math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 1.9s

Alternatives: 13

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return exp(re) * sin(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) * sin(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return exp(re) * sin(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) * sin(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := im \cdot e^{re}\\ \mathbf{if}\;re \leq \frac{-2611279770367599}{4835703278458516698824704}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq \frac{2504478220538107}{21778071482940061661655974875633165533184}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 10500000000000000815831766655872649178879125871197860277834278179049830808884244966648174743689294249984:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) \cdot \sin im\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* im (exp re))))
  (if (<= re -2611279770367599/4835703278458516698824704)
    t_0
    (if (<=
         re
         2504478220538107/21778071482940061661655974875633165533184)
      (sin im)
      (if (<=
           re
           10500000000000000815831766655872649178879125871197860277834278179049830808884244966648174743689294249984)
        t_0
        (* (+ 1 (* re (+ 1 (* re (+ 1/2 (* 1/6 re)))))) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double tmp;
	if (re <= -5.4e-10) {
		tmp = t_0;
	} else if (re <= 1.15e-25) {
		tmp = sin(im);
	} else if (re <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * sin(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 = im * exp(re)
    if (re <= (-5.4d-10)) then
        tmp = t_0
    else if (re <= 1.15d-25) then
        tmp = sin(im)
    else if (re <= 1.05d+103) then
        tmp = t_0
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (0.16666666666666666d0 * re)))))) * sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * Math.exp(re);
	double tmp;
	if (re <= -5.4e-10) {
		tmp = t_0;
	} else if (re <= 1.15e-25) {
		tmp = Math.sin(im);
	} else if (re <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * math.exp(re)
	tmp = 0
	if re <= -5.4e-10:
		tmp = t_0
	elif re <= 1.15e-25:
		tmp = math.sin(im)
	elif re <= 1.05e+103:
		tmp = t_0
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * math.sin(im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	tmp = 0.0
	if (re <= -5.4e-10)
		tmp = t_0;
	elseif (re <= 1.15e-25)
		tmp = sin(im);
	elseif (re <= 1.05e+103)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(0.16666666666666666 * re)))))) * sin(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * exp(re);
	tmp = 0.0;
	if (re <= -5.4e-10)
		tmp = t_0;
	elseif (re <= 1.15e-25)
		tmp = sin(im);
	elseif (re <= 1.05e+103)
		tmp = t_0;
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re)))))) * sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2611279770367599/4835703278458516698824704], t$95$0, If[LessEqual[re, 2504478220538107/21778071482940061661655974875633165533184], N[Sin[im], $MachinePrecision], If[LessEqual[re, 10500000000000000815831766655872649178879125871197860277834278179049830808884244966648174743689294249984], t$95$0, N[(N[(1 + N[(re * N[(1 + N[(re * N[(1/2 + N[(1/6 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.4e-10 or 1.15e-25 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   if -5.4e-10 < re < 1.15e-25

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 50.1%

         \[\leadsto \sin im \]

   4. Applied rewrites 50.1%

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

   if 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 66.6%

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

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|im\right|\right)\\ t_1 := e^{re} \cdot t\_0\\ t_2 := \left|im\right| \cdot e^{re}\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|im\right| - -1\right)\right)\\ \mathbf{elif}\;t\_1 \leq \frac{-3602879701896397}{72057594037927936}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \frac{7872201966280717}{3936100983140358674171118325863157261303419813782882110237782515784158576702511753696331798193284779002326689610310857585686054524054270515222392815820422596546908348791339130466666204306680269934417552562141332061201544797059608540225005885713074181150932467712}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sin (fabs im)))
       (t_1 (* (exp re) t_0))
       (t_2 (* (fabs im) (exp re))))
  (*
   (copysign 1 im)
   (if (<= t_1 (- INFINITY))
     (*
      (+ 1 re)
      (* (fabs im) (- (* (* -1/6 (fabs im)) (fabs im)) -1)))
     (if (<= t_1 -3602879701896397/72057594037927936)
       t_0
       (if (<=
            t_1
            7872201966280717/3936100983140358674171118325863157261303419813782882110237782515784158576702511753696331798193284779002326689610310857585686054524054270515222392815820422596546908348791339130466666204306680269934417552562141332061201544797059608540225005885713074181150932467712)
         t_2
         (if (<= t_1 1) t_0 t_2)))))))

C

double code(double re, double im) {
	double t_0 = sin(fabs(im));
	double t_1 = exp(re) * t_0;
	double t_2 = fabs(im) * exp(re);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * (fabs(im) * (((-0.16666666666666666 * fabs(im)) * fabs(im)) - -1.0));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 2e-246) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(Math.abs(im));
	double t_1 = Math.exp(re) * t_0;
	double t_2 = Math.abs(im) * Math.exp(re);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 + re) * (Math.abs(im) * (((-0.16666666666666666 * Math.abs(im)) * Math.abs(im)) - -1.0));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 2e-246) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = math.sin(math.fabs(im))
	t_1 = math.exp(re) * t_0
	t_2 = math.fabs(im) * math.exp(re)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (1.0 + re) * (math.fabs(im) * (((-0.16666666666666666 * math.fabs(im)) * math.fabs(im)) - -1.0))
	elif t_1 <= -0.05:
		tmp = t_0
	elif t_1 <= 2e-246:
		tmp = t_2
	elif t_1 <= 1.0:
		tmp = t_0
	else:
		tmp = t_2
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = sin(abs(im))
	t_1 = Float64(exp(re) * t_0)
	t_2 = Float64(abs(im) * exp(re))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * Float64(abs(im) * Float64(Float64(Float64(-0.16666666666666666 * abs(im)) * abs(im)) - -1.0)));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 2e-246)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(abs(im));
	t_1 = exp(re) * t_0;
	t_2 = abs(im) * exp(re);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (1.0 + re) * (abs(im) * (((-0.16666666666666666 * abs(im)) * abs(im)) - -1.0));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 2e-246)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -3602879701896397/72057594037927936], t$95$0, If[LessEqual[t$95$1, 7872201966280717/3936100983140358674171118325863157261303419813782882110237782515784158576702511753696331798193284779002326689610310857585686054524054270515222392815820422596546908348791339130466666204306680269934417552562141332061201544797059608540225005885713074181150932467712], t$95$2, If[LessEqual[t$95$1, 1], t$95$0, t$95$2]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.9999999999999999e-246 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 50.1%

         \[\leadsto \sin im \]

   4. Applied rewrites 50.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-246 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|im\right|\right)\\ t_1 := e^{re} \cdot t\_0\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|im\right| - -1\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot \frac{\left(re - \left(-1 - \left(\left(\frac{1}{6} \cdot re - \frac{-1}{2}\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sin (fabs im))) (t_1 (* (exp re) t_0)))
  (*
   (copysign 1 im)
   (if (<= t_1 (- INFINITY))
     (*
      (+ 1 re)
      (* (fabs im) (- (* (* -1/6 (fabs im)) (fabs im)) -1)))
     (if (<= t_1 1)
       t_0
       (*
        (fabs im)
        (/
         (* (- re (- -1 (* (* (- (* 1/6 re) -1/2) re) re))) re)
         re)))))))

C

double code(double re, double im) {
	double t_0 = sin(fabs(im));
	double t_1 = exp(re) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * (fabs(im) * (((-0.16666666666666666 * fabs(im)) * fabs(im)) - -1.0));
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = fabs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	}
	return copysign(1.0, im) * tmp;
}

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(abs(im));
	t_1 = exp(re) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (1.0 + re) * (abs(im) * (((-0.16666666666666666 * abs(im)) * abs(im)) - -1.0));
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = abs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1], t$95$0, N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(re - N[(-1 - N[(N[(N[(N[(1/6 * re), $MachinePrecision] - -1/2), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 50.1%

         \[\leadsto \sin im \]

   4. Applied rewrites 50.1%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 39.6%

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

   7. Applied rewrites 39.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. sum-to-mult N/A

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

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

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

   9. Applied rewrites 39.5%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-lft-identity N/A

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

       9. add-flip N/A

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

       10. lower--.f64 N/A

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

       11. lift--.f64 N/A

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

       12. sub-negate-rev N/A

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

       13. lower--.f64 40.7%

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

   11. Applied rewrites 40.7%

       \[\leadsto im \cdot \frac{\left(re - \left(-1 - \left(\left(\frac{1}{6} \cdot re - \frac{-1}{2}\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{6} \cdot \left|im\right|\\ t_1 := e^{re} \cdot \sin \left(\left|im\right|\right)\\ t_2 := \left|im\right| \cdot \left|im\right|\\ t_3 := t\_0 \cdot t\_2\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq \frac{-5404319552844595}{36028797018963968}:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(t\_0 \cdot \left|im\right| - -1\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \frac{t\_3 \cdot t\_3 - t\_2}{-1 \cdot \left|im\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot \frac{\left(re - \left(-1 - \left(\left(\frac{1}{6} \cdot re - \frac{-1}{2}\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* -1/6 (fabs im)))
       (t_1 (* (exp re) (sin (fabs im))))
       (t_2 (* (fabs im) (fabs im)))
       (t_3 (* t_0 t_2)))
  (*
   (copysign 1 im)
   (if (<= t_1 -5404319552844595/36028797018963968)
     (* (+ 1 re) (* (fabs im) (- (* t_0 (fabs im)) -1)))
     (if (<= t_1 0)
       (* (+ 1 re) (/ (- (* t_3 t_3) t_2) (* -1 (fabs im))))
       (*
        (fabs im)
        (/
         (* (- re (- -1 (* (* (- (* 1/6 re) -1/2) re) re))) re)
         re)))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * fabs(im);
	double t_1 = exp(re) * sin(fabs(im));
	double t_2 = fabs(im) * fabs(im);
	double t_3 = t_0 * t_2;
	double tmp;
	if (t_1 <= -0.15) {
		tmp = (1.0 + re) * (fabs(im) * ((t_0 * fabs(im)) - -1.0));
	} else if (t_1 <= 0.0) {
		tmp = (1.0 + re) * (((t_3 * t_3) - t_2) / (-1.0 * fabs(im)));
	} else {
		tmp = fabs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * Math.abs(im);
	double t_1 = Math.exp(re) * Math.sin(Math.abs(im));
	double t_2 = Math.abs(im) * Math.abs(im);
	double t_3 = t_0 * t_2;
	double tmp;
	if (t_1 <= -0.15) {
		tmp = (1.0 + re) * (Math.abs(im) * ((t_0 * Math.abs(im)) - -1.0));
	} else if (t_1 <= 0.0) {
		tmp = (1.0 + re) * (((t_3 * t_3) - t_2) / (-1.0 * Math.abs(im)));
	} else {
		tmp = Math.abs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * math.fabs(im)
	t_1 = math.exp(re) * math.sin(math.fabs(im))
	t_2 = math.fabs(im) * math.fabs(im)
	t_3 = t_0 * t_2
	tmp = 0
	if t_1 <= -0.15:
		tmp = (1.0 + re) * (math.fabs(im) * ((t_0 * math.fabs(im)) - -1.0))
	elif t_1 <= 0.0:
		tmp = (1.0 + re) * (((t_3 * t_3) - t_2) / (-1.0 * math.fabs(im)))
	else:
		tmp = math.fabs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re)
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * abs(im))
	t_1 = Float64(exp(re) * sin(abs(im)))
	t_2 = Float64(abs(im) * abs(im))
	t_3 = Float64(t_0 * t_2)
	tmp = 0.0
	if (t_1 <= -0.15)
		tmp = Float64(Float64(1.0 + re) * Float64(abs(im) * Float64(Float64(t_0 * abs(im)) - -1.0)));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(Float64(t_3 * t_3) - t_2) / Float64(-1.0 * abs(im))));
	else
		tmp = Float64(abs(im) * Float64(Float64(Float64(re - Float64(-1.0 - Float64(Float64(Float64(Float64(0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * abs(im);
	t_1 = exp(re) * sin(abs(im));
	t_2 = abs(im) * abs(im);
	t_3 = t_0 * t_2;
	tmp = 0.0;
	if (t_1 <= -0.15)
		tmp = (1.0 + re) * (abs(im) * ((t_0 * abs(im)) - -1.0));
	elseif (t_1 <= 0.0)
		tmp = (1.0 + re) * (((t_3 * t_3) - t_2) / (-1.0 * abs(im)));
	else
		tmp = abs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -5404319552844595/36028797018963968], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(t$95$0 * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0], N[(N[(1 + re), $MachinePrecision] * N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(-1 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(re - N[(-1 - N[(N[(N[(N[(1/6 * re), $MachinePrecision] - -1/2), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.14999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if -0.14999999999999999 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. flip-+ N/A

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

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

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

   9. Applied rewrites 22.1%

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

   10. Taylor expanded in im around 0

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

       1. lower-*.f64 23.1%

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

   12. Applied rewrites 23.1%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 39.6%

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

   7. Applied rewrites 39.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. sum-to-mult N/A

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

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

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

   9. Applied rewrites 39.5%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-lft-identity N/A

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

       9. add-flip N/A

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

       10. lower--.f64 N/A

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

       11. lift--.f64 N/A

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

       12. sub-negate-rev N/A

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

       13. lower--.f64 40.7%

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

   11. Applied rewrites 40.7%

       \[\leadsto im \cdot \frac{\left(re - \left(-1 - \left(\left(\frac{1}{6} \cdot re - \frac{-1}{2}\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 46.4% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq \frac{5764607523034235}{576460752303423488}:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|im\right| - -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot \frac{\left(re - \left(-1 - \left(\left(\frac{1}{6} \cdot re - \frac{-1}{2}\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 im)
 (if (<=
      (* (exp re) (sin (fabs im)))
      5764607523034235/576460752303423488)
   (* (+ 1 re) (* (fabs im) (- (* (* -1/6 (fabs im)) (fabs im)) -1)))
   (*
    (fabs im)
    (/ (* (- re (- -1 (* (* (- (* 1/6 re) -1/2) re) re))) re) re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= 0.01) {
		tmp = (1.0 + re) * (fabs(im) * (((-0.16666666666666666 * fabs(im)) * fabs(im)) - -1.0));
	} else {
		tmp = fabs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(Math.abs(im))) <= 0.01) {
		tmp = (1.0 + re) * (Math.abs(im) * (((-0.16666666666666666 * Math.abs(im)) * Math.abs(im)) - -1.0));
	} else {
		tmp = Math.abs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(math.fabs(im))) <= 0.01:
		tmp = (1.0 + re) * (math.fabs(im) * (((-0.16666666666666666 * math.fabs(im)) * math.fabs(im)) - -1.0))
	else:
		tmp = math.fabs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re)
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= 0.01)
		tmp = Float64(Float64(1.0 + re) * Float64(abs(im) * Float64(Float64(Float64(-0.16666666666666666 * abs(im)) * abs(im)) - -1.0)));
	else
		tmp = Float64(abs(im) * Float64(Float64(Float64(re - Float64(-1.0 - Float64(Float64(Float64(Float64(0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(abs(im))) <= 0.01)
		tmp = (1.0 + re) * (abs(im) * (((-0.16666666666666666 * abs(im)) * abs(im)) - -1.0));
	else
		tmp = abs(im) * (((re - (-1.0 - ((((0.16666666666666666 * re) - -0.5) * re) * re))) * re) / re);
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5764607523034235/576460752303423488], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(re - N[(-1 - N[(N[(N[(N[(1/6 * re), $MachinePrecision] - -1/2), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.01

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if 0.01 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 39.6%

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

   7. Applied rewrites 39.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. sum-to-mult N/A

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

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

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

   9. Applied rewrites 39.5%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-lft-identity N/A

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

       9. add-flip N/A

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

       10. lower--.f64 N/A

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

       11. lift--.f64 N/A

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

       12. sub-negate-rev N/A

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

       13. lower--.f64 40.7%

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

   11. Applied rewrites 40.7%

       \[\leadsto im \cdot \frac{\left(re - \left(-1 - \left(\left(\frac{1}{6} \cdot re - \frac{-1}{2}\right) \cdot re\right) \cdot re\right)\right) \cdot re}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 45.3% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq \frac{5764607523034235}{576460752303423488}:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|im\right| - -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 im)
 (if (<=
      (* (exp re) (sin (fabs im)))
      5764607523034235/576460752303423488)
   (* (+ 1 re) (* (fabs im) (- (* (* -1/6 (fabs im)) (fabs im)) -1)))
   (* (fabs im) (+ 1 (* re (+ 1 (* re (+ 1/2 (* 1/6 re))))))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= 0.01) {
		tmp = (1.0 + re) * (fabs(im) * (((-0.16666666666666666 * fabs(im)) * fabs(im)) - -1.0));
	} else {
		tmp = fabs(im) * (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re))))));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(Math.abs(im))) <= 0.01) {
		tmp = (1.0 + re) * (Math.abs(im) * (((-0.16666666666666666 * Math.abs(im)) * Math.abs(im)) - -1.0));
	} else {
		tmp = Math.abs(im) * (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re))))));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(math.fabs(im))) <= 0.01:
		tmp = (1.0 + re) * (math.fabs(im) * (((-0.16666666666666666 * math.fabs(im)) * math.fabs(im)) - -1.0))
	else:
		tmp = math.fabs(im) * (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re))))))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= 0.01)
		tmp = Float64(Float64(1.0 + re) * Float64(abs(im) * Float64(Float64(Float64(-0.16666666666666666 * abs(im)) * abs(im)) - -1.0)));
	else
		tmp = Float64(abs(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(0.16666666666666666 * re)))))));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(abs(im))) <= 0.01)
		tmp = (1.0 + re) * (abs(im) * (((-0.16666666666666666 * abs(im)) * abs(im)) - -1.0));
	else
		tmp = abs(im) * (1.0 + (re * (1.0 + (re * (0.5 + (0.16666666666666666 * re))))));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5764607523034235/576460752303423488], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[(1 + N[(re * N[(1 + N[(re * N[(1/2 + N[(1/6 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.01

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if 0.01 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 39.6%

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

   7. Applied rewrites 39.6%

      \[\leadsto im \cdot \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 45.3% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq \frac{5764607523034235}{576460752303423488}:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|im\right| - -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot \left(1 + re \cdot \left(\left(1 + \frac{1}{3} \cdot re\right) \cdot \left(1 - \frac{-1}{2} \cdot re\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 im)
 (if (<=
      (* (exp re) (sin (fabs im)))
      5764607523034235/576460752303423488)
   (* (+ 1 re) (* (fabs im) (- (* (* -1/6 (fabs im)) (fabs im)) -1)))
   (* (fabs im) (+ 1 (* re (* (+ 1 (* 1/3 re)) (- 1 (* -1/2 re)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= 0.01) {
		tmp = (1.0 + re) * (fabs(im) * (((-0.16666666666666666 * fabs(im)) * fabs(im)) - -1.0));
	} else {
		tmp = fabs(im) * (1.0 + (re * ((1.0 + (0.3333333333333333 * re)) * (1.0 - (-0.5 * re)))));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(Math.abs(im))) <= 0.01) {
		tmp = (1.0 + re) * (Math.abs(im) * (((-0.16666666666666666 * Math.abs(im)) * Math.abs(im)) - -1.0));
	} else {
		tmp = Math.abs(im) * (1.0 + (re * ((1.0 + (0.3333333333333333 * re)) * (1.0 - (-0.5 * re)))));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(math.fabs(im))) <= 0.01:
		tmp = (1.0 + re) * (math.fabs(im) * (((-0.16666666666666666 * math.fabs(im)) * math.fabs(im)) - -1.0))
	else:
		tmp = math.fabs(im) * (1.0 + (re * ((1.0 + (0.3333333333333333 * re)) * (1.0 - (-0.5 * re)))))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= 0.01)
		tmp = Float64(Float64(1.0 + re) * Float64(abs(im) * Float64(Float64(Float64(-0.16666666666666666 * abs(im)) * abs(im)) - -1.0)));
	else
		tmp = Float64(abs(im) * Float64(1.0 + Float64(re * Float64(Float64(1.0 + Float64(0.3333333333333333 * re)) * Float64(1.0 - Float64(-0.5 * re))))));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(abs(im))) <= 0.01)
		tmp = (1.0 + re) * (abs(im) * (((-0.16666666666666666 * abs(im)) * abs(im)) - -1.0));
	else
		tmp = abs(im) * (1.0 + (re * ((1.0 + (0.3333333333333333 * re)) * (1.0 - (-0.5 * re)))));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5764607523034235/576460752303423488], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[(1 + N[(re * N[(N[(1 + N[(1/3 * re), $MachinePrecision]), $MachinePrecision] * N[(1 - N[(-1/2 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.01

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if 0.01 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 39.6%

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

   7. Applied rewrites 39.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. sum-to-mult N/A

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

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

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

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

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

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

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

      11. lower-*.f64 N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-out N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-neg-out N/A

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

   9. Applied rewrites 39.6%

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

   10. Taylor expanded in re around inf

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

       1. lower-*.f64 39.4%

          \[\leadsto im \cdot \left(1 + re \cdot \left(\left(1 + \frac{1}{3} \cdot re\right) \cdot \left(1 - \frac{-1}{2} \cdot re\right)\right)\right) \]

   12. Applied rewrites 39.4%

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

Alternative 8: 43.9% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq \frac{5764607523034235}{576460752303423488}:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|im\right| - -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| + re \cdot \left(\left|im\right| + \left|im\right| \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 im)
 (if (<=
      (* (exp re) (sin (fabs im)))
      5764607523034235/576460752303423488)
   (* (+ 1 re) (* (fabs im) (- (* (* -1/6 (fabs im)) (fabs im)) -1)))
   (+
    (fabs im)
    (* re (+ (fabs im) (* (fabs im) (* re (+ 1/2 (* 1/6 re))))))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= 0.01) {
		tmp = (1.0 + re) * (fabs(im) * (((-0.16666666666666666 * fabs(im)) * fabs(im)) - -1.0));
	} else {
		tmp = fabs(im) + (re * (fabs(im) + (fabs(im) * (re * (0.5 + (0.16666666666666666 * re))))));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(Math.abs(im))) <= 0.01) {
		tmp = (1.0 + re) * (Math.abs(im) * (((-0.16666666666666666 * Math.abs(im)) * Math.abs(im)) - -1.0));
	} else {
		tmp = Math.abs(im) + (re * (Math.abs(im) + (Math.abs(im) * (re * (0.5 + (0.16666666666666666 * re))))));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(math.fabs(im))) <= 0.01:
		tmp = (1.0 + re) * (math.fabs(im) * (((-0.16666666666666666 * math.fabs(im)) * math.fabs(im)) - -1.0))
	else:
		tmp = math.fabs(im) + (re * (math.fabs(im) + (math.fabs(im) * (re * (0.5 + (0.16666666666666666 * re))))))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= 0.01)
		tmp = Float64(Float64(1.0 + re) * Float64(abs(im) * Float64(Float64(Float64(-0.16666666666666666 * abs(im)) * abs(im)) - -1.0)));
	else
		tmp = Float64(abs(im) + Float64(re * Float64(abs(im) + Float64(abs(im) * Float64(re * Float64(0.5 + Float64(0.16666666666666666 * re)))))));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(abs(im))) <= 0.01)
		tmp = (1.0 + re) * (abs(im) * (((-0.16666666666666666 * abs(im)) * abs(im)) - -1.0));
	else
		tmp = abs(im) + (re * (abs(im) + (abs(im) * (re * (0.5 + (0.16666666666666666 * re))))));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5764607523034235/576460752303423488], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] + N[(re * N[(N[Abs[im], $MachinePrecision] + N[(N[Abs[im], $MachinePrecision] * N[(re * N[(1/2 + N[(1/6 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.01

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if 0.01 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 37.8%

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

   7. Applied rewrites 37.8%

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

   8. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 38.3%

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

   10. Applied rewrites 38.3%

       \[\leadsto im + re \cdot \left(im + im \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 43.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq \frac{5764607523034235}{576460752303423488}:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left|im\right| \cdot \left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|im\right| - -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| + \left|im\right| \cdot \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 im)
 (if (<=
      (* (exp re) (sin (fabs im)))
      5764607523034235/576460752303423488)
   (* (+ 1 re) (* (fabs im) (- (* (* -1/6 (fabs im)) (fabs im)) -1)))
   (+ (fabs im) (* (fabs im) (* re (+ 1 (* 1/2 re))))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= 0.01) {
		tmp = (1.0 + re) * (fabs(im) * (((-0.16666666666666666 * fabs(im)) * fabs(im)) - -1.0));
	} else {
		tmp = fabs(im) + (fabs(im) * (re * (1.0 + (0.5 * re))));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(Math.abs(im))) <= 0.01) {
		tmp = (1.0 + re) * (Math.abs(im) * (((-0.16666666666666666 * Math.abs(im)) * Math.abs(im)) - -1.0));
	} else {
		tmp = Math.abs(im) + (Math.abs(im) * (re * (1.0 + (0.5 * re))));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(math.fabs(im))) <= 0.01:
		tmp = (1.0 + re) * (math.fabs(im) * (((-0.16666666666666666 * math.fabs(im)) * math.fabs(im)) - -1.0))
	else:
		tmp = math.fabs(im) + (math.fabs(im) * (re * (1.0 + (0.5 * re))))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= 0.01)
		tmp = Float64(Float64(1.0 + re) * Float64(abs(im) * Float64(Float64(Float64(-0.16666666666666666 * abs(im)) * abs(im)) - -1.0)));
	else
		tmp = Float64(abs(im) + Float64(abs(im) * Float64(re * Float64(1.0 + Float64(0.5 * re)))));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(abs(im))) <= 0.01)
		tmp = (1.0 + re) * (abs(im) * (((-0.16666666666666666 * abs(im)) * abs(im)) - -1.0));
	else
		tmp = abs(im) + (abs(im) * (re * (1.0 + (0.5 * re))));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5764607523034235/576460752303423488], N[(N[(1 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] + N[(N[Abs[im], $MachinePrecision] * N[(re * N[(1 + N[(1/2 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.01

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.6%

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-out N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-out N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 31.6%

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

   9. Applied rewrites 31.6%

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

   if 0.01 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.6%

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

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.6%

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

   7. Applied rewrites 34.6%

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

   8. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 37.3%

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

   10. Applied rewrites 37.3%

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

Alternative 10: 37.3% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

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 = im + (im * (re * (1.0d0 + (0.5d0 * re))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

   2. lower-exp.f64 69.6%

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

4. Applied rewrites 69.6%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

5. Taylor expanded in re around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 34.6%

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

7. Applied rewrites 34.6%

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

8. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 37.3%

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

10. Applied rewrites 37.3%

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

Alternative 11: 29.9% accurate, 22.9× speedup

Math

\[im + im \cdot re \]

FPCore

(FPCore (re im)
  :precision binary64
  (+ im (* im re)))

C

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

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 = im + (im * re)
end function

Java

public static double code(double re, double im) {
	return im + (im * re);
}

Python

def code(re, im):
	return im + (im * re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

   2. lower-exp.f64 69.6%

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

4. Applied rewrites 69.6%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

5. Taylor expanded in re around 0

   \[\leadsto im + \color{blue}{im \cdot re} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto im + im \cdot \color{blue}{re} \]

   2. lower-*.f64 29.9%

      \[\leadsto im + im \cdot re \]

7. Applied rewrites 29.9%

   \[\leadsto im + \color{blue}{im \cdot re} \]
8. Add Preprocessing

Alternative 12: 26.7% accurate, 206.0× speedup

Math

\[im \]

FPCore

(FPCore (re im)
  :precision binary64
  im)

C

double code(double re, double im) {
	return 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 = im
end function

Java

public static double code(double re, double im) {
	return im;
}

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

function tmp = code(re, im)
	tmp = im;
end

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

   2. lower-exp.f64 69.6%

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

4. Applied rewrites 69.6%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

5. Taylor expanded in re around 0

   \[\leadsto im \]
6. Step-by-step derivation

7. Applied rewrites 26.7%

   \[\leadsto im \]
8. Add Preprocessing

Reproduce

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