math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 1.8s

Alternatives: 8

Speedup: 1.0×

• 74.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[e^{re} \cdot \cos im \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(im);
}

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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[e^{re} \cdot \cos im \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(im);
}

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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Alternative 1: 70.6% accurate, 1.3× speedup

Math

\[\left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \left(1 + re \cdot \left(\frac{1}{4} \cdot re - \frac{1}{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (+
  1
  (*
   re
   (*
    (+ 1 (* 1/6 (* (* re re) (+ 1 (* re (- (* 1/4 re) 1/2))))))
    (- (* 1/2 re) -1))))
 (cos im)))

C

double code(double re, double im) {
	return (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (1.0 + (re * ((0.25 * re) - 0.5)))))) * ((0.5 * re) - -1.0)))) * cos(im);
}

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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (1.0d0 + (re * ((1.0d0 + (0.16666666666666666d0 * ((re * re) * (1.0d0 + (re * ((0.25d0 * re) - 0.5d0)))))) * ((0.5d0 * re) - (-1.0d0))))) * cos(im)
end function

Java

public static double code(double re, double im) {
	return (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (1.0 + (re * ((0.25 * re) - 0.5)))))) * ((0.5 * re) - -1.0)))) * Math.cos(im);
}

Python

def code(re, im):
	return (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (1.0 + (re * ((0.25 * re) - 0.5)))))) * ((0.5 * re) - -1.0)))) * math.cos(im)

Julia

function code(re, im)
	return Float64(Float64(1.0 + Float64(re * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(Float64(re * re) * Float64(1.0 + Float64(re * Float64(Float64(0.25 * re) - 0.5)))))) * Float64(Float64(0.5 * re) - -1.0)))) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (1.0 + (re * ((0.25 * re) - 0.5)))))) * ((0.5 * re) - -1.0)))) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[(1 + N[(re * N[(N[(1 + N[(1/6 * N[(N[(re * re), $MachinePrecision] * N[(1 + N[(re * N[(N[(1/4 * re), $MachinePrecision] - 1/2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1/2 * re), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \cdot \cos im \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \cdot \cos im \]

   3. lower-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   4. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   5. lower-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6} \cdot re}\right)\right)\right) \cdot \cos im \]

   6. lower-*.f64 66.6%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

4. Applied rewrites 66.6%

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   3. lift-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6} \cdot re}\right)\right)\right) \cdot \cos im \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \left(\frac{1}{2} \cdot re + \color{blue}{\left(\frac{1}{6} \cdot re\right) \cdot re}\right)\right)\right) \cdot \cos im \]

   5. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \left(\frac{1}{2} \cdot re + \color{blue}{\left(\frac{1}{6} \cdot re\right)} \cdot re\right)\right)\right) \cdot \cos im \]

   6. associate-+r+ N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{2} \cdot re\right) + \color{blue}{\left(\frac{1}{6} \cdot re\right) \cdot re}\right)\right) \cdot \cos im \]

   7. lift-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{2} \cdot re\right) + \color{blue}{\left(\frac{1}{6} \cdot re\right)} \cdot re\right)\right) \cdot \cos im \]

   8. sum-to-mult N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \cdot \cos im \]

   9. lower-unsound-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \cdot \cos im \]

   10. lower-unsound-+.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(\color{blue}{1} + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   11. lower-unsound-/.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   12. lower-*.f64 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   13. lift-+.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   14. +-commutative N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re + 1}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   15. add-flip N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - \left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   16. lower--.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - \left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   17. metadata-eval 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   18. lift-+.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot re}\right)\right)\right) \cdot \cos im \]

   19. +-commutative N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re + \color{blue}{1}\right)\right)\right) \cdot \cos im \]

   20. add-flip N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \cdot \cos im \]

   21. lower--.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \cdot \cos im \]

   22. metadata-eval 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

6. Applied rewrites 66.6%

   \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot re - -1\right)}\right)\right) \cdot \cos im \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   2. mult-flip N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   3. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   4. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   5. associate-*l* N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\frac{1}{6} \cdot \left(re \cdot re\right)\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   6. associate-*l* N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   7. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   8. lower-*.f32 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   9. lower-unsound-*.f32 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   10. lower-*.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   11. lower-unsound-*.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   12. frac-2neg N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   13. metadata-eval N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   14. lower-/.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   15. lift--.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   16. sub-negate-rev N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1 - \frac{1}{2} \cdot re}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   17. lower--.f64 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1 - \frac{1}{2} \cdot re}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

8. Applied rewrites 66.6%

   \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1 - \frac{1}{2} \cdot re}\right)\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

9. Taylor expanded in re around 0

   \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \left(1 + re \cdot \left(\frac{1}{4} \cdot re - \frac{1}{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]
10. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \left(1 + re \cdot \left(\frac{1}{4} \cdot re - \frac{1}{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

    2. lower-*.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \left(1 + re \cdot \left(\frac{1}{4} \cdot re - \frac{1}{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

    3. lower--.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \left(1 + re \cdot \left(\frac{1}{4} \cdot re - \frac{1}{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

    4. lower-*.f64 70.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \left(1 + re \cdot \left(\frac{1}{4} \cdot re - \frac{1}{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

11. Applied rewrites 70.6%

    \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \left(1 + re \cdot \left(\frac{1}{4} \cdot re - \frac{1}{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]
12. Add Preprocessing

Alternative 2: 68.6% accurate, 1.3× speedup

Math

\[\left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (+
  1
  (* re (* (+ 1 (* 1/6 (* (* re re) (/ -1 -1)))) (- (* 1/2 re) -1))))
 (cos im)))

C

double code(double re, double im) {
	return (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (-1.0 / -1.0)))) * ((0.5 * re) - -1.0)))) * cos(im);
}

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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (1.0d0 + (re * ((1.0d0 + (0.16666666666666666d0 * ((re * re) * ((-1.0d0) / (-1.0d0))))) * ((0.5d0 * re) - (-1.0d0))))) * cos(im)
end function

Java

public static double code(double re, double im) {
	return (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (-1.0 / -1.0)))) * ((0.5 * re) - -1.0)))) * Math.cos(im);
}

Python

def code(re, im):
	return (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (-1.0 / -1.0)))) * ((0.5 * re) - -1.0)))) * math.cos(im)

Julia

function code(re, im)
	return Float64(Float64(1.0 + Float64(re * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(Float64(re * re) * Float64(-1.0 / -1.0)))) * Float64(Float64(0.5 * re) - -1.0)))) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (1.0 + (re * ((1.0 + (0.16666666666666666 * ((re * re) * (-1.0 / -1.0)))) * ((0.5 * re) - -1.0)))) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[(1 + N[(re * N[(N[(1 + N[(1/6 * N[(N[(re * re), $MachinePrecision] * N[(-1 / -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1/2 * re), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \cdot \cos im \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \cdot \cos im \]

   3. lower-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   4. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   5. lower-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6} \cdot re}\right)\right)\right) \cdot \cos im \]

   6. lower-*.f64 66.6%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

4. Applied rewrites 66.6%

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   3. lift-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6} \cdot re}\right)\right)\right) \cdot \cos im \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \left(\frac{1}{2} \cdot re + \color{blue}{\left(\frac{1}{6} \cdot re\right) \cdot re}\right)\right)\right) \cdot \cos im \]

   5. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \left(\frac{1}{2} \cdot re + \color{blue}{\left(\frac{1}{6} \cdot re\right)} \cdot re\right)\right)\right) \cdot \cos im \]

   6. associate-+r+ N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{2} \cdot re\right) + \color{blue}{\left(\frac{1}{6} \cdot re\right) \cdot re}\right)\right) \cdot \cos im \]

   7. lift-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{2} \cdot re\right) + \color{blue}{\left(\frac{1}{6} \cdot re\right)} \cdot re\right)\right) \cdot \cos im \]

   8. sum-to-mult N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \cdot \cos im \]

   9. lower-unsound-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \cdot \cos im \]

   10. lower-unsound-+.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(\color{blue}{1} + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   11. lower-unsound-/.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   12. lower-*.f64 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   13. lift-+.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{1 + \frac{1}{2} \cdot re}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   14. +-commutative N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re + 1}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   15. add-flip N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - \left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   16. lower--.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - \left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   17. metadata-eval 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right) \cdot \cos im \]

   18. lift-+.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot re}\right)\right)\right) \cdot \cos im \]

   19. +-commutative N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re + \color{blue}{1}\right)\right)\right) \cdot \cos im \]

   20. add-flip N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \cdot \cos im \]

   21. lower--.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \cdot \cos im \]

   22. metadata-eval 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

6. Applied rewrites 66.6%

   \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot re - -1\right)}\right)\right) \cdot \cos im \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{\left(\frac{1}{6} \cdot re\right) \cdot re}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   2. mult-flip N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   3. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   4. lift-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   5. associate-*l* N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \left(\frac{1}{6} \cdot \left(re \cdot re\right)\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   6. associate-*l* N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   7. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

   8. lower-*.f32 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   9. lower-unsound-*.f32 N/A

      \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   10. lower-*.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   11. lower-unsound-*.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{\frac{1}{2} \cdot re - -1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   12. frac-2neg N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   13. metadata-eval N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   14. lower-/.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   15. lift--.f64 N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot re - -1\right)\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   16. sub-negate-rev N/A

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1 - \frac{1}{2} \cdot re}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

   17. lower--.f64 66.6%

       \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1 - \frac{1}{2} \cdot re}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]

8. Applied rewrites 66.6%

   \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1 - \frac{1}{2} \cdot re}\right)\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{re} - -1\right)\right)\right) \cdot \cos im \]

9. Taylor expanded in re around 0

   \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]
10. Step-by-step derivation

11. Applied rewrites 68.6%

    \[\leadsto \left(1 + re \cdot \left(\left(1 + \frac{1}{6} \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{-1}\right)\right) \cdot \left(\frac{1}{2} \cdot re - -1\right)\right)\right) \cdot \cos im \]
12. Add Preprocessing

Alternative 3: 66.6% accurate, 1.6× speedup

Math

\[\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) \cdot \cos im \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (+ 1 (* re (+ 1 (* re (+ 1/2 (* 1/6 re)))))) (cos im)))

C

double code(double re, double im) {
	return (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * cos(im);
}

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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (0.16666666666666666d0 * re)))))) * cos(im)
end function

Java

public static double code(double re, double im) {
	return (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * Math.cos(im);
}

Python

def code(re, im):
	return (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * math.cos(im)

Julia

function code(re, im)
	return Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(0.16666666666666666 * re)))))) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[(1 + N[(re * N[(1 + N[(re * N[(1/2 + N[(1/6 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \cdot \cos im \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \cdot \cos im \]

   3. lower-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   4. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \cdot \cos im \]

   5. lower-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6} \cdot re}\right)\right)\right) \cdot \cos im \]

   6. lower-*.f64 66.6%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

4. Applied rewrites 66.6%

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
5. Add Preprocessing

Alternative 4: 62.9% accurate, 1.7× speedup

Math

\[\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right) \cdot \cos im \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (+ 1 (* re (+ 1 (* 1/2 re)))) (cos im)))

C

double code(double re, double im) {
	return (1.0 + (re * (1.0 + (0.5 * re)))) * cos(im);
}

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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (1.0d0 + (re * (1.0d0 + (0.5d0 * re)))) * cos(im)
end function

Java

public static double code(double re, double im) {
	return (1.0 + (re * (1.0 + (0.5 * re)))) * Math.cos(im);
}

Python

def code(re, im):
	return (1.0 + (re * (1.0 + (0.5 * re)))) * math.cos(im)

Julia

function code(re, im)
	return Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(0.5 * re)))) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (1.0 + (re * (1.0 + (0.5 * re)))) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[(1 + N[(re * N[(1 + N[(1/2 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot re\right)}\right) \cdot \cos im \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + re \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right) \cdot \cos im \]

   3. lower-+.f64 N/A

      \[\leadsto \left(1 + re \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot re}\right)\right) \cdot \cos im \]

   4. lower-*.f64 62.9%

      \[\leadsto \left(1 + re \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

4. Applied rewrites 62.9%

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) (- INFINITY))
  (* (- re -1) (- (* (* im im) -1/2) -1))
  (* (+ 1 re) (cos im))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -((double) INFINITY)) {
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0);
	} else {
		tmp = (1.0 + re) * cos(im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= -Double.POSITIVE_INFINITY) {
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0);
	} else {
		tmp = (1.0 + re) * Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= -math.inf:
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0)
	else:
		tmp = (1.0 + re) * math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= Float64(-Inf))
		tmp = Float64(Float64(re - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	else
		tmp = Float64(Float64(1.0 + re) * cos(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= -Inf)
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0);
	else
		tmp = (1.0 + re) * cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(re - -1), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -1/2), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision], N[(N[(1 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
   3. Step-by-step derivation

      1. lower-+.f64 50.8%

         \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \cos im \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}}\right) \]

      3. lower-pow.f64 31.3%

         \[\leadsto \left(1 + re\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 31.3%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re + \color{blue}{1}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      3. add-flip N/A

         \[\leadsto \left(re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      5. lower--.f64 31.3%

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(\frac{-1}{2} \cdot {im}^{2} + 1\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(add-flip, \left(\frac{-1}{2} \cdot {im}^{2} - \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\frac{-1}{2} \cdot {im}^{2} - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{-1}{2} \cdot {im}^{2} - -1\right)\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{-1}{2} \cdot {im}^{2}\right)\right) - -1\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(*-commutative, \left({im}^{2} \cdot \frac{-1}{2}\right)\right) - -1\right) \]

      13. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lower-*.f64, \left({im}^{2} \cdot \frac{-1}{2}\right)\right) - -1\right) \]

      14. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-pow.f64, \left({im}^{2}\right)\right) \cdot \frac{-1}{2} - -1\right) \]

      15. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(unpow2, \left(im \cdot im\right)\right) \cdot \frac{-1}{2} - -1\right) \]

      16. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lower-*.f64, \left(im \cdot im\right)\right) \cdot \frac{-1}{2} - -1\right) \]

   9. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2} - -1\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
   3. Step-by-step derivation

      1. lower-+.f64 50.8%

         \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \cos im \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq 1049999999999999988962860034754329451968323613331891412166240059253280871332490001883104283464506251138225297201529375222126904026096048314827562549248:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2} - -1\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     re
     1049999999999999988962860034754329451968323613331891412166240059253280871332490001883104283464506251138225297201529375222126904026096048314827562549248)
  (cos im)
  (* (- re -1) (- (* (* im im) -1/2) -1))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+150) {
		tmp = cos(im);
	} else {
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0);
	}
	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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.05d+150) then
        tmp = cos(im)
    else
        tmp = (re - (-1.0d0)) * (((im * im) * (-0.5d0)) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+150) {
		tmp = Math.cos(im);
	} else {
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e+150:
		tmp = math.cos(im)
	else:
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e+150)
		tmp = cos(im);
	else
		tmp = Float64(Float64(re - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.05e+150)
		tmp = cos(im);
	else
		tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1049999999999999988962860034754329451968323613331891412166240059253280871332490001883104283464506251138225297201529375222126904026096048314827562549248], N[Cos[im], $MachinePrecision], N[(N[(re - -1), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -1/2), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.05e150

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   3. Step-by-step derivation

      1. lower-cos.f64 49.9%

         \[\leadsto \cos im \]

   4. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\cos im} \]

   if 1.05e150 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
   3. Step-by-step derivation

      1. lower-+.f64 50.8%

         \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \cos im \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}}\right) \]

      3. lower-pow.f64 31.3%

         \[\leadsto \left(1 + re\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 31.3%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re + \color{blue}{1}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      3. add-flip N/A

         \[\leadsto \left(re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      5. lower--.f64 31.3%

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(\frac{-1}{2} \cdot {im}^{2} + 1\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(add-flip, \left(\frac{-1}{2} \cdot {im}^{2} - \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\frac{-1}{2} \cdot {im}^{2} - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{-1}{2} \cdot {im}^{2} - -1\right)\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{-1}{2} \cdot {im}^{2}\right)\right) - -1\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(*-commutative, \left({im}^{2} \cdot \frac{-1}{2}\right)\right) - -1\right) \]

      13. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lower-*.f64, \left({im}^{2} \cdot \frac{-1}{2}\right)\right) - -1\right) \]

      14. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-pow.f64, \left({im}^{2}\right)\right) \cdot \frac{-1}{2} - -1\right) \]

      15. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(unpow2, \left(im \cdot im\right)\right) \cdot \frac{-1}{2} - -1\right) \]

      16. lower--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lower-*.f64, \left(im \cdot im\right)\right) \cdot \frac{-1}{2} - -1\right) \]

   9. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2} - -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 31.3% accurate, 9.4× speedup

Math

\[\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2} - -1\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (- re -1) (- (* (* im im) -1/2) -1)))

C

double code(double re, double im) {
	return (re - -1.0) * (((im * im) * -0.5) - -1.0);
}

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(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (re - (-1.0d0)) * (((im * im) * (-0.5d0)) - (-1.0d0))
end function

Java

public static double code(double re, double im) {
	return (re - -1.0) * (((im * im) * -0.5) - -1.0);
}

Python

def code(re, im):
	return (re - -1.0) * (((im * im) * -0.5) - -1.0)

Julia

function code(re, im)
	return Float64(Float64(re - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (re - -1.0) * (((im * im) * -0.5) - -1.0);
end

Wolfram

code[re_, im_] := N[(N[(re - -1), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -1/2), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
3. Step-by-step derivation

   1. lower-+.f64 50.8%

      \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \cos im \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]

5. Taylor expanded in im around 0

   \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + re\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + re\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}}\right) \]

   3. lower-pow.f64 31.3%

      \[\leadsto \left(1 + re\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}}\right) \]

7. Applied rewrites 31.3%

   \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

   2. +-commutative N/A

      \[\leadsto \left(re + \color{blue}{1}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

   3. add-flip N/A

      \[\leadsto \left(re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(re - -1\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

   5. lower--.f64 31.3%

      \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

   7. lower--.f64 N/A

      \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(\frac{-1}{2} \cdot {im}^{2} + 1\right)\right) \]

   8. lower--.f64 N/A

      \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(add-flip, \left(\frac{-1}{2} \cdot {im}^{2} - \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\frac{-1}{2} \cdot {im}^{2} - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)\right) \]

   10. lower--.f64 N/A

       \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{-1}{2} \cdot {im}^{2} - -1\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{-1}{2} \cdot {im}^{2}\right)\right) - -1\right) \]

   12. lower--.f64 N/A

       \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(*-commutative, \left({im}^{2} \cdot \frac{-1}{2}\right)\right) - -1\right) \]

   13. lower--.f64 N/A

       \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lower-*.f64, \left({im}^{2} \cdot \frac{-1}{2}\right)\right) - -1\right) \]

   14. lower--.f64 N/A

       \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-pow.f64, \left({im}^{2}\right)\right) \cdot \frac{-1}{2} - -1\right) \]

   15. lower--.f64 N/A

       \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(unpow2, \left(im \cdot im\right)\right) \cdot \frac{-1}{2} - -1\right) \]

   16. lower--.f64 N/A

       \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lower-*.f64, \left(im \cdot im\right)\right) \cdot \frac{-1}{2} - -1\right) \]

9. Applied rewrites 31.3%

   \[\leadsto \color{blue}{\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2} - -1\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))