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

Percentage Accurate: 60.0% → 98.5%

Time: 19.1s

Alternatives: 10

Speedup: 8.9×

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: 60.0% 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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} \cdot \left(alphax \cdot alphay\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/
   (- (log1p (- u0)))
   (+ (/ (* alphax sin2phi) alphay) (/ (* alphay cos2phi) alphax)))
  (* alphax alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (-log1pf(-u0) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax))) * (alphax * alphay);
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(Float32(alphax * sin2phi) / alphay) + Float32(Float32(alphay * cos2phi) / alphax))) * Float32(alphax * alphay))
end

Derivation

1. Initial program 64.2%

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

   1. sub-neg 64.2%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-/r* 98.1%

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

   3. frac-add 97.7%

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

6. Applied egg-rr 97.7%

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

   1. +-commutative 97.7%

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

   2. *-commutative 97.7%

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

   3. *-commutative 97.7%

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

   4. fma-def 97.8%

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

8. Simplified 97.8%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphax \cdot alphay}}} \]
9. Step-by-step derivation

   1. associate-/r/ 98.3%

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

   2. associate-*r/ 98.3%

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

10. Applied egg-rr 98.3%

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

11. Taylor expanded in sin2phi around 0 98.7%

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

12. Final simplification 98.7%

    \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} \cdot \left(alphax \cdot alphay\right) \]
13. Add Preprocessing

Alternative 2: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.012000000104308128:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \left(\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \frac{-alphay}{alphax}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.012000000104308128)
   (/
    u0
    (- (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ -1.0 alphay))))
   (* (* alphax alphay) (* (/ (log1p (- u0)) sin2phi) (/ (- alphay) alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.012000000104308128f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) - ((sin2phi / alphay) * (-1.0f / alphay)));
	} else {
		tmp = (alphax * alphay) * ((log1pf(-u0) / sin2phi) * (-alphay / alphax));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.012000000104308128))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) - Float32(Float32(sin2phi / alphay) * Float32(Float32(-1.0) / alphay))));
	else
		tmp = Float32(Float32(alphax * alphay) * Float32(Float32(log1p(Float32(-u0)) / sin2phi) * Float32(Float32(-alphay) / alphax)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0120000001

   1. Initial program 60.2%

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

   3. Taylor expanded in u0 around 0 71.0%

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

      1. mul-1-neg 71.0%

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

   5. Simplified 71.0%

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

      1. associate-/r* 71.1%

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

      2. div-inv 71.1%

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

   7. Applied egg-rr 71.1%

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

   if 0.0120000001 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.8%

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

      1. sub-neg 66.8%

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

      2. log1p-def 97.6%

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

   3. Simplified 97.6%

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

      1. associate-/r* 97.6%

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

      2. associate-/r* 97.7%

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

      3. frac-add 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. +-commutative 97.4%

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

      2. *-commutative 97.4%

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

      3. *-commutative 97.4%

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

      4. fma-def 97.4%

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

   8. Simplified 97.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphax \cdot alphay}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 98.2%

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

      2. associate-*r/ 98.2%

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

   10. Applied egg-rr 98.2%

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

   11. Taylor expanded in sin2phi around inf 66.9%

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

       1. mul-1-neg 66.9%

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

       2. times-frac 67.0%

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

       3. distribute-rgt-neg-in 67.0%

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

       4. sub-neg 67.0%

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

       5. log1p-def 97.3%

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

   13. Simplified 97.3%

       \[\leadsto \color{blue}{\left(\frac{alphay}{alphax} \cdot \left(-\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi}\right)\right)} \cdot \left(alphax \cdot alphay\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.012000000104308128:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \left(\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \frac{-alphay}{alphax}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(alphax \cdot alphay\right) \cdot \frac{-\mathsf{log1p}\left(-u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \frac{alphay}{\frac{alphax}{cos2phi}}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (* alphax alphay)
  (/
   (- (log1p (- u0)))
   (+ (* alphax (/ sin2phi alphay)) (/ alphay (/ alphax cos2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * alphay) * (-log1pf(-u0) / ((alphax * (sin2phi / alphay)) + (alphay / (alphax / cos2phi))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(alphax * alphay) * Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(alphax * Float32(sin2phi / alphay)) + Float32(alphay / Float32(alphax / cos2phi)))))
end

Derivation

1. Initial program 64.2%

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

   1. sub-neg 64.2%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-/r* 98.1%

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

   3. frac-add 97.7%

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

6. Applied egg-rr 97.7%

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

   1. +-commutative 97.7%

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

   2. *-commutative 97.7%

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

   3. *-commutative 97.7%

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

   4. fma-def 97.8%

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

8. Simplified 97.8%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphax \cdot alphay}}} \]
9. Step-by-step derivation

   1. associate-/r/ 98.3%

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

   2. associate-*r/ 98.3%

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

10. Applied egg-rr 98.3%

    \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{alphay \cdot cos2phi}{alphax}\right)} \cdot \left(alphax \cdot alphay\right)} \]
11. Step-by-step derivation

    1. fma-udef 98.3%

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

    2. associate-/l* 98.2%

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

12. Applied egg-rr 98.2%

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

13. Final simplification 98.2%

    \[\leadsto \left(alphax \cdot alphay\right) \cdot \frac{-\mathsf{log1p}\left(-u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \frac{alphay}{\frac{alphax}{cos2phi}}} \]
14. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 64.2%

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

   1. sub-neg 64.2%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 64.2%

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

   1. sub-neg 64.2%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

   1. clear-num 75.7%

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

   2. associate-/r/ 75.6%

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

   3. pow2 75.6%

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

   4. pow-flip 75.7%

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

   5. metadata-eval 75.7%

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

6. Applied egg-rr 98.2%

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

   1. *-commutative 75.7%

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

   2. metadata-eval 75.7%

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

   3. pow-flip 75.6%

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

   4. pow2 75.6%

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

   5. div-inv 75.6%

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

   6. associate-/r* 75.7%

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

8. Applied egg-rr 98.1%

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

9. Final simplification 98.1%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
10. Add Preprocessing

Alternative 6: 76.3% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.2%

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

   1. sub-neg 64.2%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-/r* 98.1%

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

   3. frac-add 97.7%

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

6. Applied egg-rr 97.7%

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

   1. +-commutative 97.7%

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

   2. *-commutative 97.7%

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

   3. *-commutative 97.7%

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

   4. fma-def 97.8%

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

8. Simplified 97.8%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphax \cdot alphay}}} \]
9. Step-by-step derivation

   1. associate-/r/ 98.3%

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

   2. associate-*r/ 98.3%

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

10. Applied egg-rr 98.3%

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

11. Taylor expanded in u0 around 0 76.1%

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

12. Final simplification 76.1%

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

Alternative 7: 76.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in u0 around 0 75.6%

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

   1. mul-1-neg 75.6%

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

5. Simplified 75.6%

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

6. Final simplification 75.6%

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

Alternative 8: 76.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in u0 around 0 75.6%

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

   1. mul-1-neg 75.6%

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

5. Simplified 75.6%

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

   1. clear-num 75.7%

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

   2. associate-/r/ 75.6%

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

   3. pow2 75.6%

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

   4. pow-flip 75.7%

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

   5. metadata-eval 75.7%

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

7. Applied egg-rr 75.7%

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

   1. *-commutative 75.7%

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

   2. metadata-eval 75.7%

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

   3. pow-flip 75.6%

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

   4. pow2 75.6%

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

   5. div-inv 75.6%

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

   6. associate-/r* 75.7%

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

9. Applied egg-rr 75.7%

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

10. Final simplification 75.7%

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

Alternative 9: 59.2% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

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 * alphay) * ((alphay / alphax) * (u0 / sin2phi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 64.2%

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

   1. sub-neg 64.2%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-/r* 98.1%

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

   3. frac-add 97.7%

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

6. Applied egg-rr 97.7%

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

   1. +-commutative 97.7%

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

   2. *-commutative 97.7%

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

   3. *-commutative 97.7%

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

   4. fma-def 97.8%

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

8. Simplified 97.8%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphax \cdot alphay}}} \]
9. Step-by-step derivation

   1. associate-/r/ 98.3%

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

   2. associate-*r/ 98.3%

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

10. Applied egg-rr 98.3%

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

11. Taylor expanded in sin2phi around inf 75.0%

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

12. Taylor expanded in u0 around 0 60.7%

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

    1. times-frac 60.7%

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

14. Simplified 60.7%

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

15. Final simplification 60.7%

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

Alternative 10: 59.2% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.2%

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

   1. sub-neg 64.2%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-/r* 98.1%

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

   3. frac-add 97.7%

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

6. Applied egg-rr 97.7%

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

   1. +-commutative 97.7%

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

   2. *-commutative 97.7%

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

   3. *-commutative 97.7%

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

   4. fma-def 97.8%

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

8. Simplified 97.8%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphax \cdot alphay}}} \]
9. Step-by-step derivation

   1. associate-/r/ 98.3%

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

   2. associate-*r/ 98.3%

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

10. Applied egg-rr 98.3%

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

11. Taylor expanded in sin2phi around inf 75.0%

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

12. Taylor expanded in u0 around 0 60.7%

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

    1. *-commutative 60.7%

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

    2. times-frac 60.7%

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

14. Simplified 60.7%

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

15. Final simplification 60.7%

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

Reproduce

herbie shell --seed 2024041 
(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)))))