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

Percentage Accurate: 60.5% → 96.2%

Time: 12.0s

Alternatives: 14

Speedup: 5.4×

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: 60.5% 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: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{\frac{1}{sin2phi}} \cdot {alphay}^{-2} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax))))
   (if (<= (- 1.0 u0) 0.9970499873161316)
     (/
      (log (- 1.0 u0))
      (- (* (/ -1.0 (/ 1.0 sin2phi)) (pow alphay -2.0)) t_0))
     (/
      (- (* (+ 1.0 (* -0.5 u0)) u0) (* (- u0) u0))
      (+ (/ sin2phi (* alphay alphay)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float tmp;
	if ((1.0f - u0) <= 0.9970499873161316f) {
		tmp = logf((1.0f - u0)) / (((-1.0f / (1.0f / sin2phi)) * powf(alphay, -2.0f)) - t_0);
	} else {
		tmp = (((1.0f + (-0.5f * u0)) * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + 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)
    if ((1.0e0 - u0) <= 0.9970499873161316e0) then
        tmp = log((1.0e0 - u0)) / ((((-1.0e0) / (1.0e0 / sin2phi)) * (alphay ** (-2.0e0))) - t_0)
    else
        tmp = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + t_0)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9970499873161316))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(Float32(Float32(-1.0) / Float32(Float32(1.0) / sin2phi)) * (alphay ^ Float32(-2.0))) - t_0));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u0)) * u0) - Float32(Float32(-u0) * u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = cos2phi / (alphax * alphax);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9970499873161316))
		tmp = log((single(1.0) - u0)) / (((single(-1.0) / (single(1.0) / sin2phi)) * (alphay ^ single(-2.0))) - t_0);
	else
		tmp = (((single(1.0) + (single(-0.5) * u0)) * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997049987

   1. Initial program 92.7%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. div-inv N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. lower-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. pow2 N/A

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

      10. pow-flip N/A

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

      11. lower-pow.f32 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f32 N/A

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

      14. lower-/.f32 92.7

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

   4. Applied rewrites 92.7%

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

      1. lift-pow.f32 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f32 92.7

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

   6. Applied rewrites 92.7%

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

   if 0.997049987 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 50.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f32 86.2

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

   7. Applied rewrites 86.2%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 86.2

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

   10. Applied rewrites 85.9%

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

   12. Applied rewrites 97.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{\frac{1}{sin2phi}} \cdot {alphay}^{-2} - \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{alphax \cdot alphax} \cdot cos2phi - \frac{1}{alphay \cdot alphay} \cdot sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9970499873161316)
   (/
    (log (- 1.0 u0))
    (-
     (* (/ -1.0 (* alphax alphax)) cos2phi)
     (* (/ 1.0 (* alphay alphay)) sin2phi)))
   (/
    (- (* (+ 1.0 (* -0.5 u0)) u0) (* (- u0) u0))
    (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9970499873161316f) {
		tmp = logf((1.0f - u0)) / (((-1.0f / (alphax * alphax)) * cos2phi) - ((1.0f / (alphay * alphay)) * sin2phi));
	} else {
		tmp = (((1.0f + (-0.5f * u0)) * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	}
	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 ((1.0e0 - u0) <= 0.9970499873161316e0) then
        tmp = log((1.0e0 - u0)) / ((((-1.0e0) / (alphax * alphax)) * cos2phi) - ((1.0e0 / (alphay * alphay)) * sin2phi))
    else
        tmp = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9970499873161316))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(Float32(Float32(-1.0) / Float32(alphax * alphax)) * cos2phi) - Float32(Float32(Float32(1.0) / Float32(alphay * alphay)) * sin2phi)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u0)) * u0) - Float32(Float32(-u0) * u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9970499873161316))
		tmp = log((single(1.0) - u0)) / (((single(-1.0) / (alphax * alphax)) * cos2phi) - ((single(1.0) / (alphay * alphay)) * sin2phi));
	else
		tmp = (((single(1.0) + (single(-0.5) * u0)) * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997049987

   1. Initial program 92.7%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f32 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f32 N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 92.6

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

   6. Applied rewrites 92.6%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f32 N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-/.f32 92.7

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

   8. Applied rewrites 92.7%

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

   if 0.997049987 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 50.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f32 86.2

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

   7. Applied rewrites 86.2%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 86.2

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

   10. Applied rewrites 85.9%

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

   12. Applied rewrites 97.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{alphax \cdot alphax} \cdot cos2phi - \frac{1}{alphay \cdot alphay} \cdot sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{alphax \cdot alphax} \cdot cos2phi - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- 1.0 u0) 0.9970499873161316)
     (/ (log (- 1.0 u0)) (- (* (/ -1.0 (* alphax alphax)) cos2phi) t_0))
     (/
      (- (* (+ 1.0 (* -0.5 u0)) u0) (* (- u0) u0))
      (+ t_0 (/ cos2phi (* alphax alphax)))))))

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) <= 0.9970499873161316f) {
		tmp = logf((1.0f - u0)) / (((-1.0f / (alphax * alphax)) * cos2phi) - t_0);
	} else {
		tmp = (((1.0f + (-0.5f * u0)) * u0) - (-u0 * u0)) / (t_0 + (cos2phi / (alphax * alphax)));
	}
	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) <= 0.9970499873161316e0) then
        tmp = log((1.0e0 - u0)) / ((((-1.0e0) / (alphax * alphax)) * cos2phi) - t_0)
    else
        tmp = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / (t_0 + (cos2phi / (alphax * alphax)))
    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(0.9970499873161316))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(Float32(Float32(-1.0) / Float32(alphax * alphax)) * cos2phi) - t_0));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u0)) * u0) - Float32(Float32(-u0) * u0)) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	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(0.9970499873161316))
		tmp = log((single(1.0) - u0)) / (((single(-1.0) / (alphax * alphax)) * cos2phi) - t_0);
	else
		tmp = (((single(1.0) + (single(-0.5) * u0)) * u0) - (-u0 * u0)) / (t_0 + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997049987

   1. Initial program 92.7%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f32 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f32 N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 92.6

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

   6. Applied rewrites 92.6%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f32 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-/.f32 92.7

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

   8. Applied rewrites 92.7%

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

   if 0.997049987 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 50.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f32 86.2

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

   7. Applied rewrites 86.2%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 86.2

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

   10. Applied rewrites 85.9%

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

   12. Applied rewrites 97.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{alphax \cdot alphax} \cdot cos2phi - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (if (<= (- 1.0 u0) 0.9970499873161316)
     (/ (- (log (- 1.0 u0))) t_0)
     (/ (- (* (+ 1.0 (* -0.5 u0)) u0) (* (- u0) u0)) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	float tmp;
	if ((1.0f - u0) <= 0.9970499873161316f) {
		tmp = -logf((1.0f - u0)) / t_0;
	} else {
		tmp = (((1.0f + (-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 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))
    if ((1.0e0 - u0) <= 0.9970499873161316e0) then
        tmp = -log((1.0e0 - u0)) / t_0
    else
        tmp = (((1.0e0 + ((-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(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9970499873161316))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u0)) * u0) - Float32(Float32(-u0) * u0)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9970499873161316))
		tmp = -log((single(1.0) - u0)) / t_0;
	else
		tmp = (((single(1.0) + (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) < 0.997049987

   1. Initial program 92.7%

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

   if 0.997049987 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 50.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f32 86.2

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

   7. Applied rewrites 86.2%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 86.2

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

   10. Applied rewrites 85.9%

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

   12. Applied rewrites 97.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9970499873161316:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (((1.0f + (-0.5f * u0)) * u0) - (-u0 * 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 = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower-log1p.f32 N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-log1p.f32 76.3

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

4. Applied rewrites 76.3%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. unpow2 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-*.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f32 76.3

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

7. Applied rewrites 76.3%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 76.3

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

10. Applied rewrites 76.1%

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

12. Applied rewrites 88.1%

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

13. Final simplification 88.1%

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

Alternative 6: 76.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((1.0f * u0) - (-u0 * 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 = ((1.0e0 * u0) - (-u0 * u0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower-log1p.f32 N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-log1p.f32 76.3

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

4. Applied rewrites 76.3%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. unpow2 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-*.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f32 76.3

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

7. Applied rewrites 76.3%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 76.3

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

10. Applied rewrites 76.1%

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

11. Taylor expanded in u0 around 0

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

13. Applied rewrites 76.3%

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

14. Final simplification 76.3%

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

Alternative 7: 76.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.9

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

5. Applied rewrites 75.9%

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

7. Applied rewrites 58.4%

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

9. Applied rewrites 76.0%

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

10. Final simplification 76.0%

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

Alternative 8: 76.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.9

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

5. Applied rewrites 75.9%

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

7. Applied rewrites 58.4%

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

9. Applied rewrites 75.9%

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

10. Final simplification 75.9%

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

Alternative 9: 76.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} - \frac{-1}{alphax \cdot alphax} \cdot cos2phi} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((sin2phi / (alphay * alphay)) - ((-1.0f / (alphax * alphax)) * 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 = u0 / ((sin2phi / (alphay * alphay)) - (((-1.0e0) / (alphax * alphax)) * cos2phi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.9

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

5. Applied rewrites 75.9%

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

7. Applied rewrites 75.9%

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

8. Final simplification 75.9%

   \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} - \frac{-1}{alphax \cdot alphax} \cdot cos2phi} \]
9. Add Preprocessing

Alternative 10: 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 60.7%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.9

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

5. Applied rewrites 75.9%

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

Alternative 11: 67.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\frac{\left(alphax \cdot u0\right) \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.999999682655225e-22)
   (/ (* (* 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 <= 9.999999682655225e-22f) {
		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 <= 9.999999682655225e-22) 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 (sin2phi <= Float32(9.999999682655225e-22))
		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 <= single(9.999999682655225e-22))
		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 sin2phi < 9.9999997e-22

   1. Initial program 53.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 62.8%

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

   10. Applied rewrites 62.8%

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

   if 9.9999997e-22 < sin2phi

   1. Initial program 62.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.0%

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

Alternative 12: 23.6% accurate, 6.9× speedup

Math

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

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(Float32(Float32(alphax * alphax) * u0) / cos2phi)
end

MATLAB

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

Derivation

1. Initial program 60.7%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.9

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

5. Applied rewrites 75.9%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 24.2%

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

Alternative 13: 23.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

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) / cos2phi) * u0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.9

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

5. Applied rewrites 75.9%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 24.2%

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

10. Applied rewrites 24.2%

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

11. Final simplification 24.2%

    \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]
12. Add Preprocessing

Alternative 14: 23.6% accurate, 6.9× speedup

Math

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

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(Float32(u0 / cos2phi) * alphax) * alphax)
end

MATLAB

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

Derivation

1. Initial program 60.7%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.9

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

5. Applied rewrites 75.9%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 24.2%

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

10. Applied rewrites 24.1%

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

11. Final simplification 24.1%

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

Reproduce

herbie shell --seed 2024295 
(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)))))