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

Percentage Accurate: 60.5% → 98.3%

Time: 7.8s

Alternatives: 17

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. flip3-- N/A

      \[\leadsto \frac{-\log \color{blue}{\left(\frac{{1}^{3} - {u0}^{3}}{1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. log-div N/A

      \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower--.f32 N/A

      \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. lower-log.f32 N/A

      \[\leadsto \frac{-\left(\color{blue}{\log \left({1}^{3} - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{-\left(\log \left(\color{blue}{1} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower--.f32 N/A

      \[\leadsto \frac{-\left(\log \color{blue}{\left(1 - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. lower-pow.f32 N/A

      \[\leadsto \frac{-\left(\log \left(1 - \color{blue}{{u0}^{3}}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \log \left(\color{blue}{1} + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. lower-log1p.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   12. lower-fma.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   13. lower-*.f32 95.3

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 95.3%

   \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
5. Step-by-step derivation

6. Applied rewrites 98.4%

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

Alternative 2: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (* (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0) u0))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -(((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 = -((((((((-0.25e0) * u0) - 0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - 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 = -(((((((single(-0.25) * u0) - single(0.3333333333333333)) * u0) - single(0.5)) * u0) - single(1.0)) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{-\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{-\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{-\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{-\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. lower--.f32 N/A

      \[\leadsto \frac{-\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{-\left(\left(\left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) \cdot u0 - \frac{1}{2}\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{-\left(\left(\left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) \cdot u0 - \frac{1}{2}\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{-\left(\left(\left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) \cdot u0 - \frac{1}{2}\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-*.f32 93.6

       \[\leadsto \frac{-\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 93.6%

   \[\leadsto \frac{-\color{blue}{\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 3: 92.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0 + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u0 + \frac{1}{3}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 93.6

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 93.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 4: 78.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(0.5 \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) + alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
   (/ (* u0 (+ (* 0.5 (* (* alphax alphax) u0)) (* alphax alphax))) cos2phi)
   (/
    (*
     u0
     (fma
      u0
      (* (* alphay alphay) (+ 0.5 (* 0.3333333333333333 u0)))
      (* alphay alphay)))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-15f) {
		tmp = (u0 * ((0.5f * ((alphax * alphax) * u0)) + (alphax * alphax))) / cos2phi;
	} else {
		tmp = (u0 * fmaf(u0, ((alphay * alphay) * (0.5f + (0.3333333333333333f * u0))), (alphay * alphay))) / sin2phi;
	}
	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(u0 * Float32(Float32(Float32(0.5) * Float32(Float32(alphax * alphax) * u0)) + Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(u0 * fma(u0, Float32(Float32(alphay * alphay) * Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0))), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.4%

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

   3. Taylor expanded in alphax around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      2. lower--.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      5. pow2 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(\mathsf{neg}\left({alphax}^{2}\right)\right)\right)}{cos2phi} \]

      8. lower-neg.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(-{alphax}^{2}\right)\right)}{cos2phi} \]

      9. pow2 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(-alphax \cdot alphax\right)\right)}{cos2phi} \]

      10. lift-*.f32 63.8

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

   8. Applied rewrites 63.8%

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

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

   1. Initial program 64.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 61.8

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

   4. Applied rewrites 61.8%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{2} \cdot {alphay}^{2}\right) + {alphay}^{2}\right)}{sin2phi} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{3} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{2} \cdot {alphay}^{2}, {alphay}^{2}\right)}{sin2phi} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, {alphay}^{2} \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, {alphay}^{2} \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      5. pow2 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      8. pow2 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), {alphay}^{2}\right)}{sin2phi} \]

      9. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), {alphay}^{2}\right)}{sin2phi} \]

      10. pow2 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

      11. lift-*.f32 85.0

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

   10. Applied rewrites 85.0%

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

   11. Taylor expanded in alphay around 0

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

       1. lower-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, {alphay}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       2. pow2 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       3. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       4. lower-+.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       5. lower-*.f32 85.0

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

   13. Applied rewrites 85.0%

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(0.5 \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) + alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(0.5 \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) + alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
   (/ (* u0 (+ (* 0.5 (* (* alphax alphax) u0)) (* alphax alphax))) cos2phi)
   (/
    (*
     u0
     (* (* alphay alphay) (+ 1.0 (* u0 (+ 0.5 (* 0.3333333333333333 u0))))))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-15f) {
		tmp = (u0 * ((0.5f * ((alphax * alphax) * u0)) + (alphax * alphax))) / cos2phi;
	} else {
		tmp = (u0 * ((alphay * alphay) * (1.0f + (u0 * (0.5f + (0.3333333333333333f * u0)))))) / sin2phi;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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)) <= 9.9999998245167e-15) then
        tmp = (u0 * ((0.5e0 * ((alphax * alphax) * u0)) + (alphax * alphax))) / cos2phi
    else
        tmp = (u0 * ((alphay * alphay) * (1.0e0 + (u0 * (0.5e0 + (0.3333333333333333e0 * u0)))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.9999998245167e-15))
		tmp = Float32(Float32(u0 * Float32(Float32(Float32(0.5) * Float32(Float32(alphax * alphax) * u0)) + Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(u0 * Float32(Float32(alphay * alphay) * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0)))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(9.9999998245167e-15))
		tmp = (u0 * ((single(0.5) * ((alphax * alphax) * u0)) + (alphax * alphax))) / cos2phi;
	else
		tmp = (u0 * ((alphay * alphay) * (single(1.0) + (u0 * (single(0.5) + (single(0.3333333333333333) * u0)))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.4%

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

   3. Taylor expanded in alphax around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      2. lower--.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      5. pow2 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(\mathsf{neg}\left({alphax}^{2}\right)\right)\right)}{cos2phi} \]

      8. lower-neg.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(-{alphax}^{2}\right)\right)}{cos2phi} \]

      9. pow2 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(-alphax \cdot alphax\right)\right)}{cos2phi} \]

      10. lift-*.f32 63.8

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

   8. Applied rewrites 63.8%

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

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

   1. Initial program 64.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 61.8

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

   4. Applied rewrites 61.8%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{2} \cdot {alphay}^{2}\right) + {alphay}^{2}\right)}{sin2phi} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{3} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{2} \cdot {alphay}^{2}, {alphay}^{2}\right)}{sin2phi} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, {alphay}^{2} \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, {alphay}^{2} \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      5. pow2 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      8. pow2 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), {alphay}^{2}\right)}{sin2phi} \]

      9. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), {alphay}^{2}\right)}{sin2phi} \]

      10. pow2 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

      11. lift-*.f32 85.0

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

   10. Applied rewrites 85.0%

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

   11. Taylor expanded in alphay around 0

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

       1. lower-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left({alphay}^{2} \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}{sin2phi} \]

       2. pow2 N/A

          \[\leadsto \frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}{sin2phi} \]

       3. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}{sin2phi} \]

       4. lower-+.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}{sin2phi} \]

       5. lower-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}{sin2phi} \]

       6. lower-+.f32 N/A

          \[\leadsto \frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}{sin2phi} \]

       7. lower-*.f32 84.9

          \[\leadsto \frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)\right)}{sin2phi} \]

   13. Applied rewrites 84.9%

       \[\leadsto \frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(0.5 \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) + alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000058430487e-8)
   (/
    (* (fma 0.5 u0 1.0) u0)
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     (* alphay alphay)
     (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* 0.25 u0))))))))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000058430487e-8f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((alphay * alphay) * (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * u0)))))))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000058430487e-8))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u0)))))))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000006e-8

   1. Initial program 56.5%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 89.1

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

   5. Applied rewrites 89.1%

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

   if 5.00000006e-8 < sin2phi

   1. Initial program 67.7%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in u0 around 0

      \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      7. lower-*.f32 93.2

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi} \]

   10. Applied rewrites 93.2%

       \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* (fma (fma 0.3333333333333333 u0 0.5) u0 1.0) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (fmaf(fmaf(0.3333333333333333f, u0, 0.5f), u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) \cdot \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right) + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\left(\frac{1}{2} + \frac{1}{3} \cdot u0\right) \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{3} \cdot u0, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3} \cdot u0 + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. lower-fma.f32 92.0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 8: 76.1% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
   (/ (* u0 (+ (* 0.5 (* (* alphax alphax) u0)) (* alphax alphax))) cos2phi)
   (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-15f) {
		tmp = (u0 * ((0.5f * ((alphax * alphax) * u0)) + (alphax * alphax))) / cos2phi;
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	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(u0 * Float32(Float32(Float32(0.5) * Float32(Float32(alphax * alphax) * u0)) + Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.4%

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

   3. Taylor expanded in alphax around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      2. lower--.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      5. pow2 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - -1 \cdot {alphax}^{2}\right)}{cos2phi} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(\mathsf{neg}\left({alphax}^{2}\right)\right)\right)}{cos2phi} \]

      8. lower-neg.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(-{alphax}^{2}\right)\right)}{cos2phi} \]

      9. pow2 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left(\left(alphax \cdot alphax\right) \cdot u0\right) - \left(-alphax \cdot alphax\right)\right)}{cos2phi} \]

      10. lift-*.f32 63.8

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

   8. Applied rewrites 63.8%

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

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

   1. Initial program 64.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 61.8

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

   4. Applied rewrites 61.8%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right) + {alphay}^{2}\right)}{sin2phi} \]

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 81.5

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

   10. Applied rewrites 81.5%

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

4. Final simplification 76.3%

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

Alternative 9: 74.0% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
   (/ (* (* alphax alphax) u0) cos2phi)
   (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-15f) {
		tmp = ((alphax * alphax) * u0) / cos2phi;
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	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(alphax * alphax) * u0) / cos2phi);
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.4%

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

   3. Taylor expanded in alphax around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 57.2

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

   8. Applied rewrites 57.2%

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

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

   1. Initial program 64.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 61.8

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

   4. Applied rewrites 61.8%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right) + {alphay}^{2}\right)}{sin2phi} \]

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 81.5

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

   10. Applied rewrites 81.5%

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

Alternative 10: 84.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000058430487e-8)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     (* alphay alphay)
     (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* 0.25 u0))))))))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000058430487e-8f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((alphay * alphay) * (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * u0)))))))) / sin2phi;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 <= 5.000000058430487e-8) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = ((alphay * alphay) * (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (0.25e0 * u0)))))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000058430487e-8))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u0)))))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000058430487e-8))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = ((alphay * alphay) * (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (single(0.25) * u0)))))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000006e-8

   1. Initial program 56.5%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 75.3%

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

   if 5.00000006e-8 < sin2phi

   1. Initial program 67.7%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in u0 around 0

      \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{sin2phi} \]

      7. lower-*.f32 93.2

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi} \]

   10. Applied rewrites 93.2%

       \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000058430487e-8)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (*
    (* alphay alphay)
    (*
     u0
     (/
      (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* 0.25 u0))))))
      sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000058430487e-8f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (alphay * alphay) * (u0 * ((1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * u0)))))) / sin2phi));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 <= 5.000000058430487e-8) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = (alphay * alphay) * (u0 * ((1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (0.25e0 * u0)))))) / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000058430487e-8))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u0)))))) / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000058430487e-8))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = (alphay * alphay) * (u0 * ((single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (single(0.25) * u0)))))) / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000006e-8

   1. Initial program 56.5%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 75.3%

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

   if 5.00000006e-8 < sin2phi

   1. Initial program 67.7%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{-\log \color{blue}{\left(\frac{{1}^{3} - {u0}^{3}}{1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. log-div N/A

         \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower--.f32 N/A

         \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-log.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\log \left({1}^{3} - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\log \left(\color{blue}{1} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower--.f32 N/A

         \[\leadsto \frac{-\left(\log \color{blue}{\left(1 - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. lower-pow.f32 N/A

         \[\leadsto \frac{-\left(\log \left(1 - \color{blue}{{u0}^{3}}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \log \left(\color{blue}{1} + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lower-log1p.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. lower-*.f32 95.3

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 95.3%

      \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

   7. Applied rewrites 93.0%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right), \frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right), \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   8. Taylor expanded in alphay around 0

      \[\leadsto {alphay}^{2} \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \frac{1}{sin2phi}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto {alphay}^{2} \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \frac{1}{sin2phi}\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right)} + \frac{1}{sin2phi}\right)\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right)} + \frac{1}{sin2phi}\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \color{blue}{\frac{1}{sin2phi}}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \color{blue}{\frac{1}{2} \cdot \frac{1}{sin2phi}}, \frac{1}{sin2phi}\right)\right) \]

   10. Applied rewrites 93.1%

       \[\leadsto \left(alphay \cdot alphay\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{sin2phi}, 0.3333333333333333 \cdot \frac{1}{sin2phi}\right), 0.5 \cdot \frac{1}{sin2phi}\right), \frac{1}{sin2phi}\right)\right)} \]

   11. Taylor expanded in sin2phi around 0

       \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{sin2phi}\right) \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{sin2phi}\right) \]

       2. lower-+.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{sin2phi}\right) \]

       3. lower-*.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{sin2phi}\right) \]

       4. lower-+.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{sin2phi}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{sin2phi}\right) \]

       6. lower-+.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{sin2phi}\right) \]

       7. lower-*.f32 93.0

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)}{sin2phi}\right) \]

   13. Applied rewrites 93.0%

       \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \frac{1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)}{sin2phi}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 84.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000058430487e-8)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (*
    (* alphay alphay)
    (/
     (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* 0.25 u0)))))))
     sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000058430487e-8f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (alphay * alphay) * ((u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * u0))))))) / sin2phi);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 <= 5.000000058430487e-8) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = (alphay * alphay) * ((u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (0.25e0 * u0))))))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000058430487e-8))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u0))))))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000058430487e-8))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = (alphay * alphay) * ((u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (single(0.25) * u0))))))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000006e-8

   1. Initial program 56.5%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 75.3%

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

   if 5.00000006e-8 < sin2phi

   1. Initial program 67.7%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{-\log \color{blue}{\left(\frac{{1}^{3} - {u0}^{3}}{1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. log-div N/A

         \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower--.f32 N/A

         \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-log.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\log \left({1}^{3} - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\log \left(\color{blue}{1} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower--.f32 N/A

         \[\leadsto \frac{-\left(\log \color{blue}{\left(1 - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. lower-pow.f32 N/A

         \[\leadsto \frac{-\left(\log \left(1 - \color{blue}{{u0}^{3}}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \log \left(\color{blue}{1} + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lower-log1p.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. lower-*.f32 95.3

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 95.3%

      \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

   7. Applied rewrites 93.0%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right), \frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right), \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   8. Taylor expanded in alphay around 0

      \[\leadsto {alphay}^{2} \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \frac{1}{sin2phi}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto {alphay}^{2} \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \frac{1}{sin2phi}\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right)} + \frac{1}{sin2phi}\right)\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right)} + \frac{1}{sin2phi}\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \color{blue}{\frac{1}{sin2phi}}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \color{blue}{\frac{1}{2} \cdot \frac{1}{sin2phi}}, \frac{1}{sin2phi}\right)\right) \]

   10. Applied rewrites 93.1%

       \[\leadsto \left(alphay \cdot alphay\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{sin2phi}, 0.3333333333333333 \cdot \frac{1}{sin2phi}\right), 0.5 \cdot \frac{1}{sin2phi}\right), \frac{1}{sin2phi}\right)\right)} \]

   11. Taylor expanded in sin2phi around 0

       \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]

       2. lower-*.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]

       3. lower-+.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]

       4. lower-*.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]

       5. lower-+.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]

       6. lower-*.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]

       7. lower-+.f32 N/A

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{sin2phi} \]

       8. lower-*.f32 93.0

          \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}{sin2phi} \]

   13. Applied rewrites 93.0%

       \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
   (/ (* (* alphax alphax) u0) cos2phi)
   (/ (* (* alphay alphay) (* u0 (+ 1.0 (* 0.5 u0)))) sin2phi)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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)) <= 9.9999998245167e-15) then
        tmp = ((alphax * alphax) * u0) / cos2phi
    else
        tmp = ((alphay * alphay) * (u0 * (1.0e0 + (0.5e0 * u0)))) / sin2phi
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(9.9999998245167e-15))
		tmp = ((alphax * alphax) * u0) / cos2phi;
	else
		tmp = ((alphay * alphay) * (u0 * (single(1.0) + (single(0.5) * u0)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.4%

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

   3. Taylor expanded in alphax around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 57.2

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

   8. Applied rewrites 57.2%

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

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

   1. Initial program 64.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 61.8

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

   4. Applied rewrites 61.8%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}{sin2phi} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}{sin2phi} \]

      3. lower-*.f32 81.4

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

   10. Applied rewrites 81.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + 0.5 \cdot u0\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000058430487e-8)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     u0
     (fma
      u0
      (* (* alphay alphay) (+ 0.5 (* 0.3333333333333333 u0)))
      (* alphay alphay)))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000058430487e-8f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (u0 * fmaf(u0, ((alphay * alphay) * (0.5f + (0.3333333333333333f * u0))), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000058430487e-8))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(u0 * fma(u0, Float32(Float32(alphay * alphay) * Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0))), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000006e-8

   1. Initial program 56.5%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 75.3%

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

   if 5.00000006e-8 < sin2phi

   1. Initial program 67.7%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{2} \cdot {alphay}^{2}\right) + {alphay}^{2}\right)}{sin2phi} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{3} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{2} \cdot {alphay}^{2}, {alphay}^{2}\right)}{sin2phi} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, {alphay}^{2} \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, {alphay}^{2} \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      5. pow2 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot {alphay}^{2}\right), {alphay}^{2}\right)}{sin2phi} \]

      8. pow2 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), {alphay}^{2}\right)}{sin2phi} \]

      9. lift-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), {alphay}^{2}\right)}{sin2phi} \]

      10. pow2 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\frac{1}{3}, \left(alphay \cdot alphay\right) \cdot u0, \frac{1}{2} \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

      11. lift-*.f32 91.6

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

   10. Applied rewrites 91.6%

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \]

   11. Taylor expanded in alphay around 0

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

       1. lower-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, {alphay}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       2. pow2 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       3. lift-*.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       4. lower-+.f32 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

       5. lower-*.f32 91.6

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]

   13. Applied rewrites 91.6%

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 66.7% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
   (/ (* (* alphax alphax) u0) cos2phi)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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)) <= 9.9999998245167e-15) then
        tmp = ((alphax * alphax) * u0) / cos2phi
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(9.9999998245167e-15))
		tmp = ((alphax * alphax) * u0) / cos2phi;
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.4%

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

   3. Taylor expanded in alphax around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 57.2

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

   8. Applied rewrites 57.2%

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

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

   1. Initial program 64.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 61.8

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

   4. Applied rewrites 61.8%

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

   5. Taylor expanded in alphax around inf

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

      1. neg-log N/A

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

      2. distribute-frac-neg N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. diff-log N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u0 around 0

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

   10. Applied rewrites 70.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.7% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-15)
   (/ (* (* alphax alphax) u0) cos2phi)
   (* (* alphay alphay) (/ u0 sin2phi))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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)) <= 9.9999998245167e-15) then
        tmp = ((alphax * alphax) * u0) / cos2phi
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(9.9999998245167e-15))
		tmp = ((alphax * alphax) * u0) / cos2phi;
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.4%

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

   3. Taylor expanded in alphax around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 57.2

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

   8. Applied rewrites 57.2%

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

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

   1. Initial program 64.3%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{-\log \color{blue}{\left(\frac{{1}^{3} - {u0}^{3}}{1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. log-div N/A

         \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower--.f32 N/A

         \[\leadsto \frac{-\color{blue}{\left(\log \left({1}^{3} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-log.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\log \left({1}^{3} - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-\left(\log \left(\color{blue}{1} - {u0}^{3}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower--.f32 N/A

         \[\leadsto \frac{-\left(\log \color{blue}{\left(1 - {u0}^{3}\right)} - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. lower-pow.f32 N/A

         \[\leadsto \frac{-\left(\log \left(1 - \color{blue}{{u0}^{3}}\right) - \log \left(1 \cdot 1 + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \log \left(\color{blue}{1} + \left(u0 \cdot u0 + 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lower-log1p.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. lower-*.f32 95.3

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 95.3%

      \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

   7. Applied rewrites 93.9%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right), \frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right), \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   8. Taylor expanded in alphay around 0

      \[\leadsto {alphay}^{2} \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \frac{1}{sin2phi}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto {alphay}^{2} \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \frac{1}{sin2phi}\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right)} + \frac{1}{sin2phi}\right)\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right)} + \frac{1}{sin2phi}\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \color{blue}{\frac{1}{sin2phi}}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \color{blue}{\frac{1}{2} \cdot \frac{1}{sin2phi}}, \frac{1}{sin2phi}\right)\right) \]

   10. Applied rewrites 86.4%

       \[\leadsto \left(alphay \cdot alphay\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{sin2phi}, 0.3333333333333333 \cdot \frac{1}{sin2phi}\right), 0.5 \cdot \frac{1}{sin2phi}\right), \frac{1}{sin2phi}\right)\right)} \]

   11. Taylor expanded in u0 around 0

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

       1. lift-/.f32 70.2

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

   13. Applied rewrites 70.2%

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

Alternative 17: 23.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 = ((alphax * alphax) * u0) / cos2phi
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in alphax around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left({alphax}^{2} \cdot \log \left(1 - u0\right)\right)}{cos2phi} \]

   4. lower-neg.f32 N/A

      \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{-{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi} \]

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. lift--.f32 21.4

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

5. Applied rewrites 21.4%

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

6. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 25.6

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

8. Applied rewrites 25.6%

   \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025071 
(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)))))