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

Percentage Accurate: 60.5% → 98.5%

Time: 15.5s

Alternatives: 11

Speedup: 8.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphax \cdot sin2phi}{alphay} + \left(alphay \cdot cos2phi\right) \cdot \frac{1}{alphax}} \cdot \left(alphax \cdot alphay\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/
   (- (log1p (- u0)))
   (+ (/ (* alphax sin2phi) alphay) (* (* alphay cos2phi) (/ 1.0 alphax))))
  (* alphax alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (-log1pf(-u0) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) * (1.0f / 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) * Float32(Float32(1.0) / alphax)))) * Float32(alphax * alphay))
end

Derivation

1. Initial program 62.4%

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

   1. sub-neg 62.4%

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

   2. log1p-def 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* 97.8%

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

   2. associate-/r* 97.8%

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

   3. frac-add 97.4%

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

5. Applied egg-rr 97.4%

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

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

   3. fma-def 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*l/ 97.4%

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

   6. associate-/l* 97.5%

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

7. Simplified 97.5%

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

   1. associate-/r/ 98.1%

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

   2. associate-/r/ 98.0%

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

9. Applied egg-rr 98.0%

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

10. Taylor expanded in alphax around 0 98.3%

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

    1. div-inv 98.3%

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

12. Applied egg-rr 98.3%

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

13. Final simplification 98.3%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.4%

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

   1. sub-neg 62.4%

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

   2. log1p-def 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* 97.8%

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

   2. associate-/r* 97.8%

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

   3. frac-add 97.4%

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

5. Applied egg-rr 97.4%

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

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

   3. fma-def 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*l/ 97.4%

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

   6. associate-/l* 97.5%

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

7. Simplified 97.5%

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

   1. associate-/r/ 98.1%

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

   2. associate-/r/ 98.0%

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

9. Applied egg-rr 98.0%

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

10. Taylor expanded in alphax around 0 98.3%

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

11. Final simplification 98.3%

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

Alternative 3: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.7999999499807018 \cdot 10^{-6}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0}{\frac{alphay \cdot cos2phi}{alphax} + \frac{alphax}{\frac{alphay}{sin2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{alphax}}{sin2phi} \cdot \left(alphax \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.7999999499807018e-6)
   (*
    (* alphax alphay)
    (/ u0 (+ (/ (* alphay cos2phi) alphax) (/ alphax (/ alphay sin2phi)))))
   (* (/ (* (log1p (- u0)) (/ alphay alphax)) sin2phi) (* alphax (- alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.7999999499807018e-6f) {
		tmp = (alphax * alphay) * (u0 / (((alphay * cos2phi) / alphax) + (alphax / (alphay / sin2phi))));
	} else {
		tmp = ((log1pf(-u0) * (alphay / alphax)) / sin2phi) * (alphax * -alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.7999999499807018e-6))
		tmp = Float32(Float32(alphax * alphay) * Float32(u0 / Float32(Float32(Float32(alphay * cos2phi) / alphax) + Float32(alphax / Float32(alphay / sin2phi)))));
	else
		tmp = Float32(Float32(Float32(log1p(Float32(-u0)) * Float32(alphay / alphax)) / sin2phi) * Float32(alphax * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.79999995e-6

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. log1p-def 98.4%

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

   3. Simplified 98.4%

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

      1. associate-/r* 98.4%

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

      2. associate-/r* 98.3%

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

      3. frac-add 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. +-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. fma-def 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*l/ 97.8%

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

      6. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

      1. associate-/r/ 98.3%

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

      2. associate-/r/ 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in u0 around 0 74.6%

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

       1. *-un-lft-identity 74.6%

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

       2. associate-/l* 74.7%

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

   12. Applied egg-rr 74.7%

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

   if 1.79999995e-6 < sin2phi

   1. Initial program 68.2%

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

      1. sub-neg 68.2%

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

      2. log1p-def 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* 97.3%

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

      2. associate-/r* 97.4%

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

      3. frac-add 96.9%

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

   5. Applied egg-rr 96.9%

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

      1. +-commutative 96.9%

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

      2. *-commutative 96.9%

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

      3. fma-def 97.0%

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

      4. *-commutative 97.0%

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

      5. associate-*l/ 97.0%

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

      6. associate-/l* 97.0%

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

   7. Simplified 97.0%

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

      1. associate-/r/ 97.9%

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

      2. associate-/r/ 98.0%

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

   9. Applied egg-rr 98.0%

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

   10. Taylor expanded in alphax around inf 68.6%

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

       1. mul-1-neg 68.6%

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

       2. times-frac 68.6%

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

   12. Simplified 68.6%

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

   13. Taylor expanded in alphay around 0 68.6%

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

       1. times-frac 68.6%

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

       2. associate-*r/ 68.5%

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

       3. sub-neg 68.5%

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

       4. mul-1-neg 68.5%

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

       5. log1p-def 96.2%

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

       6. mul-1-neg 96.2%

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

   15. Simplified 96.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.7999999499807018 \cdot 10^{-6}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0}{\frac{alphay \cdot cos2phi}{alphax} + \frac{alphax}{\frac{alphay}{sin2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{alphax}}{sin2phi} \cdot \left(alphax \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 4: 98.3% 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 62.4%

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

   1. sub-neg 62.4%

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

   2. log1p-def 97.8%

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

3. Simplified 97.8%

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

4. Final simplification 97.8%

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

Alternative 5: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0}{\frac{alphay \cdot cos2phi}{alphax} + \frac{alphax}{\frac{alphay}{sin2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999873689376e-6)
   (*
    (* alphax alphay)
    (/ u0 (+ (/ (* alphay cos2phi) alphax) (/ alphax (/ alphay sin2phi)))))
   (/ (- (pow alphay 2.0)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999873689376e-6f) {
		tmp = (alphax * alphay) * (u0 / (((alphay * cos2phi) / alphax) + (alphax / (alphay / sin2phi))));
	} else {
		tmp = -powf(alphay, 2.0f) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	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 <= 4.999999873689376e-6) then
        tmp = (alphax * alphay) * (u0 / (((alphay * cos2phi) / alphax) + (alphax / (alphay / sin2phi))))
    else
        tmp = -(alphay ** 2.0e0) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999873689376e-6))
		tmp = Float32(Float32(alphax * alphay) * Float32(u0 / Float32(Float32(Float32(alphay * cos2phi) / alphax) + Float32(alphax / Float32(alphay / sin2phi)))));
	else
		tmp = Float32(Float32(-(alphay ^ Float32(2.0))) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999873689376e-6))
		tmp = (alphax * alphay) * (u0 / (((alphay * cos2phi) / alphax) + (alphax / (alphay / sin2phi))));
	else
		tmp = -(alphay ^ single(2.0)) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999987e-6

   1. Initial program 56.1%

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

      1. sub-neg 56.1%

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

      2. log1p-def 98.4%

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

   3. Simplified 98.4%

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

      1. associate-/r* 98.4%

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

      2. associate-/r* 98.3%

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

      3. frac-add 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. +-commutative 98.0%

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

      2. *-commutative 98.0%

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

      3. fma-def 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*l/ 97.8%

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

      6. associate-/l* 98.1%

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

   7. Simplified 98.1%

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

      1. associate-/r/ 98.2%

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

      2. associate-/r/ 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in u0 around 0 73.9%

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

       1. *-un-lft-identity 73.9%

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

       2. associate-/l* 74.0%

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

   12. Applied egg-rr 74.0%

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

   if 4.99999987e-6 < sin2phi

   1. Initial program 67.8%

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

      1. sub-neg 67.8%

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

      2. log1p-def 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in cos2phi around 0 69.0%

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

      1. mul-1-neg 69.0%

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

      2. associate-/l* 67.9%

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

      3. distribute-neg-frac 67.9%

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

      4. sub-neg 67.9%

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

      5. log1p-def 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in u0 around 0 86.5%

      \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
   8. Step-by-step derivation

      1. neg-mul-1 86.5%

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

      2. +-commutative 86.5%

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

      3. unsub-neg 86.5%

         \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{0.5 \cdot sin2phi - \frac{sin2phi}{u0}}} \]

      4. *-commutative 86.5%

         \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5} - \frac{sin2phi}{u0}} \]

   9. Simplified 86.5%

      \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0}{\frac{alphay \cdot cos2phi}{alphax} + \frac{alphax}{\frac{alphay}{sin2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 6: 76.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.4%

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

   1. sub-neg 62.4%

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

   2. log1p-def 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* 97.8%

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

   2. associate-/r* 97.8%

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

   3. frac-add 97.4%

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

5. Applied egg-rr 97.4%

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

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

   3. fma-def 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*l/ 97.4%

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

   6. associate-/l* 97.5%

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

7. Simplified 97.5%

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

   1. associate-/r/ 98.1%

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

   2. associate-/r/ 98.0%

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

9. Applied egg-rr 98.0%

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

10. Taylor expanded in alphax around 0 98.3%

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

11. Taylor expanded in u0 around 0 74.7%

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

    1. associate-/l* 74.7%

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

    2. associate-/l* 74.7%

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

    3. associate-/r/ 74.7%

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

    4. *-commutative 74.7%

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

    5. associate-/l* 74.7%

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

13. Simplified 74.7%

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

14. Final simplification 74.7%

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

Alternative 7: 76.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.4%

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

   1. sub-neg 62.4%

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

   2. log1p-def 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* 97.8%

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

   2. associate-/r* 97.8%

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

   3. frac-add 97.4%

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

5. Applied egg-rr 97.4%

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

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

   3. fma-def 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*l/ 97.4%

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

   6. associate-/l* 97.5%

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

7. Simplified 97.5%

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

   1. associate-/r/ 98.1%

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

   2. associate-/r/ 98.0%

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

9. Applied egg-rr 98.0%

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

10. Taylor expanded in u0 around 0 74.7%

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

    1. associate-*l/ 74.7%

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

    2. associate-/l* 74.8%

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

    3. associate-/l* 74.8%

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

12. Applied egg-rr 74.8%

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

13. Final simplification 74.8%

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

Alternative 8: 66.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 7.599999731107421 \cdot 10^{-13}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \left(\frac{alphax}{alphay} \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 7.599999731107421e-13)
   (* (* alphax alphay) (* (/ alphax alphay) (/ u0 cos2phi)))
   (* (/ alphay sin2phi) (/ alphay (/ 1.0 u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 7.599999731107421e-13f) {
		tmp = (alphax * alphay) * ((alphax / alphay) * (u0 / cos2phi));
	} else {
		tmp = (alphay / sin2phi) * (alphay / (1.0f / u0));
	}
	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 <= 7.599999731107421e-13) then
        tmp = (alphax * alphay) * ((alphax / alphay) * (u0 / cos2phi))
    else
        tmp = (alphay / sin2phi) * (alphay / (1.0e0 / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(7.599999731107421e-13))
		tmp = Float32(Float32(alphax * alphay) * Float32(Float32(alphax / alphay) * Float32(u0 / cos2phi)));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay / Float32(Float32(1.0) / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(7.599999731107421e-13))
		tmp = (alphax * alphay) * ((alphax / alphay) * (u0 / cos2phi));
	else
		tmp = (alphay / sin2phi) * (alphay / (single(1.0) / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 7.59999973e-13

   1. Initial program 55.9%

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

      1. sub-neg 55.9%

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

      2. log1p-def 98.3%

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

   3. Simplified 98.3%

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

      1. associate-/r* 98.2%

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

      2. associate-/r* 98.2%

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

      3. frac-add 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. +-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. fma-def 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*l/ 97.8%

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

      6. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

      1. associate-/r/ 98.4%

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

      2. associate-/r/ 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in u0 around 0 74.1%

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

   11. Taylor expanded in alphax around 0 52.6%

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

       1. times-frac 52.5%

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

   13. Simplified 52.5%

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

   if 7.59999973e-13 < sin2phi

   1. Initial program 65.7%

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

      1. sub-neg 65.7%

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

      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. 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* 97.6%

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

      3. frac-add 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. +-commutative 97.1%

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

      2. *-commutative 97.1%

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

      3. fma-def 97.2%

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

      4. *-commutative 97.2%

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

      5. associate-*l/ 97.2%

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

      6. associate-/l* 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in u0 around 0 75.0%

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

   9. Taylor expanded in alphax around inf 70.3%

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

       1. associate-/l* 69.7%

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

   11. Simplified 69.7%

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

       1. unpow2 69.7%

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

       2. div-inv 69.6%

          \[\leadsto \frac{alphay \cdot alphay}{\color{blue}{sin2phi \cdot \frac{1}{u0}}} \]

       3. times-frac 70.3%

          \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}} \]

   13. Applied egg-rr 70.3%

       \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 7.599999731107421 \cdot 10^{-13}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \left(\frac{alphax}{alphay} \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}\\ \end{array} \]

Alternative 9: 66.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 7.599999731107421 \cdot 10^{-13}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0 \cdot alphax}{alphay \cdot cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 7.599999731107421e-13)
   (* (* alphax alphay) (/ (* u0 alphax) (* alphay cos2phi)))
   (* (/ alphay sin2phi) (/ alphay (/ 1.0 u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 7.599999731107421e-13f) {
		tmp = (alphax * alphay) * ((u0 * alphax) / (alphay * cos2phi));
	} else {
		tmp = (alphay / sin2phi) * (alphay / (1.0f / u0));
	}
	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 <= 7.599999731107421e-13) then
        tmp = (alphax * alphay) * ((u0 * alphax) / (alphay * cos2phi))
    else
        tmp = (alphay / sin2phi) * (alphay / (1.0e0 / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(7.599999731107421e-13))
		tmp = Float32(Float32(alphax * alphay) * Float32(Float32(u0 * alphax) / Float32(alphay * cos2phi)));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay / Float32(Float32(1.0) / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(7.599999731107421e-13))
		tmp = (alphax * alphay) * ((u0 * alphax) / (alphay * cos2phi));
	else
		tmp = (alphay / sin2phi) * (alphay / (single(1.0) / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 7.59999973e-13

   1. Initial program 55.9%

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

      1. sub-neg 55.9%

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

      2. log1p-def 98.3%

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

   3. Simplified 98.3%

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

      1. associate-/r* 98.2%

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

      2. associate-/r* 98.2%

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

      3. frac-add 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. +-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. fma-def 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*l/ 97.8%

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

      6. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

      1. associate-/r/ 98.4%

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

      2. associate-/r/ 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in u0 around 0 74.1%

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

   11. Taylor expanded in alphax around 0 52.6%

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

   if 7.59999973e-13 < sin2phi

   1. Initial program 65.7%

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

      1. sub-neg 65.7%

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

      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. 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* 97.6%

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

      3. frac-add 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. +-commutative 97.1%

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

      2. *-commutative 97.1%

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

      3. fma-def 97.2%

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

      4. *-commutative 97.2%

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

      5. associate-*l/ 97.2%

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

      6. associate-/l* 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in u0 around 0 75.0%

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

   9. Taylor expanded in alphax around inf 70.3%

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

       1. associate-/l* 69.7%

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

   11. Simplified 69.7%

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

       1. unpow2 69.7%

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

       2. div-inv 69.6%

          \[\leadsto \frac{alphay \cdot alphay}{\color{blue}{sin2phi \cdot \frac{1}{u0}}} \]

       3. times-frac 70.3%

          \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}} \]

   13. Applied egg-rr 70.3%

       \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 7.599999731107421 \cdot 10^{-13}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0 \cdot alphax}{alphay \cdot cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}\\ \end{array} \]

Alternative 10: 59.6% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.4%

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

   1. sub-neg 62.4%

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

   2. log1p-def 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* 97.8%

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

   2. associate-/r* 97.8%

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

   3. frac-add 97.4%

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

5. Applied egg-rr 97.4%

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

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

   3. fma-def 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*l/ 97.4%

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

   6. associate-/l* 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in u0 around 0 74.7%

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

9. Taylor expanded in alphax around inf 56.2%

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

    1. associate-/l* 55.7%

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

11. Simplified 55.7%

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

    1. unpow2 55.7%

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

    2. div-inv 55.7%

       \[\leadsto \frac{alphay \cdot alphay}{\color{blue}{sin2phi \cdot \frac{1}{u0}}} \]

    3. times-frac 56.2%

       \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}} \]

13. Applied egg-rr 56.2%

    \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}} \]

14. Final simplification 56.2%

    \[\leadsto \frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}} \]

Alternative 11: 59.1% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.4%

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

   1. sub-neg 62.4%

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

   2. log1p-def 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* 97.8%

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

   2. associate-/r* 97.8%

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

   3. frac-add 97.4%

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

5. Applied egg-rr 97.4%

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

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

   3. fma-def 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*l/ 97.4%

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

   6. associate-/l* 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in u0 around 0 74.7%

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

9. Taylor expanded in alphax around inf 56.2%

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

    1. associate-/l* 55.7%

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

11. Simplified 55.7%

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

    1. unpow2 55.7%

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

    2. *-un-lft-identity 55.7%

       \[\leadsto \frac{alphay \cdot alphay}{\color{blue}{1 \cdot \frac{sin2phi}{u0}}} \]

    3. times-frac 55.7%

       \[\leadsto \color{blue}{\frac{alphay}{1} \cdot \frac{alphay}{\frac{sin2phi}{u0}}} \]

13. Applied egg-rr 55.7%

    \[\leadsto \color{blue}{\frac{alphay}{1} \cdot \frac{alphay}{\frac{sin2phi}{u0}}} \]

14. Final simplification 55.7%

    \[\leadsto alphay \cdot \frac{alphay}{\frac{sin2phi}{u0}} \]

Reproduce

herbie shell --seed 2023319 
(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)))))