math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 12

Speedup: 1.0×

• 75.5% 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: 70.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\\ \mathbf{if}\;re \leq 10^{+103}:\\ \;\;\;\;\left(1 + \frac{t\_0 \cdot t\_0 - re \cdot re}{t\_0 - re}\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right) \cdot \cos im\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (* (- (* 0.16666666666666666 re) -0.5) re) re)))
  (if (<= re 1e+103)
    (* (+ 1.0 (/ (- (* t_0 t_0) (* re re)) (- t_0 re))) (cos im))
    (*
     (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* 0.16666666666666666 re))))))
     (cos im)))))

C

double code(double re, double im) {
	double t_0 = (((0.16666666666666666 * re) - -0.5) * re) * re;
	double tmp;
	if (re <= 1e+103) {
		tmp = (1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re))) * cos(im);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * cos(im);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (((0.16666666666666666d0 * re) - (-0.5d0)) * re) * re
    if (re <= 1d+103) then
        tmp = (1.0d0 + (((t_0 * t_0) - (re * re)) / (t_0 - re))) * cos(im)
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (0.16666666666666666d0 * re)))))) * cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (((0.16666666666666666 * re) - -0.5) * re) * re;
	double tmp;
	if (re <= 1e+103) {
		tmp = (1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re))) * Math.cos(im);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (((0.16666666666666666 * re) - -0.5) * re) * re
	tmp = 0
	if re <= 1e+103:
		tmp = (1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re))) * math.cos(im)
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * math.cos(im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(Float64(0.16666666666666666 * re) - -0.5) * re) * re)
	tmp = 0.0
	if (re <= 1e+103)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(t_0 * t_0) - Float64(re * re)) / Float64(t_0 - re))) * cos(im));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(0.16666666666666666 * re)))))) * cos(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (((0.16666666666666666 * re) - -0.5) * re) * re;
	tmp = 0.0;
	if (re <= 1e+103)
		tmp = (1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re))) * cos(im);
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - -0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[re, 1e+103], N[(N[(1.0 + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < 1e103

   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 67.2%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. flip-+ N/A

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

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

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

   6. Applied rewrites 54.3%

      \[\leadsto \left(1 + \frac{\left(\left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right) \cdot \left(\left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right) - re \cdot re}{\color{blue}{\left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re - re}}\right) \cdot \cos im \]

   if 1e103 < re

   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 67.2%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 + 0.16666666666666666 \cdot re\\ t_1 := e^{re} \cdot \cos im\\ t_2 := 1 + re \cdot \left(1 + re \cdot t\_0\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot \left(1 + -0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;t\_1 \leq 0.99998:\\ \;\;\;\;t\_2 \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (+ 0.5 (* 0.16666666666666666 re)))
       (t_1 (* (exp re) (cos im)))
       (t_2 (+ 1.0 (* re (+ 1.0 (* re t_0))))))
  (if (<= t_1 (- INFINITY))
    (* t_2 (+ 1.0 (* -0.5 (pow im 2.0))))
    (if (<= t_1 0.99998)
      (* t_2 (cos im))
      (-
       (+ 1.0 re)
       (* -1.0 (* (sqrt (* (* re re) (* re re))) t_0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (0.16666666666666666 * re);
	double t_1 = exp(re) * cos(im);
	double t_2 = 1.0 + (re * (1.0 + (re * t_0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * (1.0 + (-0.5 * pow(im, 2.0)));
	} else if (t_1 <= 0.99998) {
		tmp = t_2 * cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * t_0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (0.16666666666666666 * re);
	double t_1 = Math.exp(re) * Math.cos(im);
	double t_2 = 1.0 + (re * (1.0 + (re * t_0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2 * (1.0 + (-0.5 * Math.pow(im, 2.0)));
	} else if (t_1 <= 0.99998) {
		tmp = t_2 * Math.cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (Math.sqrt(((re * re) * (re * re))) * t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (0.16666666666666666 * re)
	t_1 = math.exp(re) * math.cos(im)
	t_2 = 1.0 + (re * (1.0 + (re * t_0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2 * (1.0 + (-0.5 * math.pow(im, 2.0)))
	elif t_1 <= 0.99998:
		tmp = t_2 * math.cos(im)
	else:
		tmp = (1.0 + re) - (-1.0 * (math.sqrt(((re * re) * (re * re))) * t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(0.16666666666666666 * re))
	t_1 = Float64(exp(re) * cos(im))
	t_2 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * t_0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 * Float64(1.0 + Float64(-0.5 * (im ^ 2.0))));
	elseif (t_1 <= 0.99998)
		tmp = Float64(t_2 * cos(im));
	else
		tmp = Float64(Float64(1.0 + re) - Float64(-1.0 * Float64(sqrt(Float64(Float64(re * re) * Float64(re * re))) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (0.16666666666666666 * re);
	t_1 = exp(re) * cos(im);
	t_2 = 1.0 + (re * (1.0 + (re * t_0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2 * (1.0 + (-0.5 * (im ^ 2.0)));
	elseif (t_1 <= 0.99998)
		tmp = t_2 * cos(im);
	else
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(re * N[(1.0 + N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * N[(1.0 + N[(-0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.99998], N[(t$95$2 * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] - N[(-1.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
   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 67.2%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 40.0%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 40.0%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99997999999999998

   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 67.2%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]

   if 0.99997999999999998 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

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

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{{re}^{2} \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 41.9%

           \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   11. Applied rewrites 41.9%

       \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + \color{blue}{0.16666666666666666} \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := 0.5 + 0.16666666666666666 \cdot re\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99998:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot t\_1\right)\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot t\_1\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (cos im)))
       (t_1 (+ 0.5 (* 0.16666666666666666 re))))
  (if (<= t_0 (- INFINITY))
    (*
     (- (* (- (* 0.5 re) -1.0) re) -1.0)
     (- (* (* im im) -0.5) -1.0))
    (if (<= t_0 0.99998)
      (* (+ 1.0 (* re (+ 1.0 (* re t_1)))) (cos im))
      (-
       (+ 1.0 re)
       (* -1.0 (* (sqrt (* (* re re) (* re re))) t_1)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = 0.5 + (0.16666666666666666 * re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (1.0 + (re * (1.0 + (re * t_1)))) * cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * t_1));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double t_1 = 0.5 + (0.16666666666666666 * re);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (1.0 + (re * (1.0 + (re * t_1)))) * Math.cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (Math.sqrt(((re * re) * (re * re))) * t_1));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	t_1 = 0.5 + (0.16666666666666666 * re)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0)
	elif t_0 <= 0.99998:
		tmp = (1.0 + (re * (1.0 + (re * t_1)))) * math.cos(im)
	else:
		tmp = (1.0 + re) - (-1.0 * (math.sqrt(((re * re) * (re * re))) * t_1))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(0.5 + Float64(0.16666666666666666 * re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * re) - -1.0) * re) - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	elseif (t_0 <= 0.99998)
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * t_1)))) * cos(im));
	else
		tmp = Float64(Float64(1.0 + re) - Float64(-1.0 * Float64(sqrt(Float64(Float64(re * re) * Float64(re * re))) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	t_1 = 0.5 + (0.16666666666666666 * re);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	elseif (t_0 <= 0.99998)
		tmp = (1.0 + (re * (1.0 + (re * t_1)))) * cos(im);
	else
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.5 * re), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99998], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] - N[(-1.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.6%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 37.6%

         \[\leadsto \left(re \cdot \left(1 + 0.5 \cdot re\right) - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6%

         \[\leadsto \left(\left(1 + 0.5 \cdot re\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 37.6%

          \[\leadsto \left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 N/A

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

   9. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99997999999999998

   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 67.2%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]

   if 0.99997999999999998 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

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

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{{re}^{2} \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 41.9%

           \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   11. Applied rewrites 41.9%

       \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + \color{blue}{0.16666666666666666} \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99998:\\ \;\;\;\;\left(re + \left(\left(0.5 \cdot re\right) \cdot re - -1\right)\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (cos im))))
  (if (<= t_0 (- INFINITY))
    (*
     (- (* (- (* 0.5 re) -1.0) re) -1.0)
     (- (* (* im im) -0.5) -1.0))
    (if (<= t_0 0.99998)
      (* (+ re (- (* (* 0.5 re) re) -1.0)) (cos im))
      (-
       (+ 1.0 re)
       (*
        -1.0
        (*
         (sqrt (* (* re re) (* re re)))
         (+ 0.5 (* 0.16666666666666666 re)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (re + (((0.5 * re) * re) - -1.0)) * cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (re + (((0.5 * re) * re) - -1.0)) * Math.cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (Math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0)
	elif t_0 <= 0.99998:
		tmp = (re + (((0.5 * re) * re) - -1.0)) * math.cos(im)
	else:
		tmp = (1.0 + re) - (-1.0 * (math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * re) - -1.0) * re) - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	elseif (t_0 <= 0.99998)
		tmp = Float64(Float64(re + Float64(Float64(Float64(0.5 * re) * re) - -1.0)) * cos(im));
	else
		tmp = Float64(Float64(1.0 + re) - Float64(-1.0 * Float64(sqrt(Float64(Float64(re * re) * Float64(re * re))) * Float64(0.5 + Float64(0.16666666666666666 * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	elseif (t_0 <= 0.99998)
		tmp = (re + (((0.5 * re) * re) - -1.0)) * cos(im);
	else
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.5 * re), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99998], N[(N[(re + N[(N[(N[(0.5 * re), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] - N[(-1.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.6%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 37.6%

         \[\leadsto \left(re \cdot \left(1 + 0.5 \cdot re\right) - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6%

         \[\leadsto \left(\left(1 + 0.5 \cdot re\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 37.6%

          \[\leadsto \left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 N/A

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

   9. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99997999999999998

   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 63.2%

          \[\leadsto \left(re + \left(\left(0.5 \cdot re\right) \cdot re - -1\right)\right) \cdot \cos im \]

   6. Applied rewrites 63.2%

      \[\leadsto \left(re + \color{blue}{\left(\left(0.5 \cdot re\right) \cdot re - -1\right)}\right) \cdot \cos im \]

   if 0.99997999999999998 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

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

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{{re}^{2} \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 41.9%

           \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   11. Applied rewrites 41.9%

       \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + \color{blue}{0.16666666666666666} \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99998:\\ \;\;\;\;\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (cos im))))
  (if (<= t_0 (- INFINITY))
    (*
     (- (* (- (* 0.5 re) -1.0) re) -1.0)
     (- (* (* im im) -0.5) -1.0))
    (if (<= t_0 0.99998)
      (* (+ 1.0 (* re (+ 1.0 (* 0.5 re)))) (cos im))
      (-
       (+ 1.0 re)
       (*
        -1.0
        (*
         (sqrt (* (* re re) (* re re)))
         (+ 0.5 (* 0.16666666666666666 re)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (1.0 + (re * (1.0 + (0.5 * re)))) * cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (1.0 + (re * (1.0 + (0.5 * re)))) * Math.cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (Math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0)
	elif t_0 <= 0.99998:
		tmp = (1.0 + (re * (1.0 + (0.5 * re)))) * math.cos(im)
	else:
		tmp = (1.0 + re) - (-1.0 * (math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * re) - -1.0) * re) - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	elseif (t_0 <= 0.99998)
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(0.5 * re)))) * cos(im));
	else
		tmp = Float64(Float64(1.0 + re) - Float64(-1.0 * Float64(sqrt(Float64(Float64(re * re) * Float64(re * re))) * Float64(0.5 + Float64(0.16666666666666666 * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	elseif (t_0 <= 0.99998)
		tmp = (1.0 + (re * (1.0 + (0.5 * re)))) * cos(im);
	else
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.5 * re), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99998], N[(N[(1.0 + N[(re * N[(1.0 + N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] - N[(-1.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.6%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 37.6%

         \[\leadsto \left(re \cdot \left(1 + 0.5 \cdot re\right) - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6%

         \[\leadsto \left(\left(1 + 0.5 \cdot re\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 37.6%

          \[\leadsto \left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 N/A

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

   9. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99997999999999998

   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   if 0.99997999999999998 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

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

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{{re}^{2} \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 41.9%

           \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   11. Applied rewrites 41.9%

       \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + \color{blue}{0.16666666666666666} \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99998:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (cos im))))
  (if (<= t_0 (- INFINITY))
    (*
     (- (* (- (* 0.5 re) -1.0) re) -1.0)
     (- (* (* im im) -0.5) -1.0))
    (if (<= t_0 0.99998)
      (* (+ 1.0 re) (cos im))
      (-
       (+ 1.0 re)
       (*
        -1.0
        (*
         (sqrt (* (* re re) (* re re)))
         (+ 0.5 (* 0.16666666666666666 re)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (1.0 + re) * cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = (1.0 + re) * Math.cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (Math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0)
	elif t_0 <= 0.99998:
		tmp = (1.0 + re) * math.cos(im)
	else:
		tmp = (1.0 + re) - (-1.0 * (math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * re) - -1.0) * re) - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	elseif (t_0 <= 0.99998)
		tmp = Float64(Float64(1.0 + re) * cos(im));
	else
		tmp = Float64(Float64(1.0 + re) - Float64(-1.0 * Float64(sqrt(Float64(Float64(re * re) * Float64(re * re))) * Float64(0.5 + Float64(0.16666666666666666 * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	elseif (t_0 <= 0.99998)
		tmp = (1.0 + re) * cos(im);
	else
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.5 * re), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99998], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] - N[(-1.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.6%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 37.6%

         \[\leadsto \left(re \cdot \left(1 + 0.5 \cdot re\right) - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6%

         \[\leadsto \left(\left(1 + 0.5 \cdot re\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 37.6%

          \[\leadsto \left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 N/A

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

   9. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99997999999999998

   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 51.5%

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

   4. Applied rewrites 51.5%

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

   if 0.99997999999999998 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

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

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{{re}^{2} \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 41.9%

           \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   11. Applied rewrites 41.9%

       \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + \color{blue}{0.16666666666666666} \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99998:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (cos im))))
  (if (<= t_0 (- INFINITY))
    (*
     (- (* (- (* 0.5 re) -1.0) re) -1.0)
     (- (* (* im im) -0.5) -1.0))
    (if (<= t_0 0.99998)
      (cos im)
      (-
       (+ 1.0 re)
       (*
        -1.0
        (*
         (sqrt (* (* re re) (* re re)))
         (+ 0.5 (* 0.16666666666666666 re)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else if (t_0 <= 0.99998) {
		tmp = Math.cos(im);
	} else {
		tmp = (1.0 + re) - (-1.0 * (Math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0)
	elif t_0 <= 0.99998:
		tmp = math.cos(im)
	else:
		tmp = (1.0 + re) - (-1.0 * (math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * re) - -1.0) * re) - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	elseif (t_0 <= 0.99998)
		tmp = cos(im);
	else
		tmp = Float64(Float64(1.0 + re) - Float64(-1.0 * Float64(sqrt(Float64(Float64(re * re) * Float64(re * re))) * Float64(0.5 + Float64(0.16666666666666666 * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	elseif (t_0 <= 0.99998)
		tmp = cos(im);
	else
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.5 * re), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99998], N[Cos[im], $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] - N[(-1.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.6%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 37.6%

         \[\leadsto \left(re \cdot \left(1 + 0.5 \cdot re\right) - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6%

         \[\leadsto \left(\left(1 + 0.5 \cdot re\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 37.6%

          \[\leadsto \left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 N/A

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

   9. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99997999999999998

   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 50.7%

         \[\leadsto \cos im \]

   4. Applied rewrites 50.7%

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

   if 0.99997999999999998 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

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

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{{re}^{2} \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 41.9%

           \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   11. Applied rewrites 41.9%

       \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + \color{blue}{0.16666666666666666} \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.1% accurate, 1.4× speedup

Math

\[\frac{\left(re - \left(-1 - \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re} \cdot \cos im \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (/
  (*
   (- re (- -1.0 (* (* (- (* 0.16666666666666666 re) -0.5) re) re)))
   re)
  re)
 (cos im)))

C

double code(double re, double im) {
	return (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / 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 = (((re - ((-1.0d0) - ((((0.16666666666666666d0 * re) - (-0.5d0)) * re) * re))) * re) / re) * cos(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[(N[(re - N[(-1.0 - N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - -0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] / re), $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 67.2%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot \color{blue}{re}\right)\right)\right) \cdot \cos im \]

4. Applied rewrites 67.2%

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]
5. Step-by-step derivation

   1. lift-+.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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. associate-+l+ N/A

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

   8. sum-to-mult N/A

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

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

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

6. Applied rewrites 67.0%

   \[\leadsto \left(\left(1 + \frac{\left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re - -1}{re}\right) \cdot \color{blue}{re}\right) \cdot \cos im \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. add-to-fraction N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-lft-identity N/A

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

   9. add-flip N/A

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

   10. lower--.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lower--.f64 69.1%

       \[\leadsto \frac{\left(re - \left(-1 - \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re} \cdot \cos im \]

8. Applied rewrites 69.1%

   \[\leadsto \frac{\left(re - \left(-1 - \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right)\right) \cdot re}{\color{blue}{re}} \cdot \cos im \]
9. Add Preprocessing

Alternative 9: 47.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) -0.05)
  (* (- (* (- (* 0.5 re) -1.0) re) -1.0) (- (* (* im im) -0.5) -1.0))
  (-
   (+ 1.0 re)
   (*
    -1.0
    (*
     (sqrt (* (* re re) (* re re)))
     (+ 0.5 (* 0.16666666666666666 re)))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.05) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else {
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	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 ((exp(re) * cos(im)) <= (-0.05d0)) then
        tmp = ((((0.5d0 * re) - (-1.0d0)) * re) - (-1.0d0)) * (((im * im) * (-0.5d0)) - (-1.0d0))
    else
        tmp = (1.0d0 + re) - ((-1.0d0) * (sqrt(((re * re) * (re * re))) * (0.5d0 + (0.16666666666666666d0 * re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= -0.05) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else {
		tmp = (1.0 + re) - (-1.0 * (Math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= -0.05:
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0)
	else:
		tmp = (1.0 + re) - (-1.0 * (math.sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.05)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * re) - -1.0) * re) - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	else
		tmp = Float64(Float64(1.0 + re) - Float64(-1.0 * Float64(sqrt(Float64(Float64(re * re) * Float64(re * re))) * Float64(0.5 + Float64(0.16666666666666666 * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= -0.05)
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	else
		tmp = (1.0 + re) - (-1.0 * (sqrt(((re * re) * (re * re))) * (0.5 + (0.16666666666666666 * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(N[(0.5 * re), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] - N[(-1.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.6%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 37.6%

         \[\leadsto \left(re \cdot \left(1 + 0.5 \cdot re\right) - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6%

         \[\leadsto \left(\left(1 + 0.5 \cdot re\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 37.6%

          \[\leadsto \left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 N/A

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

   9. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)} \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

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

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{{re}^{2} \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 41.9%

          \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot {re}^{2}} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 41.9%

           \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   11. Applied rewrites 41.9%

       \[\leadsto \left(1 + re\right) - -1 \cdot \left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)} \cdot \left(0.5 + \color{blue}{0.16666666666666666} \cdot re\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right) + 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) -0.05)
  (* (- (* (- (* 0.5 re) -1.0) re) -1.0) (- (* (* im im) -0.5) -1.0))
  (+ (+ re (* (* (- (* 0.16666666666666666 re) -0.5) re) re)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.05) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else {
		tmp = (re + ((((0.16666666666666666 * re) - -0.5) * re) * re)) + 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 ((exp(re) * cos(im)) <= (-0.05d0)) then
        tmp = ((((0.5d0 * re) - (-1.0d0)) * re) - (-1.0d0)) * (((im * im) * (-0.5d0)) - (-1.0d0))
    else
        tmp = (re + ((((0.16666666666666666d0 * re) - (-0.5d0)) * re) * re)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= -0.05) {
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	} else {
		tmp = (re + ((((0.16666666666666666 * re) - -0.5) * re) * re)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= -0.05:
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0)
	else:
		tmp = (re + ((((0.16666666666666666 * re) - -0.5) * re) * re)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.05)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * re) - -1.0) * re) - -1.0) * Float64(Float64(Float64(im * im) * -0.5) - -1.0));
	else
		tmp = Float64(Float64(re + Float64(Float64(Float64(Float64(0.16666666666666666 * re) - -0.5) * re) * re)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= -0.05)
		tmp = ((((0.5 * re) - -1.0) * re) - -1.0) * (((im * im) * -0.5) - -1.0);
	else
		tmp = (re + ((((0.16666666666666666 * re) - -0.5) * re) * re)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(N[(0.5 * re), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(re + N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - -0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   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 63.2%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot \color{blue}{re}\right)\right) \cdot \cos im \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)} \cdot \cos im \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.6%

         \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 37.6%

         \[\leadsto \left(re \cdot \left(1 + 0.5 \cdot re\right) - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6%

         \[\leadsto \left(\left(1 + 0.5 \cdot re\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 37.6%

          \[\leadsto \left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 N/A

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

   9. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot re - -1\right) \cdot re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5 - -1\right)} \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 67.2%

          \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

   6. Applied rewrites 67.2%

      \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 40.4%

         \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   9. Applied rewrites 40.4%

      \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. associate--l+ N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

   11. Applied rewrites 40.4%

       \[\leadsto \left(re + \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right) + \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 40.4% accurate, 8.2× speedup

Math

\[\left(re + \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right) + 1 \]

FPCore

(FPCore (re im)
  :precision binary64
  (+ (+ re (* (* (- (* 0.16666666666666666 re) -0.5) re) re)) 1.0))

C

double code(double re, double im) {
	return (re + ((((0.16666666666666666 * re) - -0.5) * re) * re)) + 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 + ((((0.16666666666666666d0 * re) - (-0.5d0)) * re) * re)) + 1.0d0
end function

Java

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

Python

def code(re, im):
	return (re + ((((0.16666666666666666 * re) - -0.5) * re) * re)) + 1.0

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(re + N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - -0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 67.2%

       \[\leadsto \cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right) \]

4. Applied rewrites 67.2%

   \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]

6. Applied rewrites 67.2%

   \[\leadsto \left(re - -1\right) \cdot \cos im - \color{blue}{\left(re \cdot re\right) \cdot \left(\cos im \cdot \left(-0.5 - 0.16666666666666666 \cdot re\right)\right)} \]

7. Taylor expanded in im around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 40.4%

      \[\leadsto \left(1 + re\right) - -1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

9. Applied rewrites 40.4%

   \[\leadsto \left(1 + re\right) - \color{blue}{-1 \cdot \left({re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. associate--l+ N/A

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

    4. +-commutative N/A

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

    5. lower-+.f64 N/A

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

11. Applied rewrites 40.4%

    \[\leadsto \left(re + \left(\left(0.16666666666666666 \cdot re - -0.5\right) \cdot re\right) \cdot re\right) + \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))