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

Percentage Accurate: 60.4% → 98.6%

Time: 10.3s

Alternatives: 30

Speedup: 1.4×

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

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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)))))

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)
use fmin_fmax_functions
    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: 60.4% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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)))))

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)
use fmin_fmax_functions
    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: 98.6% accurate, 0.9× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.4%

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

3. Applied rewrites 60.8%

   \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
4. Step-by-step derivation

5. Applied rewrites 60.8%

   \[\leadsto \frac{alphay \cdot \left(-\log \left(1 - u0\right)\right)}{\mathsf{fma}\left(alphay, \frac{cos2phi}{alphax \cdot alphax}, \frac{sin2phi}{alphay}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

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

Alternative 2: 98.6% accurate, 0.9× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.4%

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

3. Applied rewrites 60.8%

   \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
4. Step-by-step derivation

5. Applied rewrites 60.8%

   \[\leadsto \frac{alphay \cdot \left(-\log \left(1 - u0\right)\right)}{\mathsf{fma}\left(alphay, \frac{cos2phi}{alphax \cdot alphax}, \frac{sin2phi}{alphay}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 98.6%

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

Alternative 3: 98.2% accurate, 0.9× speedup

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

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (/
 (- (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 60.4%

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

3. Applied rewrites 98.2%

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

5. Applied rewrites 98.2%

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

Alternative 4: 98.2% accurate, 0.9× speedup

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

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (/
 (- (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 60.4%

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

3. Applied rewrites 98.2%

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

Alternative 5: 96.6% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9975000023841858:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\mathsf{fma}\left(alphay, t\_0, \frac{sin2phi}{alphay}\right)} \cdot alphay\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{t\_0 + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ cos2phi (* alphax alphax))))
  (if (<= (- 1.0 u0) 0.9975000023841858)
    (*
     (/ (- (log (- 1.0 u0))) (fma alphay t_0 (/ sin2phi alphay)))
     alphay)
    (/ (fma u0 (* 0.5 u0) u0) (+ t_0 (/ (/ sin2phi alphay) alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float tmp;
	if ((1.0f - u0) <= 0.9975000023841858f) {
		tmp = (-logf((1.0f - u0)) / fmaf(alphay, t_0, (sin2phi / alphay))) * alphay;
	} else {
		tmp = fmaf(u0, (0.5f * u0), u0) / (t_0 + ((sin2phi / alphay) / alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9975000023841858))
		tmp = Float32(Float32(Float32(-log(Float32(Float32(1.0) - u0))) / fma(alphay, t_0, Float32(sin2phi / alphay))) * alphay);
	else
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(t_0 + Float32(Float32(sin2phi / alphay) / alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997500002

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   if 0.997500002 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 87.1%

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

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.6% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} t_0 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{alphay}{\mathsf{fma}\left(cos2phi, \frac{alphay}{alphax \cdot alphax}, \frac{sin2phi}{alphay}\right)}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (- (log (- 1.0 u0)))))
  (if (<= t_0 0.0024999999441206455)
    (/
     (fma u0 (* 0.5 u0) u0)
     (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
    (*
     t_0
     (/
      alphay
      (fma
       cos2phi
       (/ alphay (* alphax alphax))
       (/ sin2phi alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -logf((1.0f - u0));
	float tmp;
	if (t_0 <= 0.0024999999441206455f) {
		tmp = fmaf(u0, (0.5f * u0), u0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = t_0 * (alphay / fmaf(cos2phi, (alphay / (alphax * alphax)), (sin2phi / alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0024999999441206455))
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(t_0 * Float32(alphay / fma(cos2phi, Float32(alphay / Float32(alphax * alphax)), Float32(sin2phi / alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00249999994

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 87.1%

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

   8. Applied rewrites 87.1%

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

   if 0.00249999994 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

Alternative 7: 96.2% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} t_0 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (- (log (- 1.0 u0)))))
  (if (<= t_0 0.0024999999441206455)
    (/
     (fma u0 (* 0.5 u0) u0)
     (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
    (/
     t_0
     (+
      (/ (/ cos2phi alphax) alphax)
      (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -logf((1.0f - u0));
	float tmp;
	if (t_0 <= 0.0024999999441206455f) {
		tmp = fmaf(u0, (0.5f * u0), u0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = t_0 / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0024999999441206455))
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(t_0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00249999994

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 87.1%

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

   8. Applied rewrites 87.1%

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

   if 0.00249999994 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

Alternative 8: 96.2% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9975000023841858:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{t\_0 + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ cos2phi (* alphax alphax))))
  (if (<= (- 1.0 u0) 0.9975000023841858)
    (/ (- (log (- 1.0 u0))) (+ t_0 (/ sin2phi (* alphay alphay))))
    (/ (fma u0 (* 0.5 u0) u0) (+ t_0 (/ (/ sin2phi alphay) alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float tmp;
	if ((1.0f - u0) <= 0.9975000023841858f) {
		tmp = -logf((1.0f - u0)) / (t_0 + (sin2phi / (alphay * alphay)));
	} else {
		tmp = fmaf(u0, (0.5f * u0), u0) / (t_0 + ((sin2phi / alphay) / alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9975000023841858))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(t_0 + Float32(Float32(sin2phi / alphay) / alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997500002

   1. Initial program 60.4%

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

   if 0.997500002 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 87.1%

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

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 92.6% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 120000:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 120000.0)
  (/
   (fma u0 (* 0.5 u0) u0)
   (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
  (* (* (/ (- (log1p (- u0))) sin2phi) alphay) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 120000.0f) {
		tmp = fmaf(u0, (0.5f * u0), u0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = ((-log1pf(-u0) / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(120000.0))
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e5

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 87.1%

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

   8. Applied rewrites 87.1%

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

   if 1.2e5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 10: 92.6% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 120000:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ sin2phi (* alphay alphay))))
  (if (<= t_0 120000.0)
    (/ (fma u0 (* 0.5 u0) u0) (+ (/ cos2phi (* alphax alphax)) t_0))
    (* (* (/ (- (log1p (- u0))) sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 120000.0f) {
		tmp = fmaf(u0, (0.5f * u0), u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((-log1pf(-u0) / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(120000.0))
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e5

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 87.1%

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

   if 1.2e5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 11: 92.6% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 120000:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ sin2phi (* alphay alphay))))
  (if (<= t_0 120000.0)
    (/ (* (fma 0.5 u0 1.0) u0) (+ t_0 (/ cos2phi (* alphax alphax))))
    (* (* (/ (- (log1p (- u0))) sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 120000.0f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((-log1pf(-u0) / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(120000.0))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e5

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 87.0%

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

   if 1.2e5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 12: 92.5% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 120000:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ sin2phi (* alphay alphay))))
  (if (<= t_0 120000.0)
    (* (/ (fma 0.5 u0 1.0) (+ t_0 (/ cos2phi (* alphax alphax)))) u0)
    (* (* (/ (- (log1p (- u0))) sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 120000.0f) {
		tmp = (fmaf(0.5f, u0, 1.0f) / (t_0 + (cos2phi / (alphax * alphax)))) * u0;
	} else {
		tmp = ((-log1pf(-u0) / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(120000.0))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax)))) * u0);
	else
		tmp = Float32(Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e5

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.0%

      \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

   6. Applied rewrites 87.0%

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

   if 1.2e5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 13: 87.4% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} t_0 := -\mathsf{log1p}\left(-u0\right)\\ \mathbf{if}\;sin2phi \leq 1.825378024189646 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_0}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (- (log1p (- u0)))))
  (if (<= sin2phi 1.825378024189646e-15)
    (/ t_0 (* cos2phi (/ 1.0 (* alphax alphax))))
    (* (* (/ t_0 sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -log1pf(-u0);
	float tmp;
	if (sin2phi <= 1.825378024189646e-15f) {
		tmp = t_0 / (cos2phi * (1.0f / (alphax * alphax)));
	} else {
		tmp = ((t_0 / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-log1p(Float32(-u0)))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.825378024189646e-15))
		tmp = Float32(t_0 / Float32(cos2phi * Float32(Float32(1.0) / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(t_0 / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.82537802e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   8. Applied rewrites 27.7%

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

   if 1.82537802e-15 < sin2phi

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 14: 87.4% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := -\mathsf{log1p}\left(-u0\right)\\ \mathbf{if}\;sin2phi \leq 1.825378024189646 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_0}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (- (log1p (- u0)))))
  (if (<= sin2phi 1.825378024189646e-15)
    (/ t_0 (/ cos2phi (* alphax alphax)))
    (* (* (/ t_0 sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -log1pf(-u0);
	float tmp;
	if (sin2phi <= 1.825378024189646e-15f) {
		tmp = t_0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = ((t_0 / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-log1p(Float32(-u0)))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.825378024189646e-15))
		tmp = Float32(t_0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(Float32(t_0 / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.82537802e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   8. Applied rewrites 21.6%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]
   9. Step-by-step derivation

   10. Applied rewrites 27.7%

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

   if 1.82537802e-15 < sin2phi

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 15: 83.6% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := -\mathsf{log1p}\left(-u0\right)\\ \mathbf{if}\;sin2phi \leq 1.825378024189646 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{cos2phi} \cdot \left(alphax \cdot alphax\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_0}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (- (log1p (- u0)))))
  (if (<= sin2phi 1.825378024189646e-15)
    (* (/ t_0 cos2phi) (* alphax alphax))
    (* (* (/ t_0 sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -log1pf(-u0);
	float tmp;
	if (sin2phi <= 1.825378024189646e-15f) {
		tmp = (t_0 / cos2phi) * (alphax * alphax);
	} else {
		tmp = ((t_0 / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-log1p(Float32(-u0)))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.825378024189646e-15))
		tmp = Float32(Float32(t_0 / cos2phi) * Float32(alphax * alphax));
	else
		tmp = Float32(Float32(Float32(t_0 / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.82537802e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 21.6%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi} \cdot \left(alphax \cdot alphax\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 27.7%

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

   if 1.82537802e-15 < sin2phi

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 16: 83.6% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ sin2phi (* alphay alphay))))
  (if (<= t_0 1.0000000116860974e-7)
    (/ (- (- u0)) (+ (/ cos2phi (* alphax alphax)) t_0))
    (* (* (/ (- (log1p (- u0))) sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.0000000116860974e-7f) {
		tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((-log1pf(-u0) / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0000000116860974e-7))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-7

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.7%

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

   if 1.00000001e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 17: 83.6% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ sin2phi (* alphay alphay))))
  (if (<= t_0 1.0000000116860974e-7)
    (/ (- (- u0)) (+ (/ cos2phi (* alphax alphax)) t_0))
    (* (/ (- (log1p (- u0))) sin2phi) (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.0000000116860974e-7f) {
		tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (-log1pf(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0000000116860974e-7))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-7

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.7%

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

   if 1.00000001e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 74.5%

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

Alternative 18: 83.2% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9974200129508972:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (- 1.0 u0) 0.9974200129508972)
  (* (/ (- (log (- 1.0 u0))) sin2phi) (* alphay alphay))
  (/
   (- (- u0))
   (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9974200129508972f) {
		tmp = (-logf((1.0f - u0)) / sin2phi) * (alphay * alphay);
	} else {
		tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    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 ((1.0e0 - u0) <= 0.9974200129508972e0) then
        tmp = (-log((1.0e0 - u0)) / sin2phi) * (alphay * alphay)
    else
        tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9974200129508972))
		tmp = Float32(Float32(Float32(-log(Float32(Float32(1.0) - u0))) / sin2phi) * Float32(alphay * alphay));
	else
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9974200129508972))
		tmp = (-log((single(1.0) - u0)) / sin2phi) * (alphay * alphay);
	else
		tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997420013

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 49.1%

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

   if 0.997420013 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.7%

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

Alternative 19: 75.4% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax}} \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, u0, \left(0.5 \cdot u0\right) \cdot u0\right) \cdot \frac{alphay}{\frac{sin2phi}{alphay}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
  (* (/ (* (fma 0.5 u0 1.0) u0) (/ cos2phi alphax)) alphax)
  (* (fma 1.0 u0 (* (* 0.5 u0) u0)) (/ alphay (/ sin2phi alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-15f) {
		tmp = ((fmaf(0.5f, u0, 1.0f) * u0) / (cos2phi / alphax)) * alphax;
	} else {
		tmp = fmaf(1.0f, u0, ((0.5f * u0) * u0)) * (alphay / (sin2phi / alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.9999998245167e-15))
		tmp = Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(cos2phi / alphax)) * alphax);
	else
		tmp = Float32(fma(Float32(1.0), u0, Float32(Float32(Float32(0.5) * u0) * u0)) * Float32(alphay / Float32(sin2phi / alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999982e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 25.6%

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

   11. Applied rewrites 25.6%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax}} \cdot alphax \]

   if 9.99999982e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.0%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.7%

      \[\leadsto \left(u0 \cdot \left(1 + 0.5 \cdot u0\right)\right) \cdot \frac{alphay}{\frac{sin2phi}{alphay}} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.8%

       \[\leadsto \mathsf{fma}\left(1, u0, \left(0.5 \cdot u0\right) \cdot u0\right) \cdot \frac{alphay}{\frac{sin2phi}{alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 75.4% accurate, 1.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.825378024189646 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax}} \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{alphay \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{sin2phi}\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.825378024189646e-15)
  (* (/ (* (fma 0.5 u0 1.0) u0) (/ cos2phi alphax)) alphax)
  (*
   (* -1.0 (/ (* alphay (* u0 (- (* -0.5 u0) 1.0))) sin2phi))
   alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.825378024189646e-15f) {
		tmp = ((fmaf(0.5f, u0, 1.0f) * u0) / (cos2phi / alphax)) * alphax;
	} else {
		tmp = (-1.0f * ((alphay * (u0 * ((-0.5f * u0) - 1.0f))) / sin2phi)) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.825378024189646e-15))
		tmp = Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(cos2phi / alphax)) * alphax);
	else
		tmp = Float32(Float32(Float32(-1.0) * Float32(Float32(alphay * Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))) / sin2phi)) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.82537802e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 25.6%

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

   11. Applied rewrites 25.6%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax}} \cdot alphax \]

   if 1.82537802e-15 < sin2phi

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

      \[\leadsto \left(-1 \cdot \frac{alphay \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}{sin2phi}\right) \cdot alphay \]
   10. Step-by-step derivation

   11. Applied rewrites 66.8%

       \[\leadsto \left(-1 \cdot \frac{alphay \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{sin2phi}\right) \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 74.7% accurate, 1.3× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.825378024189646 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax}} \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{sin2phi} \cdot u0\right) \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.825378024189646e-15)
  (* (/ (* (fma 0.5 u0 1.0) u0) (/ cos2phi alphax)) alphax)
  (* (* (* (/ (fma 0.5 u0 1.0) sin2phi) u0) alphay) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.825378024189646e-15f) {
		tmp = ((fmaf(0.5f, u0, 1.0f) * u0) / (cos2phi / alphax)) * alphax;
	} else {
		tmp = (((fmaf(0.5f, u0, 1.0f) / sin2phi) * u0) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.825378024189646e-15))
		tmp = Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(cos2phi / alphax)) * alphax);
	else
		tmp = Float32(Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) / sin2phi) * u0) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.82537802e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 25.6%

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

   11. Applied rewrites 25.6%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax}} \cdot alphax \]

   if 1.82537802e-15 < sin2phi

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.8%

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{sin2phi}, \frac{1}{sin2phi}\right)\right) \cdot \left(alphay \cdot alphay\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.8%

       \[\leadsto \left(\left(\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{sin2phi} \cdot u0\right) \cdot alphay\right) \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 74.7% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.825378024189646 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{sin2phi} \cdot u0\right) \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.825378024189646e-15)
  (* (* (fma 0.5 u0 1.0) u0) (/ (* alphax alphax) cos2phi))
  (* (* (* (/ (fma 0.5 u0 1.0) sin2phi) u0) alphay) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.825378024189646e-15f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) * ((alphax * alphax) / cos2phi);
	} else {
		tmp = (((fmaf(0.5f, u0, 1.0f) / sin2phi) * u0) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.825378024189646e-15))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) / sin2phi) * u0) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.82537802e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 25.6%

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

   11. Applied rewrites 25.6%

       \[\leadsto \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi} \]

   if 1.82537802e-15 < sin2phi

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.8%

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{sin2phi}, \frac{1}{sin2phi}\right)\right) \cdot \left(alphay \cdot alphay\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.8%

       \[\leadsto \left(\left(\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{sin2phi} \cdot u0\right) \cdot alphay\right) \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 74.7% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{sin2phi} \cdot u0\right) \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
  (* (* (fma 0.5 u0 1.0) u0) (/ (* alphax alphax) cos2phi))
  (* (* (/ (fma 0.5 u0 1.0) sin2phi) u0) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-15f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) * ((alphax * alphax) / cos2phi);
	} else {
		tmp = ((fmaf(0.5f, u0, 1.0f) / sin2phi) * u0) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.9999998245167e-15))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) / sin2phi) * u0) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999982e-15

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 25.6%

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

   11. Applied rewrites 25.6%

       \[\leadsto \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi} \]

   if 9.99999982e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.8%

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{sin2phi}, \frac{1}{sin2phi}\right)\right) \cdot \left(alphay \cdot alphay\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.8%

       \[\leadsto \left(\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{sin2phi} \cdot u0\right) \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 67.9% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot u0}{sin2phi} \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 1.000000013351432e-10)
  (* (* (fma 0.5 u0 1.0) u0) (/ (* alphax alphax) cos2phi))
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) * ((alphax * alphax) / cos2phi);
	} else {
		tmp = ((alphay * u0) / sin2phi) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.000000013351432e-10))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * u0) / sin2phi) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-10

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 25.6%

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

   11. Applied rewrites 25.6%

       \[\leadsto \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi} \]

   if 1.00000001e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 59.1%

       \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 65.8% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{-\left(-u0\right)}{cos2phi}}{\frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot u0}{sin2phi} \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 1.000000013351432e-10)
  (/ (/ (- (- u0)) cos2phi) (/ 1.0 (* alphax alphax)))
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10f) {
		tmp = (-(-u0) / cos2phi) / (1.0f / (alphax * alphax));
	} else {
		tmp = ((alphay * u0) / sin2phi) * alphay;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10) then
        tmp = (-(-u0) / cos2phi) / (1.0e0 / (alphax * alphax))
    else
        tmp = ((alphay * u0) / sin2phi) * alphay
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.000000013351432e-10))
		tmp = Float32(Float32(Float32(-Float32(-u0)) / cos2phi) / Float32(Float32(1.0) / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(Float32(alphay * u0) / sin2phi) * alphay);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.000000013351432e-10))
		tmp = (-(-u0) / cos2phi) / (single(1.0) / (alphax * alphax));
	else
		tmp = ((alphay * u0) / sin2phi) * alphay;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-10

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.2%

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

   11. Applied rewrites 23.3%

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

   if 1.00000001e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 59.1%

       \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 65.8% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{--1 \cdot u0}{\frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot u0}{sin2phi} \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 1.000000013351432e-10)
  (/ (- (* -1.0 u0)) (/ (/ cos2phi alphax) alphax))
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10f) {
		tmp = -(-1.0f * u0) / ((cos2phi / alphax) / alphax);
	} else {
		tmp = ((alphay * u0) / sin2phi) * alphay;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10) then
        tmp = -((-1.0e0) * u0) / ((cos2phi / alphax) / alphax)
    else
        tmp = ((alphay * u0) / sin2phi) * alphay
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.000000013351432e-10))
		tmp = Float32(Float32(-Float32(Float32(-1.0) * u0)) / Float32(Float32(cos2phi / alphax) / alphax));
	else
		tmp = Float32(Float32(Float32(alphay * u0) / sin2phi) * alphay);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.000000013351432e-10))
		tmp = -(single(-1.0) * u0) / ((cos2phi / alphax) / alphax);
	else
		tmp = ((alphay * u0) / sin2phi) * alphay;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-10

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.2%

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.3%

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

   if 1.00000001e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 59.1%

       \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 65.7% accurate, 1.3× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;-\frac{-u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot u0}{sin2phi} \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 1.000000013351432e-10)
  (- (/ (- u0) (/ cos2phi (* alphax alphax))))
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10f) {
		tmp = -(-u0 / (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((alphay * u0) / sin2phi) * alphay;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10) then
        tmp = -(-u0 / (cos2phi / (alphax * alphax)))
    else
        tmp = ((alphay * u0) / sin2phi) * alphay
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.000000013351432e-10))
		tmp = Float32(-Float32(Float32(-u0) / Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * u0) / sin2phi) * alphay);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.000000013351432e-10))
		tmp = -(-u0 / (cos2phi / (alphax * alphax)));
	else
		tmp = ((alphay * u0) / sin2phi) * alphay;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-10

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.2%

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

   11. Applied rewrites 23.3%

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

   if 1.00000001e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 59.1%

       \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 65.7% accurate, 1.3× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(-\left(-u0\right)\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot u0}{sin2phi} \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 1.000000013351432e-10)
  (/ (* (- (- u0)) (* alphax alphax)) cos2phi)
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10f) {
		tmp = (-(-u0) * (alphax * alphax)) / cos2phi;
	} else {
		tmp = ((alphay * u0) / sin2phi) * alphay;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10) then
        tmp = (-(-u0) * (alphax * alphax)) / cos2phi
    else
        tmp = ((alphay * u0) / sin2phi) * alphay
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.000000013351432e-10))
		tmp = Float32(Float32(Float32(-Float32(-u0)) * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * u0) / sin2phi) * alphay);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.000000013351432e-10))
		tmp = (-(-u0) * (alphax * alphax)) / cos2phi;
	else
		tmp = ((alphay * u0) / sin2phi) * alphay;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-10

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in alphax around 0

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

   9. Applied rewrites 23.3%

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

   11. Applied rewrites 23.3%

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

   if 1.00000001e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 59.1%

       \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 65.7% accurate, 1.3× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\left(-\left(-u0\right)\right) \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot u0}{sin2phi} \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (/ sin2phi (* alphay alphay)) 1.000000013351432e-10)
  (* (- (- u0)) (/ (* alphax alphax) cos2phi))
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10f) {
		tmp = -(-u0) * ((alphax * alphax) / cos2phi);
	} else {
		tmp = ((alphay * u0) / sin2phi) * alphay;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.000000013351432e-10) then
        tmp = -(-u0) * ((alphax * alphax) / cos2phi)
    else
        tmp = ((alphay * u0) / sin2phi) * alphay
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.000000013351432e-10))
		tmp = Float32(Float32(-Float32(-u0)) * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * u0) / sin2phi) * alphay);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.000000013351432e-10))
		tmp = -(-u0) * ((alphax * alphax) / cos2phi);
	else
		tmp = ((alphay * u0) / sin2phi) * alphay;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.00000001e-10

   1. Initial program 60.4%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.2%

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

   11. Applied rewrites 23.3%

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

   if 1.00000001e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   3. Applied rewrites 60.8%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 59.1%

       \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 59.1% accurate, 2.8× speedup

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.4%

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

3. Applied rewrites 60.8%

   \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \]
4. Step-by-step derivation

5. Applied rewrites 60.8%

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

6. Taylor expanded in alphax around inf

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

8. Applied rewrites 49.1%

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

9. Taylor expanded in u0 around 0

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

11. Applied rewrites 59.1%

    \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2026086 
(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)))))