math.cos on complex, imaginary part

Percentage Accurate: 66.4% → 99.9%

Time: 1.1min

Alternatives: 6

Speedup: 1.5×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 1/2 (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[(1/2 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 1/2 (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[(1/2 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

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 66.4%

   \[\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
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: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -\left|im\right|\\ t_1 := \left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot \left|im\right|\right) \cdot \frac{-1}{6}\\ t_2 := \sin \left(\left|re\right|\right)\\ t_3 := \left(\frac{1}{2} \cdot t\_2\right) \cdot \left(e^{t\_0} - e^{\left|im\right|}\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-1 \cdot \left(\left|re\right| \cdot \frac{t\_1 \cdot t\_1 - \left|im\right| \cdot \left|im\right|}{t\_1 - \left|im\right|}\right)\\ \mathbf{elif}\;t\_3 \leq 500000000:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{-1}{6} \cdot \left(\left|im\right| \cdot {\left(\left|re\right|\right)}^{3}\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (- (fabs im)))
       (t_1 (* (* (* (fabs re) (fabs re)) (fabs im)) -1/6))
       (t_2 (sin (fabs re)))
       (t_3 (* (* 1/2 t_2) (- (exp t_0) (exp (fabs im))))))
  (*
   (copysign 1 re)
   (*
    (copysign 1 im)
    (if (<= t_3 (- INFINITY))
      (*
       -1
       (*
        (fabs re)
        (/
         (- (* t_1 t_1) (* (fabs im) (fabs im)))
         (- t_1 (fabs im)))))
      (if (<= t_3 500000000)
        (* t_2 t_0)
        (* -1 (* -1/6 (* (fabs im) (pow (fabs re) 3))))))))))

C

double code(double re, double im) {
	double t_0 = -fabs(im);
	double t_1 = ((fabs(re) * fabs(re)) * fabs(im)) * -0.16666666666666666;
	double t_2 = sin(fabs(re));
	double t_3 = (0.5 * t_2) * (exp(t_0) - exp(fabs(im)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 * (fabs(re) * (((t_1 * t_1) - (fabs(im) * fabs(im))) / (t_1 - fabs(im))));
	} else if (t_3 <= 500000000.0) {
		tmp = t_2 * t_0;
	} else {
		tmp = -1.0 * (-0.16666666666666666 * (fabs(im) * pow(fabs(re), 3.0)));
	}
	return copysign(1.0, re) * (copysign(1.0, im) * tmp);
}

Java

public static double code(double re, double im) {
	double t_0 = -Math.abs(im);
	double t_1 = ((Math.abs(re) * Math.abs(re)) * Math.abs(im)) * -0.16666666666666666;
	double t_2 = Math.sin(Math.abs(re));
	double t_3 = (0.5 * t_2) * (Math.exp(t_0) - Math.exp(Math.abs(im)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -1.0 * (Math.abs(re) * (((t_1 * t_1) - (Math.abs(im) * Math.abs(im))) / (t_1 - Math.abs(im))));
	} else if (t_3 <= 500000000.0) {
		tmp = t_2 * t_0;
	} else {
		tmp = -1.0 * (-0.16666666666666666 * (Math.abs(im) * Math.pow(Math.abs(re), 3.0)));
	}
	return Math.copySign(1.0, re) * (Math.copySign(1.0, im) * tmp);
}

Python

def code(re, im):
	t_0 = -math.fabs(im)
	t_1 = ((math.fabs(re) * math.fabs(re)) * math.fabs(im)) * -0.16666666666666666
	t_2 = math.sin(math.fabs(re))
	t_3 = (0.5 * t_2) * (math.exp(t_0) - math.exp(math.fabs(im)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -1.0 * (math.fabs(re) * (((t_1 * t_1) - (math.fabs(im) * math.fabs(im))) / (t_1 - math.fabs(im))))
	elif t_3 <= 500000000.0:
		tmp = t_2 * t_0
	else:
		tmp = -1.0 * (-0.16666666666666666 * (math.fabs(im) * math.pow(math.fabs(re), 3.0)))
	return math.copysign(1.0, re) * (math.copysign(1.0, im) * tmp)

Julia

function code(re, im)
	t_0 = Float64(-abs(im))
	t_1 = Float64(Float64(Float64(abs(re) * abs(re)) * abs(im)) * -0.16666666666666666)
	t_2 = sin(abs(re))
	t_3 = Float64(Float64(0.5 * t_2) * Float64(exp(t_0) - exp(abs(im))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 * Float64(abs(re) * Float64(Float64(Float64(t_1 * t_1) - Float64(abs(im) * abs(im))) / Float64(t_1 - abs(im)))));
	elseif (t_3 <= 500000000.0)
		tmp = Float64(t_2 * t_0);
	else
		tmp = Float64(-1.0 * Float64(-0.16666666666666666 * Float64(abs(im) * (abs(re) ^ 3.0))));
	end
	return Float64(copysign(1.0, re) * Float64(copysign(1.0, im) * tmp))
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -abs(im);
	t_1 = ((abs(re) * abs(re)) * abs(im)) * -0.16666666666666666;
	t_2 = sin(abs(re));
	t_3 = (0.5 * t_2) * (exp(t_0) - exp(abs(im)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -1.0 * (abs(re) * (((t_1 * t_1) - (abs(im) * abs(im))) / (t_1 - abs(im))));
	elseif (t_3 <= 500000000.0)
		tmp = t_2 * t_0;
	else
		tmp = -1.0 * (-0.16666666666666666 * (abs(im) * (abs(re) ^ 3.0)));
	end
	tmp_2 = (sign(re) * abs(1.0)) * ((sign(im) * abs(1.0)) * tmp);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Abs[im], $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * -1/6), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1/2 * t$95$2), $MachinePrecision] * N[(N[Exp[t$95$0], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(-1 * N[(N[Abs[re], $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 500000000], N[(t$95$2 * t$95$0), $MachinePrecision], N[(-1 * N[(-1/6 * N[(N[Abs[im], $MachinePrecision] * N[Power[N[Abs[re], $MachinePrecision], 3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 37.1%

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

   7. Applied rewrites 37.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   9. Applied rewrites 35.5%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5e8

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   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 51.0%

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

   6. Applied rewrites 51.0%

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

   if 5e8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 37.1%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 24.7%

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

   10. Applied rewrites 24.7%

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

Alternative 3: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -\left|im\right|\\ t_1 := \left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot \left|im\right|\right) \cdot \frac{-1}{6}\\ t_2 := \sin \left(\left|re\right|\right)\\ t_3 := \left(\frac{1}{2} \cdot t\_2\right) \cdot \left(e^{t\_0} - e^{\left|im\right|}\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-1 \cdot \left(\left|re\right| \cdot \frac{t\_1 \cdot t\_1 - \left|im\right| \cdot \left|im\right|}{t\_1 - \left|im\right|}\right)\\ \mathbf{elif}\;t\_3 \leq 500000000:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(\left(\frac{-1}{6} \cdot \left|im\right|\right) \cdot \left|re\right|\right) \cdot \left|re\right| + \left|im\right|\right) \cdot \left|re\right|\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (- (fabs im)))
       (t_1 (* (* (* (fabs re) (fabs re)) (fabs im)) -1/6))
       (t_2 (sin (fabs re)))
       (t_3 (* (* 1/2 t_2) (- (exp t_0) (exp (fabs im))))))
  (*
   (copysign 1 re)
   (*
    (copysign 1 im)
    (if (<= t_3 (- INFINITY))
      (*
       -1
       (*
        (fabs re)
        (/
         (- (* t_1 t_1) (* (fabs im) (fabs im)))
         (- t_1 (fabs im)))))
      (if (<= t_3 500000000)
        (* t_2 t_0)
        (-
         (*
          (+ (* (* (* -1/6 (fabs im)) (fabs re)) (fabs re)) (fabs im))
          (fabs re)))))))))

C

double code(double re, double im) {
	double t_0 = -fabs(im);
	double t_1 = ((fabs(re) * fabs(re)) * fabs(im)) * -0.16666666666666666;
	double t_2 = sin(fabs(re));
	double t_3 = (0.5 * t_2) * (exp(t_0) - exp(fabs(im)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 * (fabs(re) * (((t_1 * t_1) - (fabs(im) * fabs(im))) / (t_1 - fabs(im))));
	} else if (t_3 <= 500000000.0) {
		tmp = t_2 * t_0;
	} else {
		tmp = -(((((-0.16666666666666666 * fabs(im)) * fabs(re)) * fabs(re)) + fabs(im)) * fabs(re));
	}
	return copysign(1.0, re) * (copysign(1.0, im) * tmp);
}

Java

public static double code(double re, double im) {
	double t_0 = -Math.abs(im);
	double t_1 = ((Math.abs(re) * Math.abs(re)) * Math.abs(im)) * -0.16666666666666666;
	double t_2 = Math.sin(Math.abs(re));
	double t_3 = (0.5 * t_2) * (Math.exp(t_0) - Math.exp(Math.abs(im)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -1.0 * (Math.abs(re) * (((t_1 * t_1) - (Math.abs(im) * Math.abs(im))) / (t_1 - Math.abs(im))));
	} else if (t_3 <= 500000000.0) {
		tmp = t_2 * t_0;
	} else {
		tmp = -(((((-0.16666666666666666 * Math.abs(im)) * Math.abs(re)) * Math.abs(re)) + Math.abs(im)) * Math.abs(re));
	}
	return Math.copySign(1.0, re) * (Math.copySign(1.0, im) * tmp);
}

Python

def code(re, im):
	t_0 = -math.fabs(im)
	t_1 = ((math.fabs(re) * math.fabs(re)) * math.fabs(im)) * -0.16666666666666666
	t_2 = math.sin(math.fabs(re))
	t_3 = (0.5 * t_2) * (math.exp(t_0) - math.exp(math.fabs(im)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -1.0 * (math.fabs(re) * (((t_1 * t_1) - (math.fabs(im) * math.fabs(im))) / (t_1 - math.fabs(im))))
	elif t_3 <= 500000000.0:
		tmp = t_2 * t_0
	else:
		tmp = -(((((-0.16666666666666666 * math.fabs(im)) * math.fabs(re)) * math.fabs(re)) + math.fabs(im)) * math.fabs(re))
	return math.copysign(1.0, re) * (math.copysign(1.0, im) * tmp)

Julia

function code(re, im)
	t_0 = Float64(-abs(im))
	t_1 = Float64(Float64(Float64(abs(re) * abs(re)) * abs(im)) * -0.16666666666666666)
	t_2 = sin(abs(re))
	t_3 = Float64(Float64(0.5 * t_2) * Float64(exp(t_0) - exp(abs(im))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 * Float64(abs(re) * Float64(Float64(Float64(t_1 * t_1) - Float64(abs(im) * abs(im))) / Float64(t_1 - abs(im)))));
	elseif (t_3 <= 500000000.0)
		tmp = Float64(t_2 * t_0);
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * abs(im)) * abs(re)) * abs(re)) + abs(im)) * abs(re)));
	end
	return Float64(copysign(1.0, re) * Float64(copysign(1.0, im) * tmp))
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -abs(im);
	t_1 = ((abs(re) * abs(re)) * abs(im)) * -0.16666666666666666;
	t_2 = sin(abs(re));
	t_3 = (0.5 * t_2) * (exp(t_0) - exp(abs(im)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -1.0 * (abs(re) * (((t_1 * t_1) - (abs(im) * abs(im))) / (t_1 - abs(im))));
	elseif (t_3 <= 500000000.0)
		tmp = t_2 * t_0;
	else
		tmp = -(((((-0.16666666666666666 * abs(im)) * abs(re)) * abs(re)) + abs(im)) * abs(re));
	end
	tmp_2 = (sign(re) * abs(1.0)) * ((sign(im) * abs(1.0)) * tmp);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Abs[im], $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * -1/6), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1/2 * t$95$2), $MachinePrecision] * N[(N[Exp[t$95$0], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(-1 * N[(N[Abs[re], $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 500000000], N[(t$95$2 * t$95$0), $MachinePrecision], (-N[(N[(N[(N[(N[(-1/6 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $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))) < -inf.0

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 37.1%

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

   7. Applied rewrites 37.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   9. Applied rewrites 35.5%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5e8

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   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 51.0%

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

   6. Applied rewrites 51.0%

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

   if 5e8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 37.1%

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

   7. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. lower-134-z0z1z2z3z4 N/A

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{re}\right), re, im, -1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

      15. lower-*.f64 30.7%

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

   9. Applied rewrites 30.7%

      \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{re}\right), re, im, -1\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right)\right) \]

       3. lower-neg.f64 30.7%

          \[\leadsto -\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

       4. lift-134-z0z1z2z3z4 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. mul-1-neg N/A

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

       9. add-flip-rev N/A

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

       10. lower-+.f64 N/A

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

       11. lower-*.f64 37.1%

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

   11. Applied rewrites 37.1%

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

Alternative 4: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \sin \left(\left|re\right|\right)\\ t_1 := \left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot im\right) \cdot \frac{-1}{6}\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq \frac{-5764607523034235}{144115188075855872}:\\ \;\;\;\;-\left(\left(\left(\frac{-1}{6} \cdot im\right) \cdot \left|re\right|\right) \cdot \left|re\right| + im\right) \cdot \left|re\right|\\ \mathbf{elif}\;t\_0 \leq \frac{5265614583427859}{52656145834278593348959013841835216159447547700274555627155488768}:\\ \;\;\;\;-1 \cdot \left(\left|re\right| \cdot \frac{t\_1 \cdot t\_1 - im \cdot im}{t\_1 - im}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|re\right| \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* 1/2 (sin (fabs re))))
       (t_1 (* (* (* (fabs re) (fabs re)) im) -1/6)))
  (*
   (copysign 1 re)
   (if (<= t_0 -5764607523034235/144115188075855872)
     (- (* (+ (* (* (* -1/6 im) (fabs re)) (fabs re)) im) (fabs re)))
     (if (<=
          t_0
          5265614583427859/52656145834278593348959013841835216159447547700274555627155488768)
       (* -1 (* (fabs re) (/ (- (* t_1 t_1) (* im im)) (- t_1 im))))
       (- (* (fabs re) im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(fabs(re));
	double t_1 = ((fabs(re) * fabs(re)) * im) * -0.16666666666666666;
	double tmp;
	if (t_0 <= -0.04) {
		tmp = -(((((-0.16666666666666666 * im) * fabs(re)) * fabs(re)) + im) * fabs(re));
	} else if (t_0 <= 1e-49) {
		tmp = -1.0 * (fabs(re) * (((t_1 * t_1) - (im * im)) / (t_1 - im)));
	} else {
		tmp = -(fabs(re) * im);
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(Math.abs(re));
	double t_1 = ((Math.abs(re) * Math.abs(re)) * im) * -0.16666666666666666;
	double tmp;
	if (t_0 <= -0.04) {
		tmp = -(((((-0.16666666666666666 * im) * Math.abs(re)) * Math.abs(re)) + im) * Math.abs(re));
	} else if (t_0 <= 1e-49) {
		tmp = -1.0 * (Math.abs(re) * (((t_1 * t_1) - (im * im)) / (t_1 - im)));
	} else {
		tmp = -(Math.abs(re) * im);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(math.fabs(re))
	t_1 = ((math.fabs(re) * math.fabs(re)) * im) * -0.16666666666666666
	tmp = 0
	if t_0 <= -0.04:
		tmp = -(((((-0.16666666666666666 * im) * math.fabs(re)) * math.fabs(re)) + im) * math.fabs(re))
	elif t_0 <= 1e-49:
		tmp = -1.0 * (math.fabs(re) * (((t_1 * t_1) - (im * im)) / (t_1 - im)))
	else:
		tmp = -(math.fabs(re) * im)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(abs(re)))
	t_1 = Float64(Float64(Float64(abs(re) * abs(re)) * im) * -0.16666666666666666)
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * im) * abs(re)) * abs(re)) + im) * abs(re)));
	elseif (t_0 <= 1e-49)
		tmp = Float64(-1.0 * Float64(abs(re) * Float64(Float64(Float64(t_1 * t_1) - Float64(im * im)) / Float64(t_1 - im))));
	else
		tmp = Float64(-Float64(abs(re) * im));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(abs(re));
	t_1 = ((abs(re) * abs(re)) * im) * -0.16666666666666666;
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = -(((((-0.16666666666666666 * im) * abs(re)) * abs(re)) + im) * abs(re));
	elseif (t_0 <= 1e-49)
		tmp = -1.0 * (abs(re) * (((t_1 * t_1) - (im * im)) / (t_1 - im)));
	else
		tmp = -(abs(re) * im);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1/2 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision] * -1/6), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, -5764607523034235/144115188075855872], (-N[(N[(N[(N[(N[(-1/6 * im), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$0, 5265614583427859/52656145834278593348959013841835216159447547700274555627155488768], N[(-1 * N[(N[Abs[re], $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Abs[re], $MachinePrecision] * im), $MachinePrecision])]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.040000000000000001

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 37.1%

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

   7. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. lower-134-z0z1z2z3z4 N/A

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{re}\right), re, im, -1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

      15. lower-*.f64 30.7%

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

   9. Applied rewrites 30.7%

      \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{re}\right), re, im, -1\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right)\right) \]

       3. lower-neg.f64 30.7%

          \[\leadsto -\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

       4. lift-134-z0z1z2z3z4 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. mul-1-neg N/A

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

       9. add-flip-rev N/A

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

       10. lower-+.f64 N/A

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

       11. lower-*.f64 37.1%

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

   11. Applied rewrites 37.1%

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

   if -0.040000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 9.9999999999999994e-50

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 37.1%

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

   7. Applied rewrites 37.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   9. Applied rewrites 35.5%

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

   if 9.9999999999999994e-50 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   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 32.8%

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

   7. Applied rewrites 32.8%

      \[\leadsto -1 \cdot \left(im \cdot \color{blue}{re}\right) \]
   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 32.8%

         \[\leadsto -im \cdot re \]

      4. lift-*.f64 N/A

         \[\leadsto -im \cdot re \]

      5. *-commutative N/A

         \[\leadsto -re \cdot im \]

      6. lower-*.f64 32.8%

         \[\leadsto -re \cdot im \]

   9. Applied rewrites 32.8%

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

Alternative 5: 43.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= -0.01)
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * im) * abs(re)) * abs(re)) + im) * abs(re)));
	else
		tmp = Float64(-Float64(abs(re) * im));
	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))) <= -0.01)
		tmp = -(((((-0.16666666666666666 * im) * abs(re)) * abs(re)) + im) * abs(re));
	else
		tmp = -(abs(re) * im);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.01

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 37.1%

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

   7. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. lower-134-z0z1z2z3z4 N/A

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{re}\right), re, im, -1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

      15. lower-*.f64 30.7%

          \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

   9. Applied rewrites 30.7%

      \[\leadsto -1 \cdot \mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{re}\right), re, im, -1\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right)\right) \]

       3. lower-neg.f64 30.7%

          \[\leadsto -\mathsf{134\_z0z1z2z3z4}\left(re, \left(\left(\frac{-1}{6} \cdot im\right) \cdot re\right), re, im, -1\right) \]

       4. lift-134-z0z1z2z3z4 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. mul-1-neg N/A

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

       9. add-flip-rev N/A

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

       10. lower-+.f64 N/A

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

       11. lower-*.f64 37.1%

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

   11. Applied rewrites 37.1%

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

   if -0.01 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.4%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   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 32.8%

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

   7. Applied rewrites 32.8%

      \[\leadsto -1 \cdot \left(im \cdot \color{blue}{re}\right) \]
   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 32.8%

         \[\leadsto -im \cdot re \]

      4. lift-*.f64 N/A

         \[\leadsto -im \cdot re \]

      5. *-commutative N/A

         \[\leadsto -re \cdot im \]

      6. lower-*.f64 32.8%

         \[\leadsto -re \cdot im \]

   9. Applied rewrites 32.8%

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

Alternative 6: 32.8% accurate, 39.5× speedup

Math

\[-re \cdot im \]

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 66.4%

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

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 51.0%

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

4. Applied rewrites 51.0%

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

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 32.8%

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

7. Applied rewrites 32.8%

   \[\leadsto -1 \cdot \left(im \cdot \color{blue}{re}\right) \]
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 32.8%

      \[\leadsto -im \cdot re \]

   4. lift-*.f64 N/A

      \[\leadsto -im \cdot re \]

   5. *-commutative N/A

      \[\leadsto -re \cdot im \]

   6. lower-*.f64 32.8%

      \[\leadsto -re \cdot im \]

9. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 1/2 (sin re)) (- (exp (- im)) (exp im))))