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

Percentage Accurate: 60.2% → 76.0%

Time: 12.3s

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: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 76.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 74.7%

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

Alternative 2: 53.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{cos2phi}{alphax \cdot alphax} + {\left({\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right)}^{0.5}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u0)))))
   (if (<= t_0 0.0035000001080334187)
     (/
      (-
       (*
        (*
         (fma
          (fma (fma (* -0.25 u0) u0 -0.3333333333333333) (* u0 u0) -0.5)
          (* u0 u0)
          -1.0)
         u0)
        u0)
       (* (+ (* (fma u0 0.3333333333333333 -0.5) u0) 1.0) u0))
      (- (/ (- cos2phi) (* alphax alphax)) (/ sin2phi (* alphay alphay))))
     (/
      t_0
      (+
       (/ cos2phi (* alphax alphax))
       (pow (pow (/ (/ sin2phi alphay) alphay) 0.5) 2.0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -logf((1.0f - u0));
	float tmp;
	if (t_0 <= 0.0035000001080334187f) {
		tmp = (((fmaf(fmaf(fmaf((-0.25f * u0), u0, -0.3333333333333333f), (u0 * u0), -0.5f), (u0 * u0), -1.0f) * u0) * u0) - (((fmaf(u0, 0.3333333333333333f, -0.5f) * u0) + 1.0f) * u0)) / ((-cos2phi / (alphax * alphax)) - (sin2phi / (alphay * alphay)));
	} else {
		tmp = t_0 / ((cos2phi / (alphax * alphax)) + powf(powf(((sin2phi / alphay) / alphay), 0.5f), 2.0f));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(Float32(Float32(fma(fma(fma(Float32(Float32(-0.25) * u0), u0, Float32(-0.3333333333333333)), Float32(u0 * u0), Float32(-0.5)), Float32(u0 * u0), Float32(-1.0)) * u0) * u0) - Float32(Float32(Float32(fma(u0, Float32(0.3333333333333333), Float32(-0.5)) * u0) + Float32(1.0)) * u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(t_0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + ((Float32(Float32(sin2phi / alphay) / alphay) ^ Float32(0.5)) ^ Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00350000011

   1. Initial program 48.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) + 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \left(\color{blue}{\left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) \cdot u0} + 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot u0 - \frac{1}{2}, u0, 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\frac{1}{3} \cdot u0 + \color{blue}{\frac{-1}{2}}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 85.7

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 85.7%

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right), u0, 1\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

   10. Applied rewrites 85.7%

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

   12. Applied rewrites 97.3%

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

   if 0.00350000011 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.6%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-/.f32 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. neg-mul-1 N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 92.6

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

   6. Applied rewrites 92.6%

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

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. /-rgt-identity N/A

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

      7. clear-num N/A

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

      8. lift-/.f32 N/A

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

      9. inv-pow N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f32 92.6

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

      14. lift-/.f32 N/A

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

      15. lift-/.f32 N/A

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

      16. associate-/r/ N/A

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

      17. lift-/.f32 N/A

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

      18. clear-num N/A

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

      19. lift-/.f32 N/A

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

      20. lift-/.f32 N/A

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

      21. div-inv N/A

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

      22. lower-/.f32 92.7

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

   8. Applied rewrites 92.7%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + {\left({\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right)}^{0.5}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u0)))))
   (if (<= t_0 0.0035000001080334187)
     (/
      (-
       (*
        (*
         (fma
          (fma (fma (* -0.25 u0) u0 -0.3333333333333333) (* u0 u0) -0.5)
          (* u0 u0)
          -1.0)
         u0)
        u0)
       (* (+ (* (fma u0 0.3333333333333333 -0.5) u0) 1.0) u0))
      (- (/ (- cos2phi) (* alphax alphax)) (/ sin2phi (* alphay alphay))))
     (/
      t_0
      (+
       (/ cos2phi (* alphax alphax))
       (* (/ sin2phi alphay) (/ 1.0 alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -logf((1.0f - u0));
	float tmp;
	if (t_0 <= 0.0035000001080334187f) {
		tmp = (((fmaf(fmaf(fmaf((-0.25f * u0), u0, -0.3333333333333333f), (u0 * u0), -0.5f), (u0 * u0), -1.0f) * u0) * u0) - (((fmaf(u0, 0.3333333333333333f, -0.5f) * u0) + 1.0f) * u0)) / ((-cos2phi / (alphax * alphax)) - (sin2phi / (alphay * alphay)));
	} else {
		tmp = t_0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(Float32(Float32(fma(fma(fma(Float32(Float32(-0.25) * u0), u0, Float32(-0.3333333333333333)), Float32(u0 * u0), Float32(-0.5)), Float32(u0 * u0), Float32(-1.0)) * u0) * u0) - Float32(Float32(Float32(fma(u0, Float32(0.3333333333333333), Float32(-0.5)) * u0) + Float32(1.0)) * u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(t_0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00350000011

   1. Initial program 48.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) + 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \left(\color{blue}{\left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) \cdot u0} + 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot u0 - \frac{1}{2}, u0, 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\frac{1}{3} \cdot u0 + \color{blue}{\frac{-1}{2}}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 85.7

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 85.7%

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right), u0, 1\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

   10. Applied rewrites 85.7%

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

   12. Applied rewrites 97.3%

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

   if 0.00350000011 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.6%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 92.6

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

   4. Applied rewrites 92.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;-t\_1 \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\frac{-1}{alphax \cdot \frac{alphax}{cos2phi}} - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))) (t_1 (log (- 1.0 u0))))
   (if (<= (- t_1) 0.0035000001080334187)
     (/
      (-
       (*
        (*
         (fma
          (fma (fma (* -0.25 u0) u0 -0.3333333333333333) (* u0 u0) -0.5)
          (* u0 u0)
          -1.0)
         u0)
        u0)
       (* (+ (* (fma u0 0.3333333333333333 -0.5) u0) 1.0) u0))
      (- (/ (- cos2phi) (* alphax alphax)) t_0))
     (/ t_1 (- (/ -1.0 (* alphax (/ alphax cos2phi))) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (-t_1 <= 0.0035000001080334187f) {
		tmp = (((fmaf(fmaf(fmaf((-0.25f * u0), u0, -0.3333333333333333f), (u0 * u0), -0.5f), (u0 * u0), -1.0f) * u0) * u0) - (((fmaf(u0, 0.3333333333333333f, -0.5f) * u0) + 1.0f) * u0)) / ((-cos2phi / (alphax * alphax)) - t_0);
	} else {
		tmp = t_1 / ((-1.0f / (alphax * (alphax / cos2phi))) - t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (Float32(-t_1) <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(Float32(Float32(fma(fma(fma(Float32(Float32(-0.25) * u0), u0, Float32(-0.3333333333333333)), Float32(u0 * u0), Float32(-0.5)), Float32(u0 * u0), Float32(-1.0)) * u0) * u0) - Float32(Float32(Float32(fma(u0, Float32(0.3333333333333333), Float32(-0.5)) * u0) + Float32(1.0)) * u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - t_0));
	else
		tmp = Float32(t_1 / Float32(Float32(Float32(-1.0) / Float32(alphax * Float32(alphax / cos2phi))) - t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00350000011

   1. Initial program 48.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) + 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \left(\color{blue}{\left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) \cdot u0} + 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot u0 - \frac{1}{2}, u0, 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\frac{1}{3} \cdot u0 + \color{blue}{\frac{-1}{2}}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 85.7

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 85.7%

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right), u0, 1\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

   10. Applied rewrites 85.7%

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

   12. Applied rewrites 97.3%

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

   if 0.00350000011 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.6%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-/.f32 92.7

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

   4. Applied rewrites 92.7%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{alphax \cdot \frac{alphax}{cos2phi}} - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;-t\_1 \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))) (t_1 (log (- 1.0 u0))))
   (if (<= (- t_1) 0.0035000001080334187)
     (/
      (-
       (*
        (*
         (fma
          (fma (fma (* -0.25 u0) u0 -0.3333333333333333) (* u0 u0) -0.5)
          (* u0 u0)
          -1.0)
         u0)
        u0)
       (* (+ (* (fma u0 0.3333333333333333 -0.5) u0) 1.0) u0))
      (- (/ (- cos2phi) (* alphax alphax)) t_0))
     (/ t_1 (- (/ -1.0 (/ (* alphax alphax) cos2phi)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (-t_1 <= 0.0035000001080334187f) {
		tmp = (((fmaf(fmaf(fmaf((-0.25f * u0), u0, -0.3333333333333333f), (u0 * u0), -0.5f), (u0 * u0), -1.0f) * u0) * u0) - (((fmaf(u0, 0.3333333333333333f, -0.5f) * u0) + 1.0f) * u0)) / ((-cos2phi / (alphax * alphax)) - t_0);
	} else {
		tmp = t_1 / ((-1.0f / ((alphax * alphax) / cos2phi)) - t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (Float32(-t_1) <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(Float32(Float32(fma(fma(fma(Float32(Float32(-0.25) * u0), u0, Float32(-0.3333333333333333)), Float32(u0 * u0), Float32(-0.5)), Float32(u0 * u0), Float32(-1.0)) * u0) * u0) - Float32(Float32(Float32(fma(u0, Float32(0.3333333333333333), Float32(-0.5)) * u0) + Float32(1.0)) * u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - t_0));
	else
		tmp = Float32(t_1 / Float32(Float32(Float32(-1.0) / Float32(Float32(alphax * alphax) / cos2phi)) - t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00350000011

   1. Initial program 48.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) + 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \left(\color{blue}{\left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) \cdot u0} + 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot u0 - \frac{1}{2}, u0, 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\frac{1}{3} \cdot u0 + \color{blue}{\frac{-1}{2}}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 85.7

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 85.7%

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right), u0, 1\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

   10. Applied rewrites 85.7%

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

   12. Applied rewrites 97.3%

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

   if 0.00350000011 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.6%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 92.6

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

   4. Applied rewrites 92.6%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))) (t_1 (- (log (- 1.0 u0)))))
   (if (<= t_1 0.0035000001080334187)
     (/
      (-
       (*
        (*
         (fma
          (fma (fma (* -0.25 u0) u0 -0.3333333333333333) (* u0 u0) -0.5)
          (* u0 u0)
          -1.0)
         u0)
        u0)
       (* (+ (* (fma u0 0.3333333333333333 -0.5) u0) 1.0) u0))
      (- (/ (- cos2phi) (* alphax alphax)) t_0))
     (/ t_1 (+ (/ 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 t_1 = -logf((1.0f - u0));
	float tmp;
	if (t_1 <= 0.0035000001080334187f) {
		tmp = (((fmaf(fmaf(fmaf((-0.25f * u0), u0, -0.3333333333333333f), (u0 * u0), -0.5f), (u0 * u0), -1.0f) * u0) * u0) - (((fmaf(u0, 0.3333333333333333f, -0.5f) * u0) + 1.0f) * u0)) / ((-cos2phi / (alphax * alphax)) - t_0);
	} else {
		tmp = t_1 / ((cos2phi / (alphax * alphax)) + t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(Float32(Float32(fma(fma(fma(Float32(Float32(-0.25) * u0), u0, Float32(-0.3333333333333333)), Float32(u0 * u0), Float32(-0.5)), Float32(u0 * u0), Float32(-1.0)) * u0) * u0) - Float32(Float32(Float32(fma(u0, Float32(0.3333333333333333), Float32(-0.5)) * u0) + Float32(1.0)) * u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - t_0));
	else
		tmp = Float32(t_1 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00350000011

   1. Initial program 48.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log1p.f32 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) + 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \left(\color{blue}{\left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) \cdot u0} + 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot u0 - \frac{1}{2}, u0, 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\frac{1}{3} \cdot u0 + \color{blue}{\frac{-1}{2}}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 85.7

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 85.7%

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right), u0, 1\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

   10. Applied rewrites 85.7%

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

   12. Applied rewrites 97.3%

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

   if 0.00350000011 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.6%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0035000001080334187:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), u0 \cdot u0, -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (-
   (*
    (*
     (fma
      (fma (fma (* -0.25 u0) u0 -0.3333333333333333) (* u0 u0) -0.5)
      (* u0 u0)
      -1.0)
     u0)
    u0)
   (* (+ (* (fma u0 0.3333333333333333 -0.5) u0) 1.0) u0))
  (- (/ (- cos2phi) (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (((fmaf(fmaf(fmaf((-0.25f * u0), u0, -0.3333333333333333f), (u0 * u0), -0.5f), (u0 * u0), -1.0f) * u0) * u0) - (((fmaf(u0, 0.3333333333333333f, -0.5f) * u0) + 1.0f) * u0)) / ((-cos2phi / (alphax * alphax)) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(Float32(fma(fma(fma(Float32(Float32(-0.25) * u0), u0, Float32(-0.3333333333333333)), Float32(u0 * u0), Float32(-0.5)), Float32(u0 * u0), Float32(-1.0)) * u0) * u0) - Float32(Float32(Float32(fma(u0, Float32(0.3333333333333333), Float32(-0.5)) * u0) + Float32(1.0)) * u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.6%

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

   1. lift-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 75.2

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

4. Applied rewrites 75.2%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) + 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \left(\color{blue}{\left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) \cdot u0} + 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot u0 - \frac{1}{2}, u0, 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. sub-neg N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\frac{1}{3} \cdot u0 + \color{blue}{\frac{-1}{2}}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 75.2

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 75.2%

   \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right), u0, 1\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

10. Applied rewrites 75.2%

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

12. Applied rewrites 86.8%

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

13. Final simplification 86.9%

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

Alternative 8: 15.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(-0.25 \cdot u0, u0, -0.3333333333333333\right), -0.5\right), u0 \cdot u0, -1\right) \cdot u0\right) \cdot u0 - \mathsf{fma}\left(\mathsf{fma}\left(u0, 0.3333333333333333, -0.5\right), u0, 1\right) \cdot u0}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (-
   (*
    (*
     (fma
      (fma (* u0 u0) (fma (* -0.25 u0) u0 -0.3333333333333333) -0.5)
      (* u0 u0)
      -1.0)
     u0)
    u0)
   (* (fma (fma u0 0.3333333333333333 -0.5) u0 1.0) u0))
  (- (/ (- cos2phi) (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (((fmaf(fmaf((u0 * u0), fmaf((-0.25f * u0), u0, -0.3333333333333333f), -0.5f), (u0 * u0), -1.0f) * u0) * u0) - (fmaf(fmaf(u0, 0.3333333333333333f, -0.5f), u0, 1.0f) * u0)) / ((-cos2phi / (alphax * alphax)) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(Float32(fma(fma(Float32(u0 * u0), fma(Float32(Float32(-0.25) * u0), u0, Float32(-0.3333333333333333)), Float32(-0.5)), Float32(u0 * u0), Float32(-1.0)) * u0) * u0) - Float32(fma(fma(u0, Float32(0.3333333333333333), Float32(-0.5)), u0, Float32(1.0)) * u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.6%

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

   1. lift-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 75.2

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

4. Applied rewrites 75.2%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right)\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) + 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \left(\color{blue}{\left(\frac{1}{3} \cdot u0 - \frac{1}{2}\right) \cdot u0} + 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot u0 - \frac{1}{2}, u0, 1\right)} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. sub-neg N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\frac{1}{3} \cdot u0 + \color{blue}{\frac{-1}{2}}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 75.2

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right)}, u0, 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 75.2%

   \[\leadsto \frac{-\left(\mathsf{log1p}\left(\left(-u0\right) \cdot u0\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, -0.5\right), u0, 1\right) \cdot u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

10. Applied rewrites 75.2%

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

12. Applied rewrites 75.2%

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

13. Final simplification 75.2%

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

Alternative 9: 12.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \mathsf{fma}\left(\frac{u0}{t\_0}, 0.5, \frac{1}{t\_0}\right) \cdot u0 \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (* (fma (/ u0 t_0) 0.5 (/ 1.0 t_0)) u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	return fmaf((u0 / t_0), 0.5f, (1.0f / t_0)) * u0;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	return Float32(fma(Float32(u0 / t_0), Float32(0.5), Float32(Float32(1.0) / t_0)) * u0)
end

Derivation

1. Initial program 59.6%

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

   1. lift-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 75.2

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

4. Applied rewrites 75.2%

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

5. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 74.7%

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

Alternative 10: 76.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

Alternative 11: 66.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.5%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 56.0%

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

   10. Applied rewrites 56.3%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 73.4%

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

Alternative 12: 24.0% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 24.4%

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

10. Applied rewrites 24.5%

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

Alternative 13: 24.0% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 74.7

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 24.4%

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

10. Applied rewrites 24.5%

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

Reproduce

herbie shell --seed 2024314 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))