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

Percentage Accurate: 60.8% → 98.4%

Time: 7.4s

Alternatives: 18

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 65.2%

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

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

   11. lower-/.f32 N/A

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

   12. pow2 N/A

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

   13. lift-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. pow2 N/A

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

   17. lift-*.f32 65.1

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

4. Applied rewrites 65.1%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 98.1

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

6. Applied rewrites 98.1%

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

7. Taylor expanded in alphax around inf

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

   1. pow2 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. pow2 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-+.f32 98.4

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

9. Applied rewrites 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 93.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 93.1

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

5. Applied rewrites 93.1%

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

6. Taylor expanded in u0 around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 93.1

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

8. Applied rewrites 93.1%

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

Alternative 3: 83.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi + \frac{\left(alphax \cdot alphax\right) \cdot sin2phi}{alphay \cdot alphay}}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 7.999999773744548e-9)
     (/
      u0
      (/
       (+ cos2phi (/ (* (* alphax alphax) sin2phi) (* alphay alphay)))
       (* alphax alphax)))
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 7.999999773744548e-9f) {
		tmp = u0 / ((cos2phi + (((alphax * alphax) * sin2phi) / (alphay * alphay))) / (alphax * alphax));
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(7.999999773744548e-9))
		tmp = Float32(u0 / Float32(Float32(cos2phi + Float32(Float32(Float32(alphax * alphax) * sin2phi) / Float32(alphay * alphay))) / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 7.99999977e-9

   1. Initial program 57.9%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in alphax around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 72.4

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

   8. Applied rewrites 72.4%

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

   if 7.99999977e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 68.7%

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

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 89.3

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

   8. Applied rewrites 89.3%

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

Alternative 4: 83.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 7.999999773744548e-9)
     (/
      (* alphax (* alphay u0))
      (+ (/ (* alphax sin2phi) alphay) (/ (* alphay cos2phi) alphax)))
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 7.999999773744548e-9f) {
		tmp = (alphax * (alphay * u0)) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax));
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(7.999999773744548e-9))
		tmp = Float32(Float32(alphax * Float32(alphay * u0)) / Float32(Float32(Float32(alphax * sin2phi) / alphay) + Float32(Float32(alphay * cos2phi) / alphax)));
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 7.99999977e-9

   1. Initial program 57.9%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 57.8

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 72.3

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

   7. Applied rewrites 72.3%

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

   if 7.99999977e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 68.7%

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

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 89.3

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

   8. Applied rewrites 89.3%

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

Alternative 5: 83.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + t\_0} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 7.999999773744548e-9)
     (* (/ 1.0 (+ (/ cos2phi (* alphax alphax)) t_0)) u0)
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 7.999999773744548e-9f) {
		tmp = (1.0f / ((cos2phi / (alphax * alphax)) + t_0)) * u0;
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(7.999999773744548e-9))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0)) * u0);
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 7.99999977e-9

   1. Initial program 57.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 92.4%

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

   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 72.2

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

   8. Applied rewrites 72.2%

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

   if 7.99999977e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 68.7%

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

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 89.3

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

   8. Applied rewrites 89.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 7.999999773744548e-9)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 7.999999773744548e-9f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(7.999999773744548e-9))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 7.99999977e-9

   1. Initial program 57.9%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 72.2%

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

   if 7.99999977e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 68.7%

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

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 89.3

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

   8. Applied rewrites 89.3%

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

Alternative 7: 93.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 93.1

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

5. Applied rewrites 93.1%

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

Alternative 8: 80.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 7.999999773744548e-9)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) 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 <= 7.999999773744548e-9f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(7.999999773744548e-9))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 7.99999977e-9

   1. Initial program 57.9%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 72.2%

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

   if 7.99999977e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 68.7%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 84.2

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

   8. Applied rewrites 84.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (/
    (*
     (* alphax alphax)
     (* u0 (- (* u0 (- (* -0.3333333333333333 u0) 0.5)) 1.0)))
    (- cos2phi))
   (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = ((alphax * alphax) * (u0 * ((u0 * ((-0.3333333333333333f * u0) - 0.5f)) - 1.0f))) / -cos2phi;
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.3333333333333333) * u0) - Float32(0.5))) - Float32(1.0)))) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 59.4%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 59.3

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

   4. Applied rewrites 59.3%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in alphax around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 73.3

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

   10. Applied rewrites 73.3%

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

   11. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 68.3

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

   13. Applied rewrites 68.3%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 80.8

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

   8. Applied rewrites 80.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 90.9

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

5. Applied rewrites 90.9%

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

Alternative 11: 76.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right)}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (* (/ (fma 0.5 (* (* alphax alphax) u0) (* alphax alphax)) cos2phi) u0)
   (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = (fmaf(0.5f, ((alphax * alphax) * u0), (alphax * alphax)) / cos2phi) * u0;
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(fma(Float32(0.5), Float32(Float32(alphax * alphax) * u0), Float32(alphax * alphax)) / cos2phi) * u0);
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 59.4%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 64.6

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

   8. Applied rewrites 64.6%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 80.8

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

   8. Applied rewrites 80.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right)}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (/ (* (* alphax alphax) (* u0 (- (* -0.5 u0) 1.0))) (- cos2phi))
   (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = ((alphax * alphax) * (u0 * ((-0.5f * u0) - 1.0f))) / -cos2phi;
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 59.4%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 59.3

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

   4. Applied rewrites 59.3%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in alphax around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 73.3

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

   10. Applied rewrites 73.3%

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

   11. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 64.5

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

   13. Applied rewrites 64.5%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 80.8

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

   8. Applied rewrites 80.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (/ (* (* alphax alphax) (* u0 (- (* -0.5 u0) 1.0))) (- cos2phi))
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = ((alphax * alphax) * (u0 * ((-0.5f * u0) - 1.0f))) / -cos2phi;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14) then
        tmp = ((alphax * alphax) * (u0 * (((-0.5e0) * u0) - 1.0e0))) / -cos2phi
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.99999996490334e-14))
		tmp = ((alphax * alphax) * (u0 * ((single(-0.5) * u0) - single(1.0)))) / -cos2phi;
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 59.4%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 59.3

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

   4. Applied rewrites 59.3%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in alphax around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 73.3

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

   10. Applied rewrites 73.3%

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

   11. Taylor expanded in u0 around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 64.5

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

   13. Applied rewrites 64.5%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 67.0

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in alphax around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. pow2 N/A

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

      4. lift-*.f32 70.6

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

   10. Applied rewrites 70.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 86.0

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

5. Applied rewrites 86.0%

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

Alternative 15: 67.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.99999996490334e-14))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 59.4%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 71.1%

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

   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 55.4

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

   8. Applied rewrites 55.4%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 67.0

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in alphax around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. pow2 N/A

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

      4. lift-*.f32 70.6

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

   10. Applied rewrites 70.6%

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

Alternative 16: 67.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot alphax}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (* (/ (* alphax alphax) cos2phi) u0)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = ((alphax * alphax) / cos2phi) * u0;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14) then
        tmp = ((alphax * alphax) / cos2phi) * u0
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) / cos2phi) * u0);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.99999996490334e-14))
		tmp = ((alphax * alphax) / cos2phi) * u0;
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 59.4%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 92.2%

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

   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 71.1

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in alphax around 0

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

       1. lower-/.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 55.4

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

   11. Applied rewrites 55.4%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 67.0

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in alphax around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. pow2 N/A

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

      4. lift-*.f32 70.6

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

   10. Applied rewrites 70.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot alphax}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot alphax}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (* (/ (* alphax alphax) cos2phi) u0)
   (* (/ (* alphay alphay) sin2phi) u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = ((alphax * alphax) / cos2phi) * u0;
	} 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 / (alphay * alphay)) <= 1.99999996490334e-14) then
        tmp = ((alphax * alphax) / cos2phi) * u0
    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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) / cos2phi) * u0);
	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 / (alphay * alphay)) <= single(1.99999996490334e-14))
		tmp = ((alphax * alphax) / cos2phi) * u0;
	else
		tmp = ((alphay * alphay) / sin2phi) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 59.4%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 92.2%

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

   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 71.1

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in alphax around 0

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

       1. lower-/.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 55.4

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

   11. Applied rewrites 55.4%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 90.5%

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

   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 74.2

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in alphax around inf

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

       1. lower-/.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 70.6

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

   11. Applied rewrites 70.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot alphax}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 23.7% accurate, 6.9× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((alphax * alphax) / cos2phi) * 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 = ((alphax * alphax) / cos2phi) * u0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 90.9%

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

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 73.4

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

8. Applied rewrites 73.4%

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

9. Taylor expanded in alphax around 0

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

    1. lower-/.f32 N/A

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

    2. pow2 N/A

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

    3. lift-*.f32 22.2

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

11. Applied rewrites 22.2%

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

12. Final simplification 22.2%

    \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025084 
(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)))))