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

Percentage Accurate: 61.4% → 97.9%

Time: 8.8s

Alternatives: 17

Speedup: 4.5×

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: 61.4% 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: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq -0.03200000151991844:\\ \;\;\;\;\frac{-t\_0}{\frac{sin2phi + \frac{\left(alphay \cdot alphay\right) \cdot cos2phi}{alphax \cdot alphax}}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))))
   (if (<= t_0 -0.03200000151991844)
     (/
      (- t_0)
      (/
       (+ sin2phi (/ (* (* alphay alphay) cos2phi) (* alphax alphax)))
       (* alphay alphay)))
     (/
      (* (+ (* (+ (* (+ (* 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) {
	float t_0 = logf((1.0f - u0));
	float tmp;
	if (t_0 <= -0.03200000151991844f) {
		tmp = -t_0 / ((sin2phi + (((alphay * alphay) * cos2phi) / (alphax * alphax))) / (alphay * alphay));
	} else {
		tmp = (((((((0.25f * u0) + 0.3333333333333333f) * u0) + 0.5f) * u0) + 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	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 = log((1.0e0 - u0))
    if (t_0 <= (-0.03200000151991844e0)) then
        tmp = -t_0 / ((sin2phi + (((alphay * alphay) * cos2phi) / (alphax * alphax))) / (alphay * alphay))
    else
        tmp = (((((((0.25e0 * u0) + 0.3333333333333333e0) * u0) + 0.5e0) * u0) + 1.0e0) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03200000151991844))
		tmp = Float32(Float32(-t_0) / Float32(Float32(sin2phi + Float32(Float32(Float32(alphay * alphay) * cos2phi) / Float32(alphax * alphax))) / Float32(alphay * alphay)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u0) + Float32(0.3333333333333333)) * u0) + Float32(0.5)) * u0) + Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log((single(1.0) - u0));
	tmp = single(0.0);
	if (t_0 <= single(-0.03200000151991844))
		tmp = -t_0 / ((sin2phi + (((alphay * alphay) * cos2phi) / (alphax * alphax))) / (alphay * alphay));
	else
		tmp = (((((((single(0.25) * u0) + single(0.3333333333333333)) * u0) + single(0.5)) * u0) + single(1.0)) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0320000015

   1. Initial program 93.5%

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

      1. lift-+.f32 N/A

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

      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. associate-/r* N/A

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

      7. pow2 N/A

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

      8. frac-add N/A

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

      9. lower-/.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. pow2 N/A

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

      14. lift-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. pow2 N/A

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

      18. lift-*.f32 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in alphay around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 93.7

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

   7. Applied rewrites 93.7%

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

   if -0.0320000015 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 57.8%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0 + 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq -0.03200000151991844:\\ \;\;\;\;\frac{-t\_1}{\frac{\left(alphay \cdot alphay\right) \cdot t\_0 + sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0 + 1\right) \cdot u0}{t\_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax))) (t_1 (log (- 1.0 u0))))
   (if (<= t_1 -0.03200000151991844)
     (/ (- t_1) (/ (+ (* (* alphay alphay) t_0) sin2phi) (* alphay alphay)))
     (/
      (* (+ (* (+ (* (+ (* 0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0) u0)
      (+ t_0 (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = logf((1.0f - u0));
	float tmp;
	if (t_1 <= -0.03200000151991844f) {
		tmp = -t_1 / ((((alphay * alphay) * t_0) + sin2phi) / (alphay * alphay));
	} else {
		tmp = (((((((0.25f * u0) + 0.3333333333333333f) * u0) + 0.5f) * u0) + 1.0f) * u0) / (t_0 + (sin2phi / (alphay * alphay)));
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = cos2phi / (alphax * alphax)
    t_1 = log((1.0e0 - u0))
    if (t_1 <= (-0.03200000151991844e0)) then
        tmp = -t_1 / ((((alphay * alphay) * t_0) + sin2phi) / (alphay * alphay))
    else
        tmp = (((((((0.25e0 * u0) + 0.3333333333333333e0) * u0) + 0.5e0) * u0) + 1.0e0) * u0) / (t_0 + (sin2phi / (alphay * alphay)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.03200000151991844))
		tmp = Float32(Float32(-t_1) / Float32(Float32(Float32(Float32(alphay * alphay) * t_0) + sin2phi) / Float32(alphay * alphay)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u0) + Float32(0.3333333333333333)) * u0) + Float32(0.5)) * u0) + Float32(1.0)) * u0) / Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = cos2phi / (alphax * alphax);
	t_1 = log((single(1.0) - u0));
	tmp = single(0.0);
	if (t_1 <= single(-0.03200000151991844))
		tmp = -t_1 / ((((alphay * alphay) * t_0) + sin2phi) / (alphay * alphay));
	else
		tmp = (((((((single(0.25) * u0) + single(0.3333333333333333)) * u0) + single(0.5)) * u0) + single(1.0)) * u0) / (t_0 + (sin2phi / (alphay * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0320000015

   1. Initial program 93.5%

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

   3. Taylor expanded in alphay around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-/.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 93.5

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

   5. Applied rewrites 93.5%

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

   if -0.0320000015 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 57.8%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0 + 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq -0.031199999153614044:\\ \;\;\;\;\frac{-t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0 + 1\right) \cdot u0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0)))
        (t_1 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (if (<= t_0 -0.031199999153614044)
     (/ (- t_0) t_1)
     (/
      (* (+ (* (+ (* (+ (* 0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0) u0)
      t_1))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float t_1 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	float tmp;
	if (t_0 <= -0.031199999153614044f) {
		tmp = -t_0 / t_1;
	} else {
		tmp = (((((((0.25f * u0) + 0.3333333333333333f) * u0) + 0.5f) * u0) + 1.0f) * u0) / t_1;
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = log((1.0e0 - u0))
    t_1 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))
    if (t_0 <= (-0.031199999153614044e0)) then
        tmp = -t_0 / t_1
    else
        tmp = (((((((0.25e0 * u0) + 0.3333333333333333e0) * u0) + 0.5e0) * u0) + 1.0e0) * u0) / t_1
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.031199999153614044))
		tmp = Float32(Float32(-t_0) / t_1);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u0) + Float32(0.3333333333333333)) * u0) + Float32(0.5)) * u0) + Float32(1.0)) * u0) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log((single(1.0) - u0));
	t_1 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	tmp = single(0.0);
	if (t_0 <= single(-0.031199999153614044))
		tmp = -t_0 / t_1;
	else
		tmp = (((((((single(0.25) * u0) + single(0.3333333333333333)) * u0) + single(0.5)) * u0) + single(1.0)) * u0) / t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0311999992

   1. Initial program 93.5%

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

   if -0.0311999992 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 57.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0 + 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\\ \frac{\frac{{t\_0}^{3} + 1}{t\_0 \cdot \left(\left(0.3333333333333333 \cdot u0 + 0.5\right) \cdot u0\right) + \left(1 - t\_0\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* (+ (* (+ (* 0.25 u0) 0.3333333333333333) u0) 0.5) u0)))
   (/
    (*
     (/
      (+ (pow t_0 3.0) 1.0)
      (+ (* t_0 (* (+ (* 0.3333333333333333 u0) 0.5) u0)) (- 1.0 t_0)))
     u0)
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = ((((0.25f * u0) + 0.3333333333333333f) * u0) + 0.5f) * u0;
	return (((powf(t_0, 3.0f) + 1.0f) / ((t_0 * (((0.3333333333333333f * u0) + 0.5f) * u0)) + (1.0f - t_0))) * 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
    real(4) :: t_0
    t_0 = ((((0.25e0 * u0) + 0.3333333333333333e0) * u0) + 0.5e0) * u0
    code = ((((t_0 ** 3.0e0) + 1.0e0) / ((t_0 * (((0.3333333333333333e0 * u0) + 0.5e0) * u0)) + (1.0e0 - t_0))) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u0) + Float32(0.3333333333333333)) * u0) + Float32(0.5)) * u0)
	return Float32(Float32(Float32(Float32((t_0 ^ Float32(3.0)) + Float32(1.0)) / Float32(Float32(t_0 * Float32(Float32(Float32(Float32(0.3333333333333333) * u0) + Float32(0.5)) * u0)) + Float32(Float32(1.0) - t_0))) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = ((((single(0.25) * u0) + single(0.3333333333333333)) * u0) + single(0.5)) * u0;
	tmp = ((((t_0 ^ single(3.0)) + single(1.0)) / ((t_0 * (((single(0.3333333333333333) * u0) + single(0.5)) * u0)) + (single(1.0) - t_0))) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 93.8

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

5. Applied rewrites 93.8%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. flip3-+ N/A

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

   8. lower-/.f32 N/A

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

7. Applied rewrites 93.7%

   \[\leadsto \frac{\frac{{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right)}^{3} + 1}{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right) \cdot \left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right) + \left(1 - \left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right) \cdot 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 94.6%

    \[\leadsto \frac{\frac{{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right)}^{3} + 1}{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right) \cdot \left(\left(0.3333333333333333 \cdot u0 + 0.5\right) \cdot u0\right) + \left(1 - \left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right) \cdot 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

11. Final simplification 94.6%

    \[\leadsto \frac{\frac{{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right)}^{3} + 1}{\left(\left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right) \cdot \left(\left(0.3333333333333333 \cdot u0 + 0.5\right) \cdot u0\right) + \left(1 - \left(\left(0.25 \cdot u0 + 0.3333333333333333\right) \cdot u0 + 0.5\right) \cdot u0\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
12. Add Preprocessing

Alternative 5: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 100000:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 100000.0)
     (/ (* (+ (* 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 <= 100000.0f) {
		tmp = (((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;
}

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 <= 100000.0e0) then
        tmp = (((0.5e0 * u0) + 1.0e0) * 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(100000.0))
		tmp = Float32(Float32(Float32(Float32(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

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(100000.0))
		tmp = (((single(0.5) * u0) + single(1.0)) * 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)) < 1e5

   1. Initial program 56.6%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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-+.f32 N/A

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

      5. lower-*.f32 88.7

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

   5. Applied rewrites 88.7%

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

   if 1e5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 68.3%

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

   3. Taylor expanded in alphax around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{sin2phi} \]
   7. 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 94.9

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

   8. Applied rewrites 94.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 100000:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (((((((0.25e0 * u0) + 0.3333333333333333e0) * u0) + 0.5e0) * u0) + 1.0e0) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 93.8

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

5. Applied rewrites 93.8%

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

Alternative 7: 84.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + t\_0} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.0)
     (* (/ 1.0 (+ (/ cos2phi (* alphax alphax)) t_0)) u0)
     (/
      (*
       (* 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 <= 2.0f) {
		tmp = (1.0f / ((cos2phi / (alphax * alphax)) + t_0)) * u0;
	} 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 <= 2.0e0) then
        tmp = (1.0e0 / ((cos2phi / (alphax * alphax)) + t_0)) * u0
    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(2.0))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0)) * u0);
	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(2.0))
		tmp = (single(1.0) / ((cos2phi / (alphax * alphax)) + t_0)) * u0;
	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)) < 2

   1. Initial program 56.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      3. pow2 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      6. pow2 N/A

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

      7. lift-/.f32 N/A

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

      8. lift-*.f32 74.7

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

   8. Applied rewrites 74.7%

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

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

   1. Initial program 67.0%

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

   3. Taylor expanded in alphax around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{sin2phi} \]
   7. 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 94.1

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

   8. Applied rewrites 94.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2:\\ \;\;\;\;\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + t\_0} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.0)
     (* (/ 1.0 (+ (/ cos2phi (* alphax alphax)) t_0)) u0)
     (*
      (/
       (*
        (* alphay alphay)
        (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* 0.25 u0)))))))
       sin2phi)
      u0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = (1.0f / ((cos2phi / (alphax * alphax)) + t_0)) * u0;
	} else {
		tmp = (((alphay * alphay) * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * u0))))))) / sin2phi) * u0;
	}
	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 <= 2.0e0) then
        tmp = (1.0e0 / ((cos2phi / (alphax * alphax)) + t_0)) * u0
    else
        tmp = (((alphay * alphay) * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (0.25e0 * u0))))))) / sin2phi) * u0
    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(2.0))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0)) * u0);
	else
		tmp = Float32(Float32(Float32(Float32(alphay * alphay) * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u0))))))) / sin2phi) * u0);
	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(2.0))
		tmp = (single(1.0) / ((cos2phi / (alphax * alphax)) + t_0)) * u0;
	else
		tmp = (((alphay * alphay) * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (single(0.25) * u0))))))) / sin2phi) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      3. pow2 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}} \cdot u0 \]

      6. pow2 N/A

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

      7. lift-/.f32 N/A

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

      8. lift-*.f32 74.7

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

   8. Applied rewrites 74.7%

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

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

   1. Initial program 67.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in alphay around 0

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lower-+.f32 N/A

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in sin2phi around 0

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

       1. lower-/.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. pow2 N/A

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

       4. lift-*.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lower-+.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-+.f32 N/A

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

       10. lift-*.f32 94.1

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

   11. Applied rewrites 94.1%

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

Alternative 9: 84.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (*
      (/
       (*
        (* alphay alphay)
        (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* 0.25 u0)))))))
       sin2phi)
      u0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (((alphay * alphay) * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (0.25f * u0))))))) / sin2phi) * u0;
	}
	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 <= 2.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = (((alphay * alphay) * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (0.25e0 * u0))))))) / sin2phi) * u0
    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(2.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(Float32(alphay * alphay) * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u0))))))) / sin2phi) * u0);
	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(2.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (((alphay * alphay) * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (single(0.25) * u0))))))) / sin2phi) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.7%

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

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

   1. Initial program 67.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in alphay around 0

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lower-+.f32 N/A

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in sin2phi around 0

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

       1. lower-/.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. pow2 N/A

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

       4. lift-*.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lower-+.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-+.f32 N/A

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

       10. lift-*.f32 94.1

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

   11. Applied rewrites 94.1%

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

Alternative 10: 83.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.0)
     (/ 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 <= 2.0f) {
		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 <= 2.0e0) 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(2.0))
		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(2.0))
		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)) < 2

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.7%

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

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

   1. Initial program 67.0%

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

   3. Taylor expanded in alphax around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}{sin2phi} \]
   7. 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 92.2

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

   8. Applied rewrites 92.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (((((0.3333333333333333e0 * u0) + 0.5e0) * u0) + 1.0e0) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 91.9

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

5. Applied rewrites 91.9%

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

Alternative 12: 78.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (/ (* (+ (* 0.5 u0) 1.0) u0) (/ cos2phi (* alphax alphax)))
   (/
    (*
     (* 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 <= 4.9999998413276127e-20f) {
		tmp = (((0.5f * u0) + 1.0f) * u0) / (cos2phi / (alphax * alphax));
	} 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) :: tmp
    if (sin2phi <= 4.9999998413276127e-20) then
        tmp = (((0.5e0 * u0) + 1.0e0) * u0) / (cos2phi / (alphax * alphax))
    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)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(Float32(Float32(Float32(0.5) * u0) + Float32(1.0)) * u0) / Float32(cos2phi / Float32(alphax * alphax)));
	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)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = (((single(0.5) * u0) + single(1.0)) * u0) / (cos2phi / (alphax * alphax));
	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 sin2phi < 4.99999984e-20

   1. Initial program 60.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in alphax around 0

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

       1. pow2 N/A

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

       2. lift-/.f32 N/A

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

       3. lift-*.f32 63.5

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

   11. Applied rewrites 63.5%

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

   if 4.99999984e-20 < sin2phi

   1. Initial program 63.5%

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

   3. Taylor expanded in alphax around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}{sin2phi} \]
   7. 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 86.0

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

   8. Applied rewrites 86.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(0.5 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right) + alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (/ (* (+ (* 0.5 u0) 1.0) u0) (/ cos2phi (* alphax alphax)))
   (/ (* u0 (+ (* 0.5 (* (* alphay alphay) u0)) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.9999998413276127e-20f) {
		tmp = (((0.5f * u0) + 1.0f) * u0) / (cos2phi / (alphax * alphax));
	} else {
		tmp = (u0 * ((0.5f * ((alphay * alphay) * u0)) + (alphay * alphay))) / 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 <= 4.9999998413276127e-20) then
        tmp = (((0.5e0 * u0) + 1.0e0) * u0) / (cos2phi / (alphax * alphax))
    else
        tmp = (u0 * ((0.5e0 * ((alphay * alphay) * u0)) + (alphay * alphay))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(Float32(Float32(Float32(0.5) * u0) + Float32(1.0)) * u0) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(Float32(0.5) * Float32(Float32(alphay * alphay) * u0)) + Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = (((single(0.5) * u0) + single(1.0)) * u0) / (cos2phi / (alphax * alphax));
	else
		tmp = (u0 * ((single(0.5) * ((alphay * alphay) * u0)) + (alphay * alphay))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999984e-20

   1. Initial program 60.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in alphax around 0

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

       1. pow2 N/A

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

       2. lift-/.f32 N/A

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

       3. lift-*.f32 63.5

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

   11. Applied rewrites 63.5%

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

   if 4.99999984e-20 < sin2phi

   1. Initial program 63.5%

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

   3. Taylor expanded in alphax around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 82.5

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

   8. Applied rewrites 82.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(0.5 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right) + alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (/ (* (+ (* 0.5 u0) 1.0) u0) (/ cos2phi (* alphax alphax)))
   (/ (* (* 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 <= 4.9999998413276127e-20f) {
		tmp = (((0.5f * u0) + 1.0f) * u0) / (cos2phi / (alphax * alphax));
	} 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 <= 4.9999998413276127e-20) then
        tmp = (((0.5e0 * u0) + 1.0e0) * u0) / (cos2phi / (alphax * alphax))
    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 (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(Float32(Float32(Float32(0.5) * u0) + Float32(1.0)) * u0) / Float32(cos2phi / Float32(alphax * alphax)));
	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 <= single(4.9999998413276127e-20))
		tmp = (((single(0.5) * u0) + single(1.0)) * u0) / (cos2phi / (alphax * alphax));
	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 sin2phi < 4.99999984e-20

   1. Initial program 60.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in alphax around 0

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

       1. pow2 N/A

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

       2. lift-/.f32 N/A

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

       3. lift-*.f32 63.5

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

   11. Applied rewrites 63.5%

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

   if 4.99999984e-20 < sin2phi

   1. Initial program 63.5%

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

   3. Taylor expanded in alphax around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}{sin2phi} \]
   7. 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 82.4

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

   8. Applied rewrites 82.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(0.5 \cdot u0 + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (/ (* (* 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 <= 4.9999998413276127e-20f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} 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 <= 4.9999998413276127e-20) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    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 (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	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 <= single(4.9999998413276127e-20))
		tmp = u0 / (cos2phi / (alphax * alphax));
	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 sin2phi < 4.99999984e-20

   1. Initial program 60.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 72.5%

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

   6. Taylor expanded in alphax around 0

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 56.3

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

   8. Applied rewrites 56.3%

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

   if 4.99999984e-20 < sin2phi

   1. Initial program 63.5%

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

   3. Taylor expanded in alphax around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}{sin2phi} \]
   7. 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 82.4

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

   8. Applied rewrites 82.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* (/ (* alphay alphay) sin2phi) u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.9999998413276127e-20f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = ((alphay * alphay) / sin2phi) * u0;
	}
	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 <= 4.9999998413276127e-20) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = ((alphay * alphay) / sin2phi) * u0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = ((alphay * alphay) / sin2phi) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999984e-20

   1. Initial program 60.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 72.5%

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

   6. Taylor expanded in alphax around 0

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 56.3

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

   8. Applied rewrites 56.3%

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

   if 4.99999984e-20 < sin2phi

   1. Initial program 63.5%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in alphax around inf

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

      1. lower-+.f32 N/A

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

   8. Applied rewrites 87.8%

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

   9. Taylor expanded in u0 around 0

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

       1. pow2 N/A

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

       2. lift-/.f32 N/A

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

       3. lift-*.f32 72.0

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

   11. Applied rewrites 72.0%

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

Alternative 17: 59.2% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

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 = ((alphay * alphay) / sin2phi) * u0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 93.7%

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

6. Taylor expanded in alphax around inf

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

   1. lower-+.f32 N/A

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

8. Applied rewrites 74.9%

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

9. Taylor expanded in u0 around 0

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

    1. pow2 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-*.f32 61.7

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

11. Applied rewrites 61.7%

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

Reproduce

herbie shell --seed 2025058 
(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)))))