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

Percentage Accurate: 60.0% → 98.5%

Time: 8.7s

Alternatives: 21

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: 60.0% 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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 95.8

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

4. Applied rewrites 95.8%

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

6. Applied rewrites 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 95.8

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

4. Applied rewrites 95.8%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. lift-log1p.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. diff-log N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. flip3-- N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. fp-cancel-sign-sub-inv N/A

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

   15. mul-1-neg N/A

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

   16. lower-log1p.f32 N/A

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

   17. lower-neg.f32 98.3

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

6. Applied rewrites 98.3%

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

Alternative 3: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 24500:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 24500.0)
     (/
      (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* 0.25 u0)))))))
      (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* (* alphay alphay) (log1p (- u0))) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 24500.0f) {
		tmp = (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * u0))))))) / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = ((alphay * alphay) * log1pf(-u0)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(24500.0))
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u0))))))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * log1p(Float32(-u0))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 24500

   1. Initial program 52.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.7%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 94.3

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

   10. Applied rewrites 94.3%

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

   if 24500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.9%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 99.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 24500:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 95.8

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

4. Applied rewrites 95.8%

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 93.4

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

9. Applied rewrites 93.4%

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

10. Final simplification 93.4%

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

Alternative 5: 93.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * 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 = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (0.25e0 * u0))))))) / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 76.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 76.3

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

7. Applied rewrites 76.3%

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

8. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 93.1

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

10. Applied rewrites 93.1%

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

Alternative 6: 90.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ (* (fma 0.5 u0 1.0) u0) (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (* alphay alphay)
       (*
        u0
        (- (* u0 (- (* u0 (- (* -0.25 u0) 0.3333333333333333)) 0.5)) 1.0)))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 500.0f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * ((-0.25f * u0) - 0.3333333333333333f)) - 0.5f)) - 1.0f))) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(500.0))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 52.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.9%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower--.f32 N/A

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

       7. lower-*.f32 93.0

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

   11. Applied rewrites 93.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot u0}{\mathsf{fma}\left(alphay, cos2phi, \frac{\left(alphax \cdot alphax\right) \cdot sin2phi}{alphay}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 500.0)
   (/
    (* (* (* alphax alphax) alphay) u0)
    (fma alphay cos2phi (/ (* (* alphax alphax) sin2phi) alphay)))
   (/
    (*
     (* alphay alphay)
     (* u0 (- (* u0 (- (* u0 (- (* -0.25 u0) 0.3333333333333333)) 0.5)) 1.0)))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 500.0f) {
		tmp = (((alphax * alphax) * alphay) * u0) / fmaf(alphay, cos2phi, (((alphax * alphax) * sin2phi) / alphay));
	} else {
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * ((-0.25f * u0) - 0.3333333333333333f)) - 0.5f)) - 1.0f))) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(500.0))
		tmp = Float32(Float32(Float32(Float32(alphax * alphax) * alphay) * u0) / fma(alphay, cos2phi, Float32(Float32(Float32(alphax * alphax) * sin2phi) / alphay)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 52.4%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. pow2 N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. pow2 N/A

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

      14. lift-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. pow2 N/A

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

      18. lift-*.f32 52.4

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in u0 around 0

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

   7. Applied rewrites 75.7%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.9%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower--.f32 N/A

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

       7. lower-*.f32 93.0

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

   11. Applied rewrites 93.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot u0}{\mathsf{fma}\left(alphay, cos2phi, \frac{\left(alphax \cdot alphax\right) \cdot sin2phi}{alphay}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/
      (*
       (* alphay alphay)
       (*
        u0
        (- (* u0 (- (* u0 (- (* -0.25 u0) 0.3333333333333333)) 0.5)) 1.0)))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 500.0f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * ((-0.25f * u0) - 0.3333333333333333f)) - 0.5f)) - 1.0f))) / -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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 500.0e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * (((-0.25e0) * u0) - 0.3333333333333333e0)) - 0.5e0)) - 1.0e0))) / -sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(500.0))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(500.0))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * ((single(-0.25) * u0) - single(0.3333333333333333))) - single(0.5))) - single(1.0)))) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 52.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.6%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 75.7

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

   7. Applied rewrites 75.7%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.9%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower--.f32 N/A

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

       7. lower-*.f32 93.0

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

   11. Applied rewrites 93.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (* alphay alphay)
       (*
        u0
        (- (* u0 (- (* u0 (- (* -0.25 u0) 0.3333333333333333)) 0.5)) 1.0)))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 500.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * ((-0.25f * u0) - 0.3333333333333333f)) - 0.5f)) - 1.0f))) / -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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 500.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * (((-0.25e0) * u0) - 0.3333333333333333e0)) - 0.5e0)) - 1.0e0))) / -sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(500.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(500.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((u0 * ((single(-0.25) * u0) - single(0.3333333333333333))) - single(0.5))) - single(1.0)))) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 52.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.6%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.9%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower--.f32 N/A

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

       7. lower-*.f32 93.0

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

   11. Applied rewrites 93.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 500.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(500.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 52.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.6%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. 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 92.1

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

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 91.9

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

   10. Applied rewrites 91.9%

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

Alternative 11: 93.2% 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 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.0

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

5. Applied rewrites 93.0%

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

Alternative 12: 83.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (* alphay alphay)
       (* u0 (- (* u0 (- (* -0.3333333333333333 u0) 0.5)) 1.0)))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 500.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((-0.3333333333333333f * u0) - 0.5f)) - 1.0f))) / -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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 500.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = ((alphay * alphay) * (u0 * ((u0 * (((-0.3333333333333333e0) * u0) - 0.5e0)) - 1.0e0))) / -sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(500.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.3333333333333333) * u0) - Float32(0.5))) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(500.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((single(-0.3333333333333333) * u0) - single(0.5))) - single(1.0)))) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 52.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.6%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.9%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 91.3

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

   11. Applied rewrites 91.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* u0 (- (* u0 (- (* -0.3333333333333333 u0) 0.5)) 1.0))))
   (if (<= (/ sin2phi (* alphay alphay)) 5.000000018137469e-16)
     (/ (* (* alphax alphax) t_0) (- cos2phi))
     (/ (* (* alphay alphay) t_0) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 * ((u0 * ((-0.3333333333333333f * u0) - 0.5f)) - 1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16f) {
		tmp = ((alphax * alphax) * t_0) / -cos2phi;
	} else {
		tmp = ((alphay * alphay) * t_0) / -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) :: t_0
    real(4) :: tmp
    t_0 = u0 * ((u0 * (((-0.3333333333333333e0) * u0) - 0.5e0)) - 1.0e0)
    if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16) then
        tmp = ((alphax * alphax) * t_0) / -cos2phi
    else
        tmp = ((alphay * alphay) * t_0) / -sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.3333333333333333) * u0) - Float32(0.5))) - Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.000000018137469e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * t_0) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * t_0) / Float32(-sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = u0 * ((u0 * ((single(-0.3333333333333333) * u0) - single(0.5))) - single(1.0));
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(5.000000018137469e-16))
		tmp = ((alphax * alphax) * t_0) / -cos2phi;
	else
		tmp = ((alphay * alphay) * t_0) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000002e-16

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 40.4

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

   5. Applied rewrites 40.4%

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

      1. distribute-frac-neg 40.4

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

      2. flip3-- 40.4

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

      3. metadata-eval 40.4

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

      4. metadata-eval 40.4

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

      5. diff-log 40.4

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

      6. distribute-frac-neg 40.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 65.1

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

   10. Applied rewrites 65.1%

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

   if 5.00000002e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 78.2%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 84.5

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

   11. Applied rewrites 84.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot \left(alphax \cdot alphax\right), u0, \left(-alphax\right) \cdot alphax\right) \cdot u0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.000000018137469e-16)
   (/
    (* (fma (* -0.5 (* alphax alphax)) u0 (* (- alphax) alphax)) u0)
    (- cos2phi))
   (/
    (*
     (* alphay alphay)
     (* u0 (- (* u0 (- (* -0.3333333333333333 u0) 0.5)) 1.0)))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16f) {
		tmp = (fmaf((-0.5f * (alphax * alphax)), u0, (-alphax * alphax)) * u0) / -cos2phi;
	} else {
		tmp = ((alphay * alphay) * (u0 * ((u0 * ((-0.3333333333333333f * u0) - 0.5f)) - 1.0f))) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.000000018137469e-16))
		tmp = Float32(Float32(fma(Float32(Float32(-0.5) * Float32(alphax * alphax)), u0, Float32(Float32(-alphax) * alphax)) * u0) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.3333333333333333) * u0) - Float32(0.5))) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000002e-16

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 62.8

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

   8. Applied rewrites 62.8%

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

   if 5.00000002e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 78.2%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 84.5

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

   11. Applied rewrites 84.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot \left(alphax \cdot alphax\right), u0, \left(-alphax\right) \cdot alphax\right) \cdot u0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot \left(alphax \cdot alphax\right), u0, \left(-alphax\right) \cdot alphax\right) \cdot u0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.000000018137469e-16)
   (/
    (* (fma (* -0.5 (* alphax alphax)) u0 (* (- alphax) alphax)) u0)
    (- cos2phi))
   (/ (* (* alphay alphay) (* u0 (- (* -0.5 u0) 1.0))) (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16f) {
		tmp = (fmaf((-0.5f * (alphax * alphax)), u0, (-alphax * alphax)) * u0) / -cos2phi;
	} else {
		tmp = ((alphay * alphay) * (u0 * ((-0.5f * u0) - 1.0f))) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.000000018137469e-16))
		tmp = Float32(Float32(fma(Float32(Float32(-0.5) * Float32(alphax * alphax)), u0, Float32(Float32(-alphax) * alphax)) * u0) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000002e-16

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 62.8

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

   8. Applied rewrites 62.8%

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

   if 5.00000002e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 78.2%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 81.4

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

   11. Applied rewrites 81.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot \left(alphax \cdot alphax\right), u0, \left(-alphax\right) \cdot alphax\right) \cdot u0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.4% 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 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.3

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

5. Applied rewrites 91.3%

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

Alternative 17: 75.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.000000018137469e-16)
   (/ (* u0 (fma 0.5 (* (* alphax alphax) u0) (* alphax alphax))) cos2phi)
   (/ (* (* alphay alphay) (* u0 (- (* -0.5 u0) 1.0))) (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16f) {
		tmp = (u0 * fmaf(0.5f, ((alphax * alphax) * u0), (alphax * alphax))) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * (u0 * ((-0.5f * u0) - 1.0f))) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.000000018137469e-16))
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphax * alphax) * u0), Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000002e-16

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 40.4

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

   5. Applied rewrites 40.4%

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

      1. distribute-frac-neg 40.4

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

      2. flip3-- 40.4

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

      3. metadata-eval 40.4

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

      4. metadata-eval 40.4

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

      5. diff-log 40.4

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

      6. distribute-frac-neg 40.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 62.8

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

   10. Applied rewrites 62.8%

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

   if 5.00000002e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 78.2%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 81.4

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

   11. Applied rewrites 81.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 \cdot \left(-0.5 \cdot u0 - 1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* u0 (- (* -0.5 u0) 1.0))))
   (if (<= (/ sin2phi (* alphay alphay)) 5.000000018137469e-16)
     (/ (* (* alphax alphax) t_0) (- cos2phi))
     (/ (* (* alphay alphay) t_0) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 * ((-0.5f * u0) - 1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16f) {
		tmp = ((alphax * alphax) * t_0) / -cos2phi;
	} else {
		tmp = ((alphay * alphay) * t_0) / -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) :: t_0
    real(4) :: tmp
    t_0 = u0 * (((-0.5e0) * u0) - 1.0e0)
    if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16) then
        tmp = ((alphax * alphax) * t_0) / -cos2phi
    else
        tmp = ((alphay * alphay) * t_0) / -sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.000000018137469e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * t_0) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * t_0) / Float32(-sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = u0 * ((single(-0.5) * u0) - single(1.0));
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(5.000000018137469e-16))
		tmp = ((alphax * alphax) * t_0) / -cos2phi;
	else
		tmp = ((alphay * alphay) * t_0) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000002e-16

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 40.4

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

   5. Applied rewrites 40.4%

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

      1. distribute-frac-neg 40.4

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

      2. flip3-- 40.4

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

      3. metadata-eval 40.4

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

      4. metadata-eval 40.4

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

      5. diff-log 40.4

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

      6. distribute-frac-neg 40.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 62.8

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

   10. Applied rewrites 62.8%

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

   if 5.00000002e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 78.2%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 81.4

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

   11. Applied rewrites 81.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 73.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.000000018137469e-16)
   (/ (* (* alphax alphax) u0) cos2phi)
   (/ (* (* alphay alphay) (* u0 (- (* -0.5 u0) 1.0))) (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.000000018137469e-16f) {
		tmp = ((alphax * alphax) * u0) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * (u0 * ((-0.5f * u0) - 1.0f))) / -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)) <= 5.000000018137469e-16) then
        tmp = ((alphax * alphax) * u0) / cos2phi
    else
        tmp = ((alphay * alphay) * (u0 * (((-0.5e0) * u0) - 1.0e0))) / -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(5.000000018137469e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * u0) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))) / Float32(-sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(5.000000018137469e-16))
		tmp = ((alphax * alphax) * u0) / cos2phi;
	else
		tmp = ((alphay * alphay) * (u0 * ((single(-0.5) * u0) - single(1.0)))) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000002e-16

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 40.4

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

   5. Applied rewrites 40.4%

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

   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 54.4

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

   8. Applied rewrites 54.4%

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

   if 5.00000002e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 78.2%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 81.4

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

   11. Applied rewrites 81.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 66.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.000000018137469e-16)
   (/ (* (* 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)) <= 5.000000018137469e-16f) {
		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)) <= 5.000000018137469e-16) 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(5.000000018137469e-16))
		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(5.000000018137469e-16))
		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)) < 5.00000002e-16

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 40.4

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

   5. Applied rewrites 40.4%

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

   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 54.4

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

   8. Applied rewrites 54.4%

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

   if 5.00000002e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 78.2%

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

   6. Taylor expanded in alphax around inf

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

      1. distribute-frac-neg N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. diff-log N/A

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

      6. distribute-frac-neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 72.2

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

   11. Applied rewrites 72.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 24.1% 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 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 21.7

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

5. Applied rewrites 21.7%

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

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 22.3

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

8. Applied rewrites 22.3%

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

9. Final simplification 22.3%

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

Reproduce

herbie shell --seed 2025057 
(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)))))