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

Percentage Accurate: 61.2% → 97.2%

Time: 12.2s

Alternatives: 13

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0 + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- 1.0 u0) 0.9850000143051147)
     (/ (- (log (- 1.0 u0))) (+ t_0 (/ (/ cos2phi alphax) alphax)))
     (/
      (-
       (*
        (* (- (+ (/ 1.0 (* u0 u0)) 0.3333333333333333) (/ 0.5 u0)) (* u0 u0))
        u0)
       (* (- u0) u0))
      (+ (/ cos2phi (* alphax alphax)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if ((1.0f - u0) <= 0.9850000143051147f) {
		tmp = -logf((1.0f - u0)) / (t_0 + ((cos2phi / alphax) / alphax));
	} else {
		tmp = ((((((1.0f / (u0 * u0)) + 0.3333333333333333f) - (0.5f / u0)) * (u0 * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if ((1.0e0 - u0) <= 0.9850000143051147e0) then
        tmp = -log((1.0e0 - u0)) / (t_0 + ((cos2phi / alphax) / alphax))
    else
        tmp = ((((((1.0e0 / (u0 * u0)) + 0.3333333333333333e0) - (0.5e0 / u0)) * (u0 * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + t_0)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9850000143051147))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(t_0 + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(u0 * u0)) + Float32(0.3333333333333333)) - Float32(Float32(0.5) / u0)) * Float32(u0 * u0)) * u0) - Float32(Float32(-u0) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9850000143051147))
		tmp = -log((single(1.0) - u0)) / (t_0 + ((cos2phi / alphax) / alphax));
	else
		tmp = ((((((single(1.0) / (u0 * u0)) + single(0.3333333333333333)) - (single(0.5) / u0)) * (u0 * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.985000014

   1. Initial program 94.3%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 94.4

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

   4. Applied rewrites 94.4%

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

   if 0.985000014 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 53.1%

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

      1. lift-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 86.1

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

   4. Applied rewrites 86.1%

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

   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 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 86.1

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

   10. Applied rewrites 85.7%

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

   11. Taylor expanded in u0 around inf

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

   13. Applied rewrites 97.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0 - \left(-u0\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (if (<= (- 1.0 u0) 0.9850000143051147)
     (/ (- (log (- 1.0 u0))) t_0)
     (/
      (-
       (*
        (* (- (+ (/ 1.0 (* u0 u0)) 0.3333333333333333) (/ 0.5 u0)) (* u0 u0))
        u0)
       (* (- u0) u0))
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	float tmp;
	if ((1.0f - u0) <= 0.9850000143051147f) {
		tmp = -logf((1.0f - u0)) / t_0;
	} else {
		tmp = ((((((1.0f / (u0 * u0)) + 0.3333333333333333f) - (0.5f / u0)) * (u0 * u0)) * u0) - (-u0 * u0)) / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))
    if ((1.0e0 - u0) <= 0.9850000143051147e0) then
        tmp = -log((1.0e0 - u0)) / t_0
    else
        tmp = ((((((1.0e0 / (u0 * u0)) + 0.3333333333333333e0) - (0.5e0 / u0)) * (u0 * u0)) * u0) - (-u0 * u0)) / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9850000143051147))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(u0 * u0)) + Float32(0.3333333333333333)) - Float32(Float32(0.5) / u0)) * Float32(u0 * u0)) * u0) - Float32(Float32(-u0) * u0)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9850000143051147))
		tmp = -log((single(1.0) - u0)) / t_0;
	else
		tmp = ((((((single(1.0) / (u0 * u0)) + single(0.3333333333333333)) - (single(0.5) / u0)) * (u0 * 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) < 0.985000014

   1. Initial program 94.3%

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

   if 0.985000014 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 53.1%

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

      1. lift-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 86.1

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

   4. Applied rewrites 86.1%

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

   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 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 86.1

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

   10. Applied rewrites 85.7%

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

   11. Taylor expanded in u0 around inf

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

   13. Applied rewrites 97.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{u0 \cdot u0} + 0.3333333333333333\right) - \frac{0.5}{u0}\right) \cdot \left(u0 \cdot u0\right)\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. 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 75.7

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

4. Applied rewrites 75.7%

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

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 75.7

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

7. Applied rewrites 75.7%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 75.7

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

10. Applied rewrites 75.7%

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

11. Taylor expanded in u0 around inf

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

13. Applied rewrites 88.6%

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

14. Final simplification 88.6%

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

Alternative 4: 90.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. 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 75.7

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

4. Applied rewrites 75.7%

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

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 75.7

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

7. Applied rewrites 75.7%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 75.7

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

10. Applied rewrites 75.4%

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

11. Taylor expanded in u0 around -inf

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

13. Applied rewrites 88.5%

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

14. Final simplification 88.5%

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

Alternative 5: 54.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. 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 75.7

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

4. Applied rewrites 75.7%

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

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 75.7

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

7. Applied rewrites 75.7%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 75.7

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

10. Applied rewrites 75.4%

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

12. Applied rewrites 84.7%

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

13. Final simplification 84.9%

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

Alternative 6: 66.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{alphax}{\frac{1}{u0}} \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.999999936531045e-21)
   (* (/ alphax (/ 1.0 u0)) (/ alphax cos2phi))
   (* (/ u0 sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21f) {
		tmp = (alphax / (1.0f / u0)) * (alphax / cos2phi);
	} else {
		tmp = (u0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21) then
        tmp = (alphax / (1.0e0 / u0)) * (alphax / cos2phi)
    else
        tmp = (u0 / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(alphax / Float32(Float32(1.0) / u0)) * Float32(alphax / cos2phi));
	else
		tmp = Float32(Float32(u0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.999999936531045e-21))
		tmp = (alphax / (single(1.0) / u0)) * (alphax / cos2phi);
	else
		tmp = (u0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.9999999e-21

   1. Initial program 56.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 61.2%

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

   10. Applied rewrites 61.3%

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

   12. Applied rewrites 61.4%

       \[\leadsto \frac{alphax}{cos2phi} \cdot \frac{alphax}{\color{blue}{\frac{1}{u0}}} \]

   if 1.9999999e-21 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in alphay around 0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in alphax around inf

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

   11. Applied rewrites 69.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{alphax}{\frac{1}{u0}} \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 75.1

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

5. Applied rewrites 75.1%

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

7. Applied rewrites 75.1%

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

Alternative 8: 75.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 75.1

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

5. Applied rewrites 75.1%

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

7. Applied rewrites 75.1%

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

Alternative 9: 66.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot u0\right) \cdot \frac{1}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.999999936531045e-21)
   (* (* (* alphax alphax) u0) (/ 1.0 cos2phi))
   (* (/ u0 sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21f) {
		tmp = ((alphax * alphax) * u0) * (1.0f / cos2phi);
	} else {
		tmp = (u0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21) then
        tmp = ((alphax * alphax) * u0) * (1.0e0 / cos2phi)
    else
        tmp = (u0 / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(Float32(alphax * alphax) * u0) * Float32(Float32(1.0) / cos2phi));
	else
		tmp = Float32(Float32(u0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.999999936531045e-21))
		tmp = ((alphax * alphax) * u0) * (single(1.0) / cos2phi);
	else
		tmp = (u0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.9999999e-21

   1. Initial program 56.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 61.2%

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

   10. Applied rewrites 61.3%

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

   if 1.9999999e-21 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in alphay around 0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in alphax around inf

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

   11. Applied rewrites 69.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot u0\right) \cdot \frac{1}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(alphax \cdot u0\right) \cdot \frac{1}{cos2phi}\right) \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.999999936531045e-21)
   (* (* (* alphax u0) (/ 1.0 cos2phi)) alphax)
   (* (/ u0 sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21f) {
		tmp = ((alphax * u0) * (1.0f / cos2phi)) * alphax;
	} else {
		tmp = (u0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21) then
        tmp = ((alphax * u0) * (1.0e0 / cos2phi)) * alphax
    else
        tmp = (u0 / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(Float32(alphax * u0) * Float32(Float32(1.0) / cos2phi)) * alphax);
	else
		tmp = Float32(Float32(u0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.999999936531045e-21))
		tmp = ((alphax * u0) * (single(1.0) / cos2phi)) * alphax;
	else
		tmp = (u0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.9999999e-21

   1. Initial program 56.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 61.2%

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

   10. Applied rewrites 61.3%

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

   if 1.9999999e-21 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in alphay around 0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in alphax around inf

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

   11. Applied rewrites 69.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(alphax \cdot u0\right) \cdot \frac{1}{cos2phi}\right) \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 75.1

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

5. Applied rewrites 75.1%

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

6. Final simplification 75.1%

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

Alternative 12: 66.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.999999936531045e-21)
   (* (* (/ u0 cos2phi) alphax) alphax)
   (* (/ u0 sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21f) {
		tmp = ((u0 / cos2phi) * alphax) * alphax;
	} else {
		tmp = (u0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21) then
        tmp = ((u0 / cos2phi) * alphax) * alphax
    else
        tmp = (u0 / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(Float32(u0 / cos2phi) * alphax) * alphax);
	else
		tmp = Float32(Float32(u0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.999999936531045e-21))
		tmp = ((u0 / cos2phi) * alphax) * alphax;
	else
		tmp = (u0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.9999999e-21

   1. Initial program 56.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 61.2%

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

   10. Applied rewrites 61.3%

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

   if 1.9999999e-21 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in alphay around 0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in alphax around inf

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

   11. Applied rewrites 69.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 23.7% 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 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 75.1

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

5. Applied rewrites 75.1%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 22.5%

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

10. Applied rewrites 22.5%

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

11. Final simplification 22.5%

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

Reproduce

herbie shell --seed 2024331 
(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)))))