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

Percentage Accurate: 61.2% → 98.4%

Time: 19.0s

Alternatives: 13

Speedup: 12.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

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(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 57.4%

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

   1. sub-neg 57.4%

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

   2. log1p-def 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. div-inv 98.3%

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

5. Applied egg-rr 98.3%

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

   1. div-inv 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. sub-neg 57.4%

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

   2. log1p-def 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%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. sub-neg 57.4%

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

   2. log1p-def 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. div-inv 98.3%

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

5. Applied egg-rr 98.3%

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

   1. div-inv 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in cos2phi around 0 98.4%

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

   1. unpow2 98.4%

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

   2. unpow2 98.4%

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

   3. associate-/r* 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 4: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;sin2phi \leq 0.029999999329447746:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= sin2phi 0.029999999329447746)
     (/ (- u0 (* u0 (* u0 -0.5))) (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (- (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 <= 0.029999999329447746f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (t_0 + (cos2phi / (alphax * alphax)));
	} 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(0.029999999329447746))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0299999993

   1. Initial program 52.1%

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

   2. Taylor expanded in u0 around 0 86.3%

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

      1. +-commutative 86.3%

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

      2. mul-1-neg 86.3%

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

      3. unsub-neg 86.3%

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

      4. *-commutative 86.3%

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

      5. unpow2 86.3%

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

      6. associate-*l* 86.3%

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

   4. Simplified 86.3%

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

   if 0.0299999993 < sin2phi

   1. Initial program 63.0%

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

      1. sub-neg 63.0%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

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

      1. *-un-lft-identity 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-/r* 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in cos2phi around 0 98.2%

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

      1. unpow2 98.2%

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

   8. Simplified 98.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.029999999329447746:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 5: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.029999999329447746:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.029999999329447746)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
   (/ (- (log1p (- u0))) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.029999999329447746f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -log1pf(-u0) / ((sin2phi / alphay) / alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.029999999329447746))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(sin2phi / alphay) / alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0299999993

   1. Initial program 52.1%

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

   2. Taylor expanded in u0 around 0 86.3%

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

      1. +-commutative 86.3%

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

      2. mul-1-neg 86.3%

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

      3. unsub-neg 86.3%

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

      4. *-commutative 86.3%

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

      5. unpow2 86.3%

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

      6. associate-*l* 86.3%

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

   4. Simplified 86.3%

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

   if 0.0299999993 < sin2phi

   1. Initial program 63.0%

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

      1. sub-neg 63.0%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   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. div-inv 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. div-inv 98.2%

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in cos2phi around 0 98.2%

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

      1. unpow2 98.2%

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

      2. associate-/r* 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.029999999329447746:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

Alternative 6: 87.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 - (u0 * (u0 * -0.5f))) / ((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 - (u0 * (u0 * (-0.5e0)))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
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(cos2phi / Float32(alphax * alphax))))
end

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u0 around 0 87.0%

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

   1. +-commutative 87.0%

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

   2. mul-1-neg 87.0%

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

   3. unsub-neg 87.0%

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

   4. *-commutative 87.0%

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

   5. unpow2 87.0%

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

   6. associate-*l* 87.0%

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

4. Simplified 87.0%

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

5. Final simplification 87.0%

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

Alternative 7: 75.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 * (1.0f / ((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 * (1.0e0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u0 around 0 76.3%

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

   1. +-commutative 76.3%

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

   2. unpow2 76.3%

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

   3. unpow2 76.3%

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

4. Simplified 76.3%

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

   1. div-inv 76.3%

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

   2. associate-/r* 76.3%

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

   3. +-commutative 76.3%

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

   4. *-rgt-identity 76.3%

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

   5. *-rgt-identity 76.3%

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

   6. div-inv 76.3%

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

   7. div-inv 76.3%

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

   8. associate-/r* 76.3%

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

6. Applied egg-rr 76.3%

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

7. Final simplification 76.3%

   \[\leadsto u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

Alternative 8: 75.5% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. sub-neg 57.4%

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

   2. log1p-def 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. div-inv 98.3%

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

5. Applied egg-rr 98.3%

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

   1. div-inv 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in u0 around 0 76.3%

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

   1. unpow2 76.3%

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

   2. unpow2 76.3%

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

   3. associate-/r* 76.2%

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

10. Simplified 76.2%

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

11. Final simplification 76.2%

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

Alternative 9: 75.5% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u0 around 0 76.3%

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

   1. +-commutative 76.3%

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

   2. unpow2 76.3%

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

   3. unpow2 76.3%

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

4. Simplified 76.3%

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

5. Final simplification 76.3%

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

Alternative 10: 65.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.0000000781659255e-25)
   (* u0 (* alphax (/ alphax cos2phi)))
   (* u0 (* alphay (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.0000000781659255e-25f) {
		tmp = u0 * (alphax * (alphax / cos2phi));
	} else {
		tmp = u0 * (alphay * (alphay / 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 <= 4.0000000781659255e-25) then
        tmp = u0 * (alphax * (alphax / cos2phi))
    else
        tmp = u0 * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.0000000781659255e-25))
		tmp = Float32(u0 * Float32(alphax * Float32(alphax / cos2phi)));
	else
		tmp = Float32(u0 * Float32(alphay * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.0000000781659255e-25))
		tmp = u0 * (alphax * (alphax / cos2phi));
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.00000008e-25

   1. Initial program 50.9%

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

   2. Taylor expanded in u0 around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. unpow2 74.6%

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

      3. unpow2 74.6%

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

   4. Simplified 74.6%

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

   5. Taylor expanded in sin2phi around 0 64.1%

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

      1. *-lft-identity 64.1%

         \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

      2. times-frac 64.1%

         \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

      3. /-rgt-identity 64.1%

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

      4. unpow2 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in alphax around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. unpow2 64.1%

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

      3. associate-*l* 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in alphax around 0 64.1%

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

       1. unpow2 64.1%

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

       2. associate-*l/ 64.2%

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

       3. associate-*r/ 64.2%

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

   13. Simplified 64.2%

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

   if 4.00000008e-25 < sin2phi

   1. Initial program 58.9%

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

   2. Taylor expanded in u0 around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. unpow2 76.7%

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

      3. unpow2 76.7%

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

   4. Simplified 76.7%

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

      1. frac-2neg 76.7%

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

      2. div-inv 76.7%

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

      3. distribute-rgt-neg-in 76.7%

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

   6. Applied egg-rr 76.7%

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

   7. Taylor expanded in sin2phi around inf 69.1%

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

      1. unpow2 69.1%

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

      2. associate-/l* 68.5%

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

      3. associate-/r/ 69.1%

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

      4. associate-/l* 69.0%

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

      5. associate-/r/ 69.1%

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

   9. Simplified 69.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 11: 23.8% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u0 around 0 76.3%

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

   1. +-commutative 76.3%

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

   2. unpow2 76.3%

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

   3. unpow2 76.3%

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

4. Simplified 76.3%

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

5. Taylor expanded in sin2phi around 0 23.5%

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

   1. *-lft-identity 23.5%

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

   2. times-frac 23.5%

      \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

   3. /-rgt-identity 23.5%

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

   4. unpow2 23.5%

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

7. Simplified 23.5%

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

8. Taylor expanded in alphax around 0 23.5%

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

   1. associate-*r/ 23.5%

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

   2. unpow2 23.5%

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

   3. associate-*l* 23.5%

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

10. Simplified 23.5%

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

11. Final simplification 23.5%

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

Alternative 12: 23.8% 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 57.4%

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

2. Taylor expanded in u0 around 0 76.3%

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

   1. +-commutative 76.3%

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

   2. unpow2 76.3%

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

   3. unpow2 76.3%

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

4. Simplified 76.3%

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

5. Taylor expanded in sin2phi around 0 23.5%

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

   1. *-lft-identity 23.5%

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

   2. times-frac 23.5%

      \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

   3. /-rgt-identity 23.5%

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

   4. unpow2 23.5%

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

7. Simplified 23.5%

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

8. Final simplification 23.5%

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

Alternative 13: 23.9% accurate, 16.6× speedup

Math

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

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(alphax * Float32(alphax / cos2phi)))
end

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u0 around 0 76.3%

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

   1. +-commutative 76.3%

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

   2. unpow2 76.3%

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

   3. unpow2 76.3%

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

4. Simplified 76.3%

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

5. Taylor expanded in sin2phi around 0 23.5%

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

   1. *-lft-identity 23.5%

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

   2. times-frac 23.5%

      \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

   3. /-rgt-identity 23.5%

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

   4. unpow2 23.5%

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

7. Simplified 23.5%

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

8. Taylor expanded in alphax around 0 23.5%

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

   1. associate-*r/ 23.5%

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

   2. unpow2 23.5%

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

   3. associate-*l* 23.5%

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

10. Simplified 23.5%

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

11. Taylor expanded in alphax around 0 23.5%

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

    1. unpow2 23.5%

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

    2. associate-*l/ 23.5%

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

    3. associate-*r/ 23.6%

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

13. Simplified 23.6%

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

14. Final simplification 23.6%

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

Reproduce

herbie shell --seed 2023279 
(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)))))