - ((single(0.5) - (single(1.0) / u0)) / u0)) * (u0 * u0)) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 59.6%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lift--.f32 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

4. Applied rewrites 60.4%

   \[\leadsto \frac{\color{blue}{\log \left(\frac{-1}{-1 + u0 \cdot u0}\right) + \mathsf{log1p}\left(u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{{u0}^{2}} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 75.2

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 75.2%

   \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 75.2

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

10. Applied rewrites 75.2%

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

11. Taylor expanded in u0 around -inf

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

13. Applied rewrites 90.2%

    \[\leadsto \frac{u0 \cdot u0 + \left(\left(0.3333333333333333 - \frac{0.5 - \frac{1}{u0}}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
14. Add Preprocessing

Alternative 6: 50.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot u0 + \left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (+ (* u0 u0) (* (+ (* (fma 0.3333333333333333 u0 -0.5) u0) 1.0) u0))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((u0 * u0) + (((fmaf(0.3333333333333333f, u0, -0.5f) * u0) + 1.0f) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(u0 * u0) + Float32(Float32(Float32(fma(Float32(0.3333333333333333), u0, Float32(-0.5)) * u0) + Float32(1.0)) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.6%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lift--.f32 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

4. Applied rewrites 60.3%

   \[\leadsto \frac{\color{blue}{\log \left(\frac{-1}{-1 + u0 \cdot u0}\right) + \mathsf{log1p}\left(u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{{u0}^{2}} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 75.2

      \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 75.2%

   \[\leadsto \frac{\color{blue}{u0 \cdot u0} + \mathsf{log1p}\left(u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 75.2

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

10. Applied rewrites 75.2%

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

12. Applied rewrites 86.9%

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

Alternative 7: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\frac{u0}{\left(alphax \cdot alphax\right) \cdot sin2phi + \left(alphay \cdot alphay\right) \cdot cos2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 / (((alphax * alphax) * sin2phi) + ((alphay * alphay) * cos2phi))) * (alphax * alphax)) * (alphay * alphay)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 57.7%

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

9. Applied rewrites 75.0%

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

10. Final simplification 75.0%

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

Alternative 8: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\frac{u0}{\left(alphax \cdot alphax\right) \cdot sin2phi + \left(alphay \cdot alphay\right) \cdot cos2phi} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 / (((alphax * alphax) * sin2phi) + ((alphay * alphay) * cos2phi))) * (alphay * alphay)) * (alphax * alphax)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 57.7%

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

9. Applied rewrites 75.0%

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

10. Final simplification 75.0%

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

Alternative 9: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\frac{u0}{\left(alphax \cdot alphax\right) \cdot sin2phi + \left(alphay \cdot alphay\right) \cdot cos2phi} \cdot alphax\right) \cdot \left(\left(alphay \cdot alphay\right) \cdot alphax\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 / (((alphax * alphax) * sin2phi) + ((alphay * alphay) * cos2phi))) * alphax) * ((alphay * alphay) * alphax)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 57.7%

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

9. Applied rewrites 74.9%

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

10. Final simplification 74.9%

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

Alternative 10: 76.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

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)
    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(u0 / Float32(Float32(Float32(sin2phi / 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 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 74.7%

   \[\leadsto \frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\color{blue}{cos2phi}}{alphax \cdot alphax}} \]
8. Add Preprocessing

Alternative 11: 76.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

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)
    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(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / 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 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 74.7%

   \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}} \]
8. Add Preprocessing

Alternative 12: 66.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-12)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-12))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.9999999920083944e-12))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-12

   1. Initial program 58.5%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in alphax around 0

      \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{{alphax}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.2%

      \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   if 1.99999999e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in alphax around inf

      \[\leadsto \frac{{alphay}^{2} \cdot u0}{\color{blue}{sin2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

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

Alternative 13: 76.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

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)
    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(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 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
6. Add Preprocessing

Alternative 14: 66.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(alphax \cdot u0\right) \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-12)
   (/ (* (* alphax u0) alphax) cos2phi)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12f) {
		tmp = ((alphax * u0) * alphax) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12) then
        tmp = ((alphax * u0) * alphax) / cos2phi
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-12))
		tmp = Float32(Float32(Float32(alphax * u0) * alphax) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.9999999920083944e-12))
		tmp = ((alphax * u0) * alphax) / cos2phi;
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-12

   1. Initial program 58.5%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in alphax around 0

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.0%

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

   10. Applied rewrites 56.1%

       \[\leadsto \frac{\left(alphax \cdot u0\right) \cdot alphax}{cos2phi} \]

   if 1.99999999e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in alphax around inf

      \[\leadsto \frac{{alphay}^{2} \cdot u0}{\color{blue}{sin2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

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

Alternative 15: 24.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 / cos2phi) * (alphax * alphax)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in alphax around 0

   \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{cos2phi}} \]
7. Step-by-step derivation

8. Applied rewrites 24.4%

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

10. Applied rewrites 24.5%

    \[\leadsto \frac{u0}{cos2phi} \cdot \left(alphax \cdot \color{blue}{alphax}\right) \]
11. Add Preprocessing

Alternative 16: 24.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \left(\frac{alphax}{cos2phi} \cdot alphax\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 * ((alphax / cos2phi) * alphax)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in alphax around 0

   \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{cos2phi}} \]
7. Step-by-step derivation

8. Applied rewrites 24.4%

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

10. Applied rewrites 24.5%

    \[\leadsto u0 \cdot \left(\frac{alphax}{cos2phi} \cdot \color{blue}{alphax}\right) \]
11. Add Preprocessing

Alternative 17: 24.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    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 = alphax * (alphax * (u0 / cos2phi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in alphax around 0

   \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{cos2phi}} \]
7. Step-by-step derivation

8. Applied rewrites 24.4%

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

10. Applied rewrites 24.5%

    \[\leadsto alphax \cdot \left(alphax \cdot \color{blue}{\frac{u0}{cos2phi}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(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)))))