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

Percentage Accurate: 60.6% → 98.5%

Time: 14.1s

Alternatives: 12

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: 60.6% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.2%

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

   1. neg-sub0 61.2%

      \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   3. --rgt-identity 61.2%

      \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. --rgt-identity 61.2%

      \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. neg-sub0 61.2%

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

   7. sub-neg 61.2%

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

   8. 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. +-commutative 98.4%

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

   2. associate-/r* 98.3%

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

   3. frac-add 98.2%

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

5. Applied egg-rr 98.2%

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

   1. expm1-log1p-u 96.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay} \cdot \left(alphax \cdot alphax\right) + alphay \cdot cos2phi}{alphay \cdot \left(alphax \cdot alphax\right)}}\right)\right)} \]

   2. expm1-udef 50.9%

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

   3. associate-/r/ 50.9%

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

   4. fma-def 50.9%

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

   5. *-commutative 50.9%

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

7. Applied egg-rr 50.9%

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

   1. expm1-def 97.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphay \cdot \left(alphax \cdot alphax\right)\right)\right)\right)} \]

   2. expm1-log1p 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 2: 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 61.2%

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

   1. neg-sub0 61.2%

      \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   3. --rgt-identity 61.2%

      \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. --rgt-identity 61.2%

      \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. neg-sub0 61.2%

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

   7. sub-neg 61.2%

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

   8. 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. Final simplification 98.4%

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

Alternative 3: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;sin2phi \leq 0.05000000074505806:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t_0 + \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= sin2phi 0.05000000074505806)
     (/
      (- u0 (* u0 (* u0 -0.5)))
      (+ t_0 (/ 1.0 (/ (* alphax alphax) cos2phi))))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (sin2phi <= 0.05000000074505806f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (t_0 + (1.0f / ((alphax * alphax) / cos2phi)));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.05000000074505806))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(Float32(1.0) / Float32(Float32(alphax * alphax) / cos2phi))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0500000007

   1. Initial program 54.5%

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

      1. neg-sub0 54.5%

         \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. div-sub 54.5%

         \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      3. --rgt-identity 54.5%

         \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. div-sub 54.5%

         \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      5. --rgt-identity 54.5%

         \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg 54.5%

         \[\leadsto \frac{0 - \log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. +-commutative 54.5%

         \[\leadsto \frac{0 - \log \color{blue}{\left(\left(-u0\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. neg-sub0 54.5%

         \[\leadsto \frac{0 - \log \left(\color{blue}{\left(0 - u0\right)} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. associate-+l- 54.5%

         \[\leadsto \frac{0 - \log \color{blue}{\left(0 - \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. sub0-neg 54.5%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-\left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. neg-mul-1 54.5%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-1 \cdot \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. log-prod -0.0%

          \[\leadsto \frac{0 - \color{blue}{\left(\log -1 + \log \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. associate--r+ -0.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \log -1\right) - \log \left(u0 - 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.6%

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

      1. associate-/r* 98.7%

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

      2. frac-2neg 98.7%

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

      3. div-inv 98.6%

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

      4. distribute-rgt-neg-in 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. un-div-inv 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

      3. frac-2neg 98.7%

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

      4. clear-num 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in u0 around 0 86.7%

      \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}} + \frac{sin2phi}{alphay \cdot alphay}} \]
   9. Step-by-step derivation

      1. +-commutative 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. *-commutative 86.7%

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

      5. unpow2 86.7%

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

      6. associate-*l* 86.7%

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

   10. Simplified 86.7%

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

   if 0.0500000007 < sin2phi

   1. Initial program 67.2%

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

      1. neg-sub0 67.2%

         \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. div-sub 67.2%

         \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      3. --rgt-identity 67.2%

         \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. div-sub 67.2%

         \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      5. --rgt-identity 67.2%

         \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg 67.2%

         \[\leadsto \frac{0 - \log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. +-commutative 67.2%

         \[\leadsto \frac{0 - \log \color{blue}{\left(\left(-u0\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. neg-sub0 67.2%

         \[\leadsto \frac{0 - \log \left(\color{blue}{\left(0 - u0\right)} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. associate-+l- 67.2%

         \[\leadsto \frac{0 - \log \color{blue}{\left(0 - \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. sub0-neg 67.2%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-\left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. neg-mul-1 67.2%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-1 \cdot \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. log-prod -0.0%

          \[\leadsto \frac{0 - \color{blue}{\left(\log -1 + \log \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. associate--r+ -0.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \log -1\right) - \log \left(u0 - 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.1%

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

      1. div-inv 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in cos2phi around 0 98.0%

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

      1. unpow2 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.05000000074505806:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 4: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.20000000298023224:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.20000000298023224)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ 1.0 (/ (* alphax alphax) cos2phi))))
   (/ (* (log1p (- u0)) (* alphay (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.20000000298023224f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + (1.0f / ((alphax * alphax) / cos2phi)));
	} else {
		tmp = (log1pf(-u0) * (alphay * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.20000000298023224))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(1.0) / Float32(Float32(alphax * alphax) / cos2phi))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.200000003

   1. Initial program 54.6%

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

      1. neg-sub0 54.6%

         \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. div-sub 54.6%

         \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      3. --rgt-identity 54.6%

         \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. div-sub 54.6%

         \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      5. --rgt-identity 54.6%

         \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg 54.6%

         \[\leadsto \frac{0 - \log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. +-commutative 54.6%

         \[\leadsto \frac{0 - \log \color{blue}{\left(\left(-u0\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. neg-sub0 54.6%

         \[\leadsto \frac{0 - \log \left(\color{blue}{\left(0 - u0\right)} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. associate-+l- 54.6%

         \[\leadsto \frac{0 - \log \color{blue}{\left(0 - \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. sub0-neg 54.6%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-\left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. neg-mul-1 54.6%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-1 \cdot \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. log-prod -0.0%

          \[\leadsto \frac{0 - \color{blue}{\left(\log -1 + \log \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. associate--r+ -0.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \log -1\right) - \log \left(u0 - 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.7%

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

      1. associate-/r* 98.7%

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

      2. frac-2neg 98.7%

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

      3. div-inv 98.6%

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

      4. distribute-rgt-neg-in 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. un-div-inv 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

      3. frac-2neg 98.7%

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

      4. clear-num 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in u0 around 0 86.8%

      \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}} + \frac{sin2phi}{alphay \cdot alphay}} \]
   9. Step-by-step derivation

      1. +-commutative 86.8%

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

      2. mul-1-neg 86.8%

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

      3. unsub-neg 86.8%

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

      4. *-commutative 86.8%

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

      5. unpow2 86.8%

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

      6. associate-*l* 86.8%

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

   10. Simplified 86.8%

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

   if 0.200000003 < sin2phi

   1. Initial program 67.3%

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

      1. neg-sub0 67.3%

         \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. div-sub 67.3%

         \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      3. --rgt-identity 67.3%

         \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. div-sub 67.3%

         \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      5. --rgt-identity 67.3%

         \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. sub-neg 67.3%

         \[\leadsto \frac{0 - \log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. +-commutative 67.3%

         \[\leadsto \frac{0 - \log \color{blue}{\left(\left(-u0\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. neg-sub0 67.3%

         \[\leadsto \frac{0 - \log \left(\color{blue}{\left(0 - u0\right)} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. associate-+l- 67.3%

         \[\leadsto \frac{0 - \log \color{blue}{\left(0 - \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. sub0-neg 67.3%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-\left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. neg-mul-1 67.3%

          \[\leadsto \frac{0 - \log \color{blue}{\left(-1 \cdot \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. log-prod -0.0%

          \[\leadsto \frac{0 - \color{blue}{\left(\log -1 + \log \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. associate--r+ -0.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \log -1\right) - \log \left(u0 - 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.1%

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

      1. associate-/r* 98.1%

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

      2. frac-2neg 98.1%

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

      3. div-inv 98.1%

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

      4. distribute-rgt-neg-in 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in cos2phi around 0 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

      3. unpow2 68.1%

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

      4. *-commutative 68.1%

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

      5. sub-neg 68.1%

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

      6. log1p-def 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.20000000298023224:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 5: 87.4% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.2%

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

   1. neg-sub0 61.2%

      \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   3. --rgt-identity 61.2%

      \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. --rgt-identity 61.2%

      \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. sub-neg 61.2%

      \[\leadsto \frac{0 - \log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. +-commutative 61.2%

      \[\leadsto \frac{0 - \log \color{blue}{\left(\left(-u0\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. neg-sub0 61.2%

      \[\leadsto \frac{0 - \log \left(\color{blue}{\left(0 - u0\right)} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. associate-+l- 61.2%

      \[\leadsto \frac{0 - \log \color{blue}{\left(0 - \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. sub0-neg 61.2%

       \[\leadsto \frac{0 - \log \color{blue}{\left(-\left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. neg-mul-1 61.2%

       \[\leadsto \frac{0 - \log \color{blue}{\left(-1 \cdot \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   12. log-prod -0.0%

       \[\leadsto \frac{0 - \color{blue}{\left(\log -1 + \log \left(u0 - 1\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   13. associate--r+ -0.0%

       \[\leadsto \frac{\color{blue}{\left(0 - \log -1\right) - \log \left(u0 - 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.4%

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

   1. associate-/r* 98.4%

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

   2. frac-2neg 98.4%

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

   3. div-inv 98.3%

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

   4. distribute-rgt-neg-in 98.3%

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

5. Applied egg-rr 98.3%

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

   1. un-div-inv 98.4%

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

   2. distribute-rgt-neg-out 98.4%

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

   3. frac-2neg 98.4%

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

   4. clear-num 98.3%

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in u0 around 0 87.1%

   \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. +-commutative 87.1%

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

   2. mul-1-neg 87.1%

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

   3. unsub-neg 87.1%

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

   4. *-commutative 87.1%

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

   5. unpow2 87.1%

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

   6. associate-*l* 87.1%

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

10. Simplified 87.1%

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

11. Final simplification 87.1%

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

Alternative 6: 76.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.2%

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

   1. neg-sub0 61.2%

      \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   3. --rgt-identity 61.2%

      \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. div-sub 61.2%

      \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. --rgt-identity 61.2%

      \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. neg-sub0 61.2%

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

   7. sub-neg 61.2%

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

   8. 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. +-commutative 98.4%

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

   2. associate-/r* 98.3%

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

   3. frac-add 98.2%

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

5. Applied egg-rr 98.2%

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

6. Taylor expanded in u0 around 0 75.4%

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

   1. unpow2 75.4%

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

   2. unpow2 75.4%

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

   3. associate-*l/ 75.3%

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

   4. fma-udef 75.3%

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

8. Simplified 75.3%

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

   1. fma-udef 75.3%

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

   2. *-commutative 75.3%

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

10. Applied egg-rr 75.3%

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

11. Final simplification 75.3%

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

Alternative 7: 67.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.040000178611157 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 6.040000178611157e-14)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 6.040000178611157e-14f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 6.040000178611157e-14) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(6.040000178611157e-14))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(6.040000178611157e-14))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 6.04000018e-14

   1. Initial program 51.8%

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

      1. associate-/r* 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in u0 around 0 76.8%

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

      1. unpow2 76.8%

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

      2. unpow2 76.8%

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

   6. Simplified 76.8%

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

   7. Taylor expanded in cos2phi around inf 59.0%

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

      1. unpow2 59.0%

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

      2. associate-/l* 59.1%

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

      3. associate-/r/ 59.1%

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

   9. Simplified 59.1%

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

   if 6.04000018e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.7%

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

      1. neg-sub0 64.7%

         \[\leadsto \frac{\color{blue}{0 - \log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. div-sub 64.7%

         \[\leadsto \color{blue}{\frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      3. --rgt-identity 64.7%

         \[\leadsto \frac{0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{\color{blue}{\log \left(1 - u0\right) - 0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. div-sub 64.7%

         \[\leadsto \color{blue}{\frac{0 - \left(\log \left(1 - u0\right) - 0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      5. --rgt-identity 64.7%

         \[\leadsto \frac{0 - \color{blue}{\log \left(1 - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. neg-sub0 64.7%

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

      7. sub-neg 64.7%

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

      8. 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. +-commutative 98.3%

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

      2. associate-/r* 98.2%

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

      3. frac-add 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in u0 around 0 74.9%

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

      1. unpow2 74.9%

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

      2. unpow2 74.9%

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

      3. associate-*l/ 74.8%

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

      4. fma-udef 74.8%

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

   8. Simplified 74.8%

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

   9. Taylor expanded in sin2phi around inf 71.1%

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

       1. unpow2 71.1%

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

       2. associate-*l/ 71.0%

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

   11. Simplified 71.0%

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

   12. Taylor expanded in u0 around 0 71.1%

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

       1. associate-/l* 70.7%

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

       2. associate-/r/ 71.1%

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

       3. unpow2 71.1%

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

   14. Simplified 71.1%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.040000178611157 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 8: 76.0% 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 61.2%

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

   1. associate-/r* 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in u0 around 0 75.1%

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

   1. unpow2 75.1%

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

   2. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Final simplification 75.1%

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

Alternative 9: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.2%

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

   1. associate-/r* 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in u0 around 0 75.1%

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

   1. mul-1-neg 75.1%

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

6. Simplified 75.1%

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

7. Final simplification 75.1%

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

Alternative 10: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.2%

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

   1. associate-/r* 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in u0 around 0 75.1%

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

   1. unpow2 75.1%

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

   2. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Taylor expanded in cos2phi around inf 23.8%

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

   1. unpow2 23.8%

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

   2. associate-/l* 23.9%

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

9. Simplified 23.9%

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

    1. div-inv 23.8%

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

    2. clear-num 23.8%

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

    3. associate-/l* 23.8%

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

11. Applied egg-rr 23.8%

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

12. Final simplification 23.8%

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

Alternative 11: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.2%

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

   1. associate-/r* 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in u0 around 0 75.1%

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

   1. unpow2 75.1%

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

   2. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Taylor expanded in cos2phi around inf 23.8%

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

   1. unpow2 23.8%

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

   2. associate-/l* 23.9%

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

9. Simplified 23.9%

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

10. Taylor expanded in u0 around 0 23.8%

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

    1. associate-/l* 23.9%

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

    2. unpow2 23.9%

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

    3. associate-/r* 23.8%

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

    4. associate-/l* 23.9%

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

    5. associate-*r/ 23.8%

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

    6. associate-/l* 23.8%

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

12. Simplified 23.8%

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

13. Final simplification 23.8%

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

Alternative 12: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.2%

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

   1. associate-/r* 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in u0 around 0 75.1%

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

   1. unpow2 75.1%

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

   2. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Taylor expanded in cos2phi around inf 23.8%

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

   1. unpow2 23.8%

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

   2. associate-/l* 23.9%

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

   3. associate-/r/ 23.9%

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

9. Simplified 23.9%

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

10. Final simplification 23.9%

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

Reproduce

herbie shell --seed 2023255 
(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)))))