math.cos on complex, imaginary part

Percentage Accurate: 66.2% → 99.9%

Time: 5.0s

Alternatives: 8

Speedup: 1.3×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sinh \left(-im\right) \cdot \sin re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sinh (- im)) (sin re)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

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

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. sinh-def N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sinh im}\right)\right) \cdot \sin re \]

   14. sinh-neg N/A

       \[\leadsto \color{blue}{\sinh \left(\mathsf{neg}\left(im\right)\right)} \cdot \sin re \]

   15. lift-neg.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sinh.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-85}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -2e-85)
     (* (sinh (- im)) re)
     (if (<= t_0 5e-6)
       (* (sin re) (- im))
       (*
        (* re (+ 0.5 (* -0.08333333333333333 (pow re 2.0))))
        (- 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -2e-85) {
		tmp = sinh(-im) * re;
	} else if (t_0 <= 5e-6) {
		tmp = sin(re) * -im;
	} else {
		tmp = (re * (0.5 + (-0.08333333333333333 * pow(re, 2.0)))) * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    if (t_0 <= (-2d-85)) then
        tmp = sinh(-im) * re
    else if (t_0 <= 5d-6) then
        tmp = sin(re) * -im
    else
        tmp = (re * (0.5d0 + ((-0.08333333333333333d0) * (re ** 2.0d0)))) * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (t_0 <= -2e-85) {
		tmp = Math.sinh(-im) * re;
	} else if (t_0 <= 5e-6) {
		tmp = Math.sin(re) * -im;
	} else {
		tmp = (re * (0.5 + (-0.08333333333333333 * Math.pow(re, 2.0)))) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	tmp = 0
	if t_0 <= -2e-85:
		tmp = math.sinh(-im) * re
	elif t_0 <= 5e-6:
		tmp = math.sin(re) * -im
	else:
		tmp = (re * (0.5 + (-0.08333333333333333 * math.pow(re, 2.0)))) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -2e-85)
		tmp = Float64(sinh(Float64(-im)) * re);
	elseif (t_0 <= 5e-6)
		tmp = Float64(sin(re) * Float64(-im));
	else
		tmp = Float64(Float64(re * Float64(0.5 + Float64(-0.08333333333333333 * (re ^ 2.0)))) * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	tmp = 0.0;
	if (t_0 <= -2e-85)
		tmp = sinh(-im) * re;
	elseif (t_0 <= 5e-6)
		tmp = sin(re) * -im;
	else
		tmp = (re * (0.5 + (-0.08333333333333333 * (re ^ 2.0)))) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-85], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[(re * N[(0.5 + N[(-0.08333333333333333 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2e-85

   1. Initial program 66.2%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 53.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-def N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sinh im} \cdot re\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sinh im\right)\right) \cdot re} \]

   6. Applied rewrites 63.6%

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

   if -2e-85 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.00000000000000041e-6

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sin re \cdot im\right) \]

      5. distribute-rgt-neg-in N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-*.f64 50.9

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

   6. Applied rewrites 50.9%

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

   if 5.00000000000000041e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.2%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 37.1

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

   10. Applied rewrites 37.1%

       \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(1 - e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-85}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -2e-85)
     (* (sinh (- im)) re)
     (if (<= t_0 5e-6)
       (* (sin re) (- im))
       (* 0.16666666666666666 (* (* re re) (* im re)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -2e-85) {
		tmp = sinh(-im) * re;
	} else if (t_0 <= 5e-6) {
		tmp = sin(re) * -im;
	} else {
		tmp = 0.16666666666666666 * ((re * re) * (im * re));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    if (t_0 <= (-2d-85)) then
        tmp = sinh(-im) * re
    else if (t_0 <= 5d-6) then
        tmp = sin(re) * -im
    else
        tmp = 0.16666666666666666d0 * ((re * re) * (im * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (t_0 <= -2e-85) {
		tmp = Math.sinh(-im) * re;
	} else if (t_0 <= 5e-6) {
		tmp = Math.sin(re) * -im;
	} else {
		tmp = 0.16666666666666666 * ((re * re) * (im * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	tmp = 0
	if t_0 <= -2e-85:
		tmp = math.sinh(-im) * re
	elif t_0 <= 5e-6:
		tmp = math.sin(re) * -im
	else:
		tmp = 0.16666666666666666 * ((re * re) * (im * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -2e-85)
		tmp = Float64(sinh(Float64(-im)) * re);
	elseif (t_0 <= 5e-6)
		tmp = Float64(sin(re) * Float64(-im));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(re * re) * Float64(im * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	tmp = 0.0;
	if (t_0 <= -2e-85)
		tmp = sinh(-im) * re;
	elseif (t_0 <= 5e-6)
		tmp = sin(re) * -im;
	else
		tmp = 0.16666666666666666 * ((re * re) * (im * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-85], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(0.16666666666666666 * N[(N[(re * re), $MachinePrecision] * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2e-85

   1. Initial program 66.2%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 53.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-def N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sinh im} \cdot re\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sinh im\right)\right) \cdot re} \]

   6. Applied rewrites 63.6%

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

   if -2e-85 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.00000000000000041e-6

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sin re \cdot im\right) \]

      5. distribute-rgt-neg-in N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-*.f64 50.9

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

   6. Applied rewrites 50.9%

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

   if 5.00000000000000041e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 36.1

         \[\leadsto re \cdot \mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{6} \cdot \left(im \cdot {re}^{\color{blue}{3}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{6} \cdot \left(im \cdot {re}^{3}\right) \]

      3. lower-pow.f64 24.0

         \[\leadsto 0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right) \]

   10. Applied rewrites 24.0%

       \[\leadsto 0.16666666666666666 \cdot \left(im \cdot \color{blue}{{re}^{3}}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot {re}^{3}\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \left({re}^{3} \cdot im\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left({re}^{3} \cdot im\right) \]

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 24.0

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 24.0

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

   12. Applied rewrites 24.0%

       \[\leadsto 0.16666666666666666 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.0002:\\ \;\;\;\;re \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot im\right) \cdot re, re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.0002)
   (* re (fma (* (* 0.16666666666666666 im) re) re (- im)))
   (* (sinh (- im)) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.0002) {
		tmp = re * fma(((0.16666666666666666 * im) * re), re, -im);
	} else {
		tmp = sinh(-im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.0002)
		tmp = Float64(re * fma(Float64(Float64(0.16666666666666666 * im) * re), re, Float64(-im)));
	else
		tmp = Float64(sinh(Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.0002], N[(re * N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * re), $MachinePrecision] * re + (-im)), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -2.0000000000000001e-4

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 36.1

         \[\leadsto re \cdot \mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   7. Applied rewrites 36.1%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 36.1

          \[\leadsto re \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot im\right) \cdot re, re, -im\right) \]

   9. Applied rewrites 36.1%

      \[\leadsto re \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot im\right) \cdot re, re, -im\right) \]

   if -2.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.2%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 53.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-def N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sinh im} \cdot re\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sinh im\right)\right) \cdot re} \]

   6. Applied rewrites 63.6%

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

Alternative 5: 34.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot im\right) \cdot re, re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - \left(1 + im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (* re (fma (* (* 0.16666666666666666 im) re) re (- im)))
   (* (* 0.5 re) (- 1.0 (+ 1.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = re * fma(((0.16666666666666666 * im) * re), re, -im);
	} else {
		tmp = (0.5 * re) * (1.0 - (1.0 + im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(re * fma(Float64(Float64(0.16666666666666666 * im) * re), re, Float64(-im)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - Float64(1.0 + im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(re * N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * re), $MachinePrecision] * re + (-im)), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 36.1

         \[\leadsto re \cdot \mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   7. Applied rewrites 36.1%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 36.1

          \[\leadsto re \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot im\right) \cdot re, re, -im\right) \]

   9. Applied rewrites 36.1%

      \[\leadsto re \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot im\right) \cdot re, re, -im\right) \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.2%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 21.9

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

   10. Applied rewrites 21.9%

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

Alternative 6: 34.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - \left(1 + im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (* (* im (fma (* re re) 0.16666666666666666 -1.0)) re)
   (* (* 0.5 re) (- 1.0 (+ 1.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = (im * fma((re * re), 0.16666666666666666, -1.0)) * re;
	} else {
		tmp = (0.5 * re) * (1.0 - (1.0 + im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(Float64(im * fma(Float64(re * re), 0.16666666666666666, -1.0)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - Float64(1.0 + im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(im * N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 36.1

         \[\leadsto re \cdot \mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   7. Applied rewrites 36.1%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.1

         \[\leadsto \mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot re \]

   9. Applied rewrites 36.1%

      \[\leadsto \left(im \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right) \cdot \color{blue}{re} \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.2%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 21.9

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

   10. Applied rewrites 21.9%

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

Alternative 7: 34.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.0002:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.0002)
   (* 0.16666666666666666 (* (* re re) (* im re)))
   (- (* re im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.0002) {
		tmp = 0.16666666666666666 * ((re * re) * (im * re));
	} else {
		tmp = -(re * 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) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.0002d0)) then
        tmp = 0.16666666666666666d0 * ((re * re) * (im * re))
    else
        tmp = -(re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.0002) {
		tmp = 0.16666666666666666 * ((re * re) * (im * re));
	} else {
		tmp = -(re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.0002:
		tmp = 0.16666666666666666 * ((re * re) * (im * re))
	else:
		tmp = -(re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.0002)
		tmp = Float64(0.16666666666666666 * Float64(Float64(re * re) * Float64(im * re)));
	else
		tmp = Float64(-Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.0002)
		tmp = 0.16666666666666666 * ((re * re) * (im * re));
	else
		tmp = -(re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.0002], N[(0.16666666666666666 * N[(N[(re * re), $MachinePrecision] * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(re * im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -2.0000000000000001e-4

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 36.1

         \[\leadsto re \cdot \mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{6} \cdot \left(im \cdot {re}^{\color{blue}{3}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{6} \cdot \left(im \cdot {re}^{3}\right) \]

      3. lower-pow.f64 24.0

         \[\leadsto 0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right) \]

   10. Applied rewrites 24.0%

       \[\leadsto 0.16666666666666666 \cdot \left(im \cdot \color{blue}{{re}^{3}}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot {re}^{3}\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \left({re}^{3} \cdot im\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left({re}^{3} \cdot im\right) \]

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 24.0

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 24.0

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

   12. Applied rewrites 24.0%

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

   if -2.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.2%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.9

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 32.7

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

   7. Applied rewrites 32.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

      3. lower-neg.f64 32.7

         \[\leadsto -im \cdot re \]

      4. lift-*.f64 N/A

         \[\leadsto -im \cdot re \]

      5. *-commutative N/A

         \[\leadsto -re \cdot im \]

      6. lower-*.f64 32.7

         \[\leadsto -re \cdot im \]

   9. Applied rewrites 32.7%

      \[\leadsto -re \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 32.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ -re \cdot im \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

2. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 50.9

      \[\leadsto -1 \cdot \left(im \cdot \sin re\right) \]

4. Applied rewrites 50.9%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 32.7

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

7. Applied rewrites 32.7%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

   3. lower-neg.f64 32.7

      \[\leadsto -im \cdot re \]

   4. lift-*.f64 N/A

      \[\leadsto -im \cdot re \]

   5. *-commutative N/A

      \[\leadsto -re \cdot im \]

   6. lower-*.f64 32.7

      \[\leadsto -re \cdot im \]

9. Applied rewrites 32.7%

   \[\leadsto -re \cdot im \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))