math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 8

Speedup: 0.7×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(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 = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(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 = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\left(e^{im} \cdot 0.5\right) \cdot \sin re - \left(-0.5 \cdot \sin re\right) \cdot e^{-im} \]

FPCore

(FPCore (re im)
  :precision binary64
  (- (* (* (exp im) 0.5) (sin re)) (* (* -0.5 (sin re)) (exp (- im)))))

C

double code(double re, double im) {
	return ((exp(im) * 0.5) * sin(re)) - ((-0.5 * sin(re)) * exp(-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(im) * 0.5d0) * sin(re)) - (((-0.5d0) * sin(re)) * exp(-im))
end function

Java

public static double code(double re, double im) {
	return ((Math.exp(im) * 0.5) * Math.sin(re)) - ((-0.5 * Math.sin(re)) * Math.exp(-im));
}

Python

def code(re, im):
	return ((math.exp(im) * 0.5) * math.sin(re)) - ((-0.5 * math.sin(re)) * math.exp(-im))

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[(N[Exp[im], $MachinePrecision] * 0.5), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im} + \left(\frac{1}{2} \cdot \sin re\right) \cdot e^{0 - im}} \]

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

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im} - \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im}} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot e^{im} - \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{e^{im} \cdot \left(\frac{1}{2} \cdot \sin re\right)} - \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im} \]

   8. lift-*.f64 N/A

      \[\leadsto e^{im} \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} - \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im} \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right) \cdot \sin re} - \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right) \cdot \sin re} - \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)} \cdot \sin re - \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im} \]

   12. lower-*.f64 N/A

       \[\leadsto \left(e^{im} \cdot \frac{1}{2}\right) \cdot \sin re - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)\right) \cdot e^{0 - im}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(e^{im} \cdot \frac{1}{2}\right) \cdot \sin re - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \sin re}\right)\right) \cdot e^{0 - im} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \left(e^{im} \cdot \frac{1}{2}\right) \cdot \sin re - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \sin re\right)} \cdot e^{0 - im} \]

   15. lower-*.f64 N/A

       \[\leadsto \left(e^{im} \cdot \frac{1}{2}\right) \cdot \sin re - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \sin re\right)} \cdot e^{0 - im} \]

   16. metadata-eval 100.0%

       \[\leadsto \left(e^{im} \cdot 0.5\right) \cdot \sin re - \left(\color{blue}{-0.5} \cdot \sin re\right) \cdot e^{0 - im} \]

   17. lift--.f64 N/A

       \[\leadsto \left(e^{im} \cdot \frac{1}{2}\right) \cdot \sin re - \left(\frac{-1}{2} \cdot \sin re\right) \cdot e^{\color{blue}{0 - im}} \]

   18. sub0-neg N/A

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

   19. lower-neg.f64 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(e^{im} \cdot 0.5\right) \cdot \sin re - \left(-0.5 \cdot \sin re\right) \cdot e^{-im}} \]
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.5× speedup

Math

\[\sin re \cdot \cosh im \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

   6. metadata-eval N/A

      \[\leadsto \sin re \cdot \left(\left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

   7. mult-flip N/A

      \[\leadsto \sin re \cdot \color{blue}{\frac{e^{0 - im} + e^{im}}{2}} \]

   8. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \frac{\color{blue}{e^{0 - im} + e^{im}}}{2} \]

   9. +-commutative N/A

      \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im} + e^{0 - im}}}{2} \]

   10. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im}} + e^{0 - im}}{2} \]

   11. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \frac{e^{im} + \color{blue}{e^{0 - im}}}{2} \]

   12. lift--.f64 N/A

       \[\leadsto \sin re \cdot \frac{e^{im} + e^{\color{blue}{0 - im}}}{2} \]

   13. sub0-neg N/A

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

   14. cosh-def N/A

       \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

   15. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   16. lower-cosh.f64 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
4. Add Preprocessing

Alternative 3: 82.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot \left|im\right|\\ \mathbf{if}\;\left|im\right| \leq 2.55 \cdot 10^{+14}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;\left|im\right| \leq 10^{+86}:\\ \;\;\;\;\left(re \cdot \left(\left(\left(\frac{-6}{re \cdot re} - -1\right) \cdot re\right) \cdot \left(-0.08333333333333333 \cdot re\right)\right)\right) \cdot 2\\ \mathbf{elif}\;\left|im\right| \leq 1.75 \cdot 10^{+148}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \frac{t\_0 \cdot t\_0 - 2 \cdot 2}{t\_0 - 2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(1 + \frac{2}{t\_0}\right) \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (fabs im))))
  (if (<= (fabs im) 2.55e+14)
    (sin re)
    (if (<= (fabs im) 1e+86)
      (*
       (*
        re
        (*
         (* (- (/ -6.0 (* re re)) -1.0) re)
         (* -0.08333333333333333 re)))
       2.0)
      (if (<= (fabs im) 1.75e+148)
        (* (* 0.5 re) (/ (- (* t_0 t_0) (* 2.0 2.0)) (- t_0 2.0)))
        (* (* 0.5 re) (* (+ 1.0 (/ 2.0 t_0)) t_0)))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * fabs(im);
	double tmp;
	if (fabs(im) <= 2.55e+14) {
		tmp = sin(re);
	} else if (fabs(im) <= 1e+86) {
		tmp = (re * ((((-6.0 / (re * re)) - -1.0) * re) * (-0.08333333333333333 * re))) * 2.0;
	} else if (fabs(im) <= 1.75e+148) {
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0));
	} else {
		tmp = (0.5 * re) * ((1.0 + (2.0 / t_0)) * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(im) * abs(im)
    if (abs(im) <= 2.55d+14) then
        tmp = sin(re)
    else if (abs(im) <= 1d+86) then
        tmp = (re * (((((-6.0d0) / (re * re)) - (-1.0d0)) * re) * ((-0.08333333333333333d0) * re))) * 2.0d0
    else if (abs(im) <= 1.75d+148) then
        tmp = (0.5d0 * re) * (((t_0 * t_0) - (2.0d0 * 2.0d0)) / (t_0 - 2.0d0))
    else
        tmp = (0.5d0 * re) * ((1.0d0 + (2.0d0 / t_0)) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.abs(im) * Math.abs(im);
	double tmp;
	if (Math.abs(im) <= 2.55e+14) {
		tmp = Math.sin(re);
	} else if (Math.abs(im) <= 1e+86) {
		tmp = (re * ((((-6.0 / (re * re)) - -1.0) * re) * (-0.08333333333333333 * re))) * 2.0;
	} else if (Math.abs(im) <= 1.75e+148) {
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0));
	} else {
		tmp = (0.5 * re) * ((1.0 + (2.0 / t_0)) * t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.fabs(im) * math.fabs(im)
	tmp = 0
	if math.fabs(im) <= 2.55e+14:
		tmp = math.sin(re)
	elif math.fabs(im) <= 1e+86:
		tmp = (re * ((((-6.0 / (re * re)) - -1.0) * re) * (-0.08333333333333333 * re))) * 2.0
	elif math.fabs(im) <= 1.75e+148:
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0))
	else:
		tmp = (0.5 * re) * ((1.0 + (2.0 / t_0)) * t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(abs(im) * abs(im))
	tmp = 0.0
	if (abs(im) <= 2.55e+14)
		tmp = sin(re);
	elseif (abs(im) <= 1e+86)
		tmp = Float64(Float64(re * Float64(Float64(Float64(Float64(-6.0 / Float64(re * re)) - -1.0) * re) * Float64(-0.08333333333333333 * re))) * 2.0);
	elseif (abs(im) <= 1.75e+148)
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(t_0 * t_0) - Float64(2.0 * 2.0)) / Float64(t_0 - 2.0)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(1.0 + Float64(2.0 / t_0)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = abs(im) * abs(im);
	tmp = 0.0;
	if (abs(im) <= 2.55e+14)
		tmp = sin(re);
	elseif (abs(im) <= 1e+86)
		tmp = (re * ((((-6.0 / (re * re)) - -1.0) * re) * (-0.08333333333333333 * re))) * 2.0;
	elseif (abs(im) <= 1.75e+148)
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0));
	else
		tmp = (0.5 * re) * ((1.0 + (2.0 / t_0)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[im], $MachinePrecision], 2.55e+14], N[Sin[re], $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 1e+86], N[(N[(re * N[(N[(N[(N[(-6.0 / N[(re * re), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * re), $MachinePrecision] * N[(-0.08333333333333333 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 1.75e+148], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(2.0 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.55e14

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

   7. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 48.3%

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\sin re} \]
   12. Step-by-step derivation

       1. lower-sin.f64 51.2%

          \[\leadsto \sin re \]

   13. Applied rewrites 51.2%

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

   if 2.55e14 < im < 1e86

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 33.2%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 33.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

      3. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

      4. associate-*r* N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

      6. lower-*.f64 33.2%

         \[\leadsto \left(re \cdot \left(0.5 + \left(-0.08333333333333333 \cdot re\right) \cdot re\right)\right) \cdot 2 \]

   9. Applied rewrites 33.2%

      \[\leadsto \left(re \cdot \left(0.5 + \left(-0.08333333333333333 \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\left(\frac{-1}{12} \cdot re\right) \cdot re}\right)\right) \cdot 2 \]

       2. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot re\right)\right) \cdot 2 \]

       4. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

       6. *-commutative N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(re \cdot re\right) \cdot \color{blue}{\frac{-1}{12}}\right)\right) \cdot 2 \]

       7. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(re \cdot re\right) \cdot \color{blue}{\frac{-1}{12}}\right)\right) \cdot 2 \]

       8. +-commutative N/A

          \[\leadsto \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 2 \]

       9. sum-to-mult N/A

          \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)}\right)\right) \cdot 2 \]

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

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)}\right)\right) \cdot 2 \]

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

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       12. lower-unsound-/.f32 21.3%

           \[\leadsto \left(re \cdot \left(\left(1 + \left( \frac{0.5}{\left( \left(re \cdot re\right) \cdot -0.08333333333333333 \right)_{\text{binary64}}} \right)_{\text{binary32}}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot -0.08333333333333333\right)\right)\right) \cdot 2 \]

       13. lower-/.f32 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       14. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot 1}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       15. associate-*l/ N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}} \cdot 1\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       16. *-inverses N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}} \cdot \frac{re}{re}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       17. times-frac N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       18. lift-*.f64 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       19. lift-*.f64 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       20. lift-/.f64 21.5%

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{0.5 \cdot re}{\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot -0.08333333333333333\right)\right)\right) \cdot 2 \]

       21. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)}\right)\right) \cdot 2 \]

   11. Applied rewrites 33.3%

       \[\leadsto \left(re \cdot \left(\left(\left(\frac{-6}{re \cdot re} - -1\right) \cdot re\right) \cdot \color{blue}{\left(-0.08333333333333333 \cdot re\right)}\right)\right) \cdot 2 \]

   if 1e86 < im < 1.7499999999999999e148

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

   7. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 48.3%

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

   10. Applied rewrites 48.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

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

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

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

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

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

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

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

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

       14. lower-unsound--.f64 35.8%

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

       15. lift-pow.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 35.8%

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

   12. Applied rewrites 35.8%

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

   if 1.7499999999999999e148 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

   7. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 48.3%

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

   10. Applied rewrites 48.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. sum-to-mult N/A

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

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

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

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

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

       6. lower-unsound-/.f64 35.8%

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 35.8%

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 35.8%

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

   12. Applied rewrites 35.8%

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

Alternative 4: 56.3% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq 5 \cdot 10^{-300}:\\ \;\;\;\;\left(\left|re\right| \cdot \left(\left(\left(\frac{-6}{\left|re\right| \cdot \left|re\right|} - -1\right) \cdot \left|re\right|\right) \cdot \left(-0.08333333333333333 \cdot \left|re\right|\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(\left(im \cdot im + 1\right) + 1\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) 5e-300)
   (*
    (*
     (fabs re)
     (*
      (* (- (/ -6.0 (* (fabs re) (fabs re))) -1.0) (fabs re))
      (* -0.08333333333333333 (fabs re))))
    2.0)
   (* (* 0.5 (fabs re)) (+ (+ (* im im) 1.0) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 5e-300) {
		tmp = (fabs(re) * ((((-6.0 / (fabs(re) * fabs(re))) - -1.0) * fabs(re)) * (-0.08333333333333333 * fabs(re)))) * 2.0;
	} else {
		tmp = (0.5 * fabs(re)) * (((im * im) + 1.0) + 1.0);
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(Math.abs(re))) <= 5e-300) {
		tmp = (Math.abs(re) * ((((-6.0 / (Math.abs(re) * Math.abs(re))) - -1.0) * Math.abs(re)) * (-0.08333333333333333 * Math.abs(re)))) * 2.0;
	} else {
		tmp = (0.5 * Math.abs(re)) * (((im * im) + 1.0) + 1.0);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= 5e-300:
		tmp = (math.fabs(re) * ((((-6.0 / (math.fabs(re) * math.fabs(re))) - -1.0) * math.fabs(re)) * (-0.08333333333333333 * math.fabs(re)))) * 2.0
	else:
		tmp = (0.5 * math.fabs(re)) * (((im * im) + 1.0) + 1.0)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 5e-300)
		tmp = Float64(Float64(abs(re) * Float64(Float64(Float64(Float64(-6.0 / Float64(abs(re) * abs(re))) - -1.0) * abs(re)) * Float64(-0.08333333333333333 * abs(re)))) * 2.0);
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(Float64(Float64(im * im) + 1.0) + 1.0));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(abs(re))) <= 5e-300)
		tmp = (abs(re) * ((((-6.0 / (abs(re) * abs(re))) - -1.0) * abs(re)) * (-0.08333333333333333 * abs(re)))) * 2.0;
	else
		tmp = (0.5 * abs(re)) * (((im * im) + 1.0) + 1.0);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-300], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(N[(N[(-6.0 / N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(-0.08333333333333333 * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5e-300

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 33.2%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 33.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

      3. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

      4. associate-*r* N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

      6. lower-*.f64 33.2%

         \[\leadsto \left(re \cdot \left(0.5 + \left(-0.08333333333333333 \cdot re\right) \cdot re\right)\right) \cdot 2 \]

   9. Applied rewrites 33.2%

      \[\leadsto \left(re \cdot \left(0.5 + \left(-0.08333333333333333 \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\left(\frac{-1}{12} \cdot re\right) \cdot re}\right)\right) \cdot 2 \]

       2. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot re\right)\right) \cdot 2 \]

       4. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

       6. *-commutative N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(re \cdot re\right) \cdot \color{blue}{\frac{-1}{12}}\right)\right) \cdot 2 \]

       7. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(re \cdot re\right) \cdot \color{blue}{\frac{-1}{12}}\right)\right) \cdot 2 \]

       8. +-commutative N/A

          \[\leadsto \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 2 \]

       9. sum-to-mult N/A

          \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)}\right)\right) \cdot 2 \]

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

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)}\right)\right) \cdot 2 \]

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

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       12. lower-unsound-/.f32 21.3%

           \[\leadsto \left(re \cdot \left(\left(1 + \left( \frac{0.5}{\left( \left(re \cdot re\right) \cdot -0.08333333333333333 \right)_{\text{binary64}}} \right)_{\text{binary32}}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot -0.08333333333333333\right)\right)\right) \cdot 2 \]

       13. lower-/.f32 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       14. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot 1}{\left(re \cdot re\right) \cdot \frac{-1}{12}}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       15. associate-*l/ N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}} \cdot 1\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       16. *-inverses N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2}}{\left(re \cdot re\right) \cdot \frac{-1}{12}} \cdot \frac{re}{re}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       17. times-frac N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       18. lift-*.f64 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       19. lift-*.f64 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2 \]

       20. lift-/.f64 21.5%

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{0.5 \cdot re}{\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re}\right) \cdot \left(\left(re \cdot \color{blue}{re}\right) \cdot -0.08333333333333333\right)\right)\right) \cdot 2 \]

       21. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \left(\left(1 + \frac{\frac{1}{2} \cdot re}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re}\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)}\right)\right) \cdot 2 \]

   11. Applied rewrites 33.3%

       \[\leadsto \left(re \cdot \left(\left(\left(\frac{-6}{re \cdot re} - -1\right) \cdot re\right) \cdot \color{blue}{\left(-0.08333333333333333 \cdot re\right)}\right)\right) \cdot 2 \]

   if 5e-300 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

   7. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 48.3%

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

   10. Applied rewrites 48.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. metadata-eval N/A

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

       4. associate-+r+ N/A

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

       5. lower-+.f64 N/A

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

       6. lower-+.f64 48.3%

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 48.3%

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

   12. Applied rewrites 48.3%

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

Alternative 5: 55.9% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(\left|re\right| \cdot \left(0.5 + \left(-0.08333333333333333 \cdot \left|re\right|\right) \cdot \left|re\right|\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(\left(im \cdot im + 1\right) + 1\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<=
      (* (* 0.5 (sin (fabs re))) (+ (exp (- 0.0 im)) (exp im)))
      -0.002)
   (*
    (*
     (fabs re)
     (+ 0.5 (* (* -0.08333333333333333 (fabs re)) (fabs re))))
    2.0)
   (* (* 0.5 (fabs re)) (+ (+ (* im im) 1.0) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - im)) + exp(im))) <= -0.002) {
		tmp = (fabs(re) * (0.5 + ((-0.08333333333333333 * fabs(re)) * fabs(re)))) * 2.0;
	} else {
		tmp = (0.5 * fabs(re)) * (((im * im) + 1.0) + 1.0);
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(Math.abs(re))) * (Math.exp((0.0 - im)) + Math.exp(im))) <= -0.002) {
		tmp = (Math.abs(re) * (0.5 + ((-0.08333333333333333 * Math.abs(re)) * Math.abs(re)))) * 2.0;
	} else {
		tmp = (0.5 * Math.abs(re)) * (((im * im) + 1.0) + 1.0);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(math.fabs(re))) * (math.exp((0.0 - im)) + math.exp(im))) <= -0.002:
		tmp = (math.fabs(re) * (0.5 + ((-0.08333333333333333 * math.fabs(re)) * math.fabs(re)))) * 2.0
	else:
		tmp = (0.5 * math.fabs(re)) * (((im * im) + 1.0) + 1.0)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.002)
		tmp = Float64(Float64(abs(re) * Float64(0.5 + Float64(Float64(-0.08333333333333333 * abs(re)) * abs(re)))) * 2.0);
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(Float64(Float64(im * im) + 1.0) + 1.0));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(abs(re))) * (exp((0.0 - im)) + exp(im))) <= -0.002)
		tmp = (abs(re) * (0.5 + ((-0.08333333333333333 * abs(re)) * abs(re)))) * 2.0;
	else
		tmp = (0.5 * abs(re)) * (((im * im) + 1.0) + 1.0);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[Abs[re], $MachinePrecision] * N[(0.5 + N[(N[(-0.08333333333333333 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e-3

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 33.2%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 33.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

      3. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \left(re \cdot \color{blue}{re}\right)\right)\right) \cdot 2 \]

      4. associate-*r* N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

      6. lower-*.f64 33.2%

         \[\leadsto \left(re \cdot \left(0.5 + \left(-0.08333333333333333 \cdot re\right) \cdot re\right)\right) \cdot 2 \]

   9. Applied rewrites 33.2%

      \[\leadsto \left(re \cdot \left(0.5 + \left(-0.08333333333333333 \cdot re\right) \cdot \color{blue}{re}\right)\right) \cdot 2 \]

   if -2e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

   7. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 48.3%

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

   10. Applied rewrites 48.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. metadata-eval N/A

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

       4. associate-+r+ N/A

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

       5. lower-+.f64 N/A

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

       6. lower-+.f64 48.3%

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 48.3%

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

   12. Applied rewrites 48.3%

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

Alternative 6: 48.3% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
3. Step-by-step derivation

4. Applied rewrites 51.2%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

5. Taylor expanded in re around 0

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

   1. lower-*.f64 26.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

7. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

8. Taylor expanded in im around 0

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

   1. lower-+.f64 N/A

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

   2. lower-pow.f64 48.3%

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

10. Applied rewrites 48.3%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. metadata-eval N/A

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

    4. associate-+r+ N/A

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

    5. lower-+.f64 N/A

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

    6. lower-+.f64 48.3%

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

    7. lift-pow.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 48.3%

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

12. Applied rewrites 48.3%

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

Alternative 7: 48.3% accurate, 16.7× speedup

Math

\[\left(im \cdot im - -2\right) \cdot \left(re \cdot 0.5\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (- (* im im) -2.0) (* re 0.5)))

C

double code(double re, double im) {
	return ((im * im) - -2.0) * (re * 0.5);
}

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) - (-2.0d0)) * (re * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return ((im * im) - -2.0) * (re * 0.5);
}

Python

def code(re, im):
	return ((im * im) - -2.0) * (re * 0.5)

Julia

function code(re, im)
	return Float64(Float64(Float64(im * im) - -2.0) * Float64(re * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = ((im * im) - -2.0) * (re * 0.5);
end

Wolfram

code[re_, im_] := N[(N[(N[(im * im), $MachinePrecision] - -2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
3. Step-by-step derivation

4. Applied rewrites 51.2%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

5. Taylor expanded in re around 0

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

   1. lower-*.f64 26.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

7. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

8. Taylor expanded in im around 0

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

   1. lower-+.f64 N/A

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

   2. lower-pow.f64 48.3%

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

10. Applied rewrites 48.3%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 48.3%

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. add-flip N/A

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

    7. lower--.f64 N/A

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

    8. lift-pow.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. metadata-eval 48.3%

        \[\leadsto \left(im \cdot im - -2\right) \cdot \left(0.5 \cdot re\right) \]

    12. metadata-eval N/A

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

    13. metadata-eval N/A

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

    14. metadata-eval N/A

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

12. Applied rewrites 48.3%

    \[\leadsto \color{blue}{\left(im \cdot im - -2\right) \cdot \left(re \cdot 0.5\right)} \]
13. Add Preprocessing

Alternative 8: 26.3% accurate, 28.8× speedup

Math

\[\left(0.5 \cdot re\right) \cdot 2 \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 re) 2.0))

C

double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * 2.0d0
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Python

def code(re, im):
	return (0.5 * re) * 2.0

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * 2.0)
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * 2.0;
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
3. Step-by-step derivation

4. Applied rewrites 51.2%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

5. Taylor expanded in re around 0

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

   1. lower-*.f64 26.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

7. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))