math.cos on complex, imaginary part

Percentage Accurate: 66.0% → 99.9%

Time: 5.0s

Alternatives: 9

Speedup: 1.3×

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 66.0% accurate, 1.0× speedup

Math

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

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

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

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 66.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
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: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -\left|im\right|\\ t_1 := e^{\left|im\right|}\\ t_2 := 1 - t\_1\\ t_3 := \sin \left(\left|re\right|\right)\\ t_4 := \left(0.5 \cdot t\_3\right) \cdot \left(e^{t\_0} - t\_1\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot t\_2\\ \mathbf{elif}\;t\_4 \leq 10^{-12}:\\ \;\;\;\;t\_3 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left|re\right| \cdot \left(0.5 + -0.08333333333333333 \cdot {\left(\left|re\right|\right)}^{2}\right)\right) \cdot t\_2\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (- (fabs im)))
       (t_1 (exp (fabs im)))
       (t_2 (- 1.0 t_1))
       (t_3 (sin (fabs re)))
       (t_4 (* (* 0.5 t_3) (- (exp t_0) t_1))))
  (*
   (copysign 1.0 re)
   (*
    (copysign 1.0 im)
    (if (<= t_4 (- INFINITY))
      (* (* 0.5 (fabs re)) t_2)
      (if (<= t_4 1e-12)
        (* t_3 t_0)
        (*
         (*
          (fabs re)
          (+ 0.5 (* -0.08333333333333333 (pow (fabs re) 2.0))))
         t_2)))))))

C

double code(double re, double im) {
	double t_0 = -fabs(im);
	double t_1 = exp(fabs(im));
	double t_2 = 1.0 - t_1;
	double t_3 = sin(fabs(re));
	double t_4 = (0.5 * t_3) * (exp(t_0) - t_1);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (0.5 * fabs(re)) * t_2;
	} else if (t_4 <= 1e-12) {
		tmp = t_3 * t_0;
	} else {
		tmp = (fabs(re) * (0.5 + (-0.08333333333333333 * pow(fabs(re), 2.0)))) * t_2;
	}
	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.exp(Math.abs(im));
	double t_2 = 1.0 - t_1;
	double t_3 = Math.sin(Math.abs(re));
	double t_4 = (0.5 * t_3) * (Math.exp(t_0) - t_1);
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 * Math.abs(re)) * t_2;
	} else if (t_4 <= 1e-12) {
		tmp = t_3 * t_0;
	} else {
		tmp = (Math.abs(re) * (0.5 + (-0.08333333333333333 * Math.pow(Math.abs(re), 2.0)))) * t_2;
	}
	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.exp(math.fabs(im))
	t_2 = 1.0 - t_1
	t_3 = math.sin(math.fabs(re))
	t_4 = (0.5 * t_3) * (math.exp(t_0) - t_1)
	tmp = 0
	if t_4 <= -math.inf:
		tmp = (0.5 * math.fabs(re)) * t_2
	elif t_4 <= 1e-12:
		tmp = t_3 * t_0
	else:
		tmp = (math.fabs(re) * (0.5 + (-0.08333333333333333 * math.pow(math.fabs(re), 2.0)))) * t_2
	return math.copysign(1.0, re) * (math.copysign(1.0, im) * tmp)

Julia

function code(re, im)
	t_0 = Float64(-abs(im))
	t_1 = exp(abs(im))
	t_2 = Float64(1.0 - t_1)
	t_3 = sin(abs(re))
	t_4 = Float64(Float64(0.5 * t_3) * Float64(exp(t_0) - t_1))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * abs(re)) * t_2);
	elseif (t_4 <= 1e-12)
		tmp = Float64(t_3 * t_0);
	else
		tmp = Float64(Float64(abs(re) * Float64(0.5 + Float64(-0.08333333333333333 * (abs(re) ^ 2.0)))) * t_2);
	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 = exp(abs(im));
	t_2 = 1.0 - t_1;
	t_3 = sin(abs(re));
	t_4 = (0.5 * t_3) * (exp(t_0) - t_1);
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = (0.5 * abs(re)) * t_2;
	elseif (t_4 <= 1e-12)
		tmp = t_3 * t_0;
	else
		tmp = (abs(re) * (0.5 + (-0.08333333333333333 * (abs(re) ^ 2.0)))) * t_2;
	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[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 * t$95$3), $MachinePrecision] * N[(N[Exp[t$95$0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 1e-12], N[(t$95$3 * t$95$0), $MachinePrecision], N[(N[(N[Abs[re], $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[Power[N[Abs[re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $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 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 9.9999999999999998e-13

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   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 51.4%

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

   6. Applied rewrites 51.4%

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

   if 9.9999999999999998e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.8%

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

   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 37.5%

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

   10. Applied rewrites 37.5%

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

Alternative 3: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -\left|im\right|\\ t_1 := e^{\left|im\right|}\\ t_2 := \sin \left(\left|re\right|\right)\\ t_3 := \left(0.5 \cdot t\_2\right) \cdot \left(e^{t\_0} - t\_1\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - t\_1\right)\\ \mathbf{elif}\;t\_3 \leq 10^{-12}:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left|im\right| \cdot {\left(\left|re\right|\right)}^{3}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (- (fabs im)))
       (t_1 (exp (fabs im)))
       (t_2 (sin (fabs re)))
       (t_3 (* (* 0.5 t_2) (- (exp t_0) t_1))))
  (*
   (copysign 1.0 re)
   (*
    (copysign 1.0 im)
    (if (<= t_3 (- INFINITY))
      (* (* 0.5 (fabs re)) (- 1.0 t_1))
      (if (<= t_3 1e-12)
        (* t_2 t_0)
        (* 0.16666666666666666 (* (fabs im) (pow (fabs re) 3.0)))))))))

C

double code(double re, double im) {
	double t_0 = -fabs(im);
	double t_1 = exp(fabs(im));
	double t_2 = sin(fabs(re));
	double t_3 = (0.5 * t_2) * (exp(t_0) - t_1);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (0.5 * fabs(re)) * (1.0 - t_1);
	} else if (t_3 <= 1e-12) {
		tmp = t_2 * t_0;
	} else {
		tmp = 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.exp(Math.abs(im));
	double t_2 = Math.sin(Math.abs(re));
	double t_3 = (0.5 * t_2) * (Math.exp(t_0) - t_1);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 * Math.abs(re)) * (1.0 - t_1);
	} else if (t_3 <= 1e-12) {
		tmp = t_2 * t_0;
	} else {
		tmp = 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.exp(math.fabs(im))
	t_2 = math.sin(math.fabs(re))
	t_3 = (0.5 * t_2) * (math.exp(t_0) - t_1)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (0.5 * math.fabs(re)) * (1.0 - t_1)
	elif t_3 <= 1e-12:
		tmp = t_2 * t_0
	else:
		tmp = 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 = exp(abs(im))
	t_2 = sin(abs(re))
	t_3 = Float64(Float64(0.5 * t_2) * Float64(exp(t_0) - t_1))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - t_1));
	elseif (t_3 <= 1e-12)
		tmp = Float64(t_2 * t_0);
	else
		tmp = 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 = exp(abs(im));
	t_2 = sin(abs(re));
	t_3 = (0.5 * t_2) * (exp(t_0) - t_1);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (0.5 * abs(re)) * (1.0 - t_1);
	elseif (t_3 <= 1e-12)
		tmp = t_2 * t_0;
	else
		tmp = 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[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * t$95$2), $MachinePrecision] * N[(N[Exp[t$95$0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-12], N[(t$95$2 * t$95$0), $MachinePrecision], N[(0.16666666666666666 * N[(N[Abs[im], $MachinePrecision] * N[Power[N[Abs[re], $MachinePrecision], 3.0], $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 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 9.9999999999999998e-13

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   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 51.4%

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

   6. Applied rewrites 51.4%

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

   if 9.9999999999999998e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

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

   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 37.4%

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

   7. Applied rewrites 37.4%

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

   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 25.1%

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

   10. Applied rewrites 25.1%

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

Alternative 4: 73.8% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq -0.02:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(0.16666666666666666 \cdot \left|re\right|, \left|re\right|, -1\right)\right) \cdot \left|re\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sinh im\right) \cdot \left(0.5 \cdot \left|re\right|\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) -0.02)
   (*
    (* im (fma (* 0.16666666666666666 (fabs re)) (fabs re) -1.0))
    (fabs re))
   (* (* -2.0 (sinh im)) (* 0.5 (fabs re))))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= -0.02) {
		tmp = (im * fma((0.16666666666666666 * fabs(re)), fabs(re), -1.0)) * fabs(re);
	} else {
		tmp = (-2.0 * sinh(im)) * (0.5 * fabs(re));
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= -0.02)
		tmp = Float64(Float64(im * fma(Float64(0.16666666666666666 * abs(re)), abs(re), -1.0)) * abs(re));
	else
		tmp = Float64(Float64(-2.0 * sinh(im)) * Float64(0.5 * abs(re)));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(im * N[(N[(0.16666666666666666 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.02

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

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

   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 37.4%

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

   7. Applied rewrites 37.4%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-neg.f64 37.4%

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

   9. Applied rewrites 37.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 37.4%

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

       4. lift-fma.f64 N/A

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

       5. lift-neg.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. mul-1-neg N/A

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

       9. distribute-rgt-out N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 37.4%

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

   11. Applied rewrites 37.4%

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

   if -0.02 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(e^{-im} - e^{im}\right) \]
   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. lower-*.f64 52.0%

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-sinh.f64 62.7%

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

   6. Applied rewrites 62.7%

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

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := -\left|im\right|\\ t_1 := e^{\left|im\right|}\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{t\_0} - t\_1\right) \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left|re\right| \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot \left|im\right|\right) \cdot \left|re\right|, \left|re\right|, t\_0\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (- (fabs im))) (t_1 (exp (fabs im))))
  (*
   (copysign 1.0 re)
   (*
    (copysign 1.0 im)
    (if (<= (* (* 0.5 (sin (fabs re))) (- (exp t_0) t_1)) -2e-9)
      (* (* 0.5 (fabs re)) (- 1.0 t_1))
      (*
       (fabs re)
       (fma
        (* (* 0.16666666666666666 (fabs im)) (fabs re))
        (fabs re)
        t_0)))))))

C

double code(double re, double im) {
	double t_0 = -fabs(im);
	double t_1 = exp(fabs(im));
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp(t_0) - t_1)) <= -2e-9) {
		tmp = (0.5 * fabs(re)) * (1.0 - t_1);
	} else {
		tmp = fabs(re) * fma(((0.16666666666666666 * fabs(im)) * fabs(re)), fabs(re), t_0);
	}
	return copysign(1.0, re) * (copysign(1.0, im) * tmp);
}

Julia

function code(re, im)
	t_0 = Float64(-abs(im))
	t_1 = exp(abs(im))
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(t_0) - t_1)) <= -2e-9)
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - t_1));
	else
		tmp = Float64(abs(re) * fma(Float64(Float64(0.16666666666666666 * abs(im)) * abs(re)), abs(re), t_0));
	end
	return Float64(copysign(1.0, re) * Float64(copysign(1.0, im) * tmp))
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Abs[im], $MachinePrecision])}, Block[{t$95$1 = N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[t$95$0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], -2e-9], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Abs[re], $MachinePrecision] * N[(N[(N[(0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2.0000000000000001e-9

   1. Initial program 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.8%

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

   if -2.0000000000000001e-9 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

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

   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 37.4%

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

   7. Applied rewrites 37.4%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   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. lift-neg.f64 37.4%

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

   9. Applied rewrites 37.4%

      \[\leadsto re \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot im\right) \cdot re, re, -im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 44.1% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq 0.002:\\ \;\;\;\;\left(\left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot im\right) \cdot 0.16666666666666666 - im\right) \cdot \left|re\right|\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - \left(1 + im\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) 0.002)
   (*
    (- (* (* (* (fabs re) (fabs re)) im) 0.16666666666666666) im)
    (fabs re))
   (* (* 0.5 (fabs re)) (- 1.0 (+ 1.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 0.002) {
		tmp = ((((fabs(re) * fabs(re)) * im) * 0.16666666666666666) - im) * fabs(re);
	} else {
		tmp = (0.5 * fabs(re)) * (1.0 - (1.0 + 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))) <= 0.002) {
		tmp = ((((Math.abs(re) * Math.abs(re)) * im) * 0.16666666666666666) - im) * Math.abs(re);
	} else {
		tmp = (0.5 * Math.abs(re)) * (1.0 - (1.0 + im));
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= 0.002:
		tmp = ((((math.fabs(re) * math.fabs(re)) * im) * 0.16666666666666666) - im) * math.fabs(re)
	else:
		tmp = (0.5 * math.fabs(re)) * (1.0 - (1.0 + im))
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 0.002)
		tmp = Float64(Float64(Float64(Float64(Float64(abs(re) * abs(re)) * im) * 0.16666666666666666) - im) * abs(re));
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - Float64(1.0 + 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))) <= 0.002)
		tmp = ((((abs(re) * abs(re)) * im) * 0.16666666666666666) - im) * abs(re);
	else
		tmp = (0.5 * abs(re)) * (1.0 - (1.0 + im));
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

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

   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 37.4%

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

   7. Applied rewrites 37.4%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   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 37.4%

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-flip-reverse N/A

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

      8. lower--.f64 37.4%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 37.4%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 37.4%

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 37.4%

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

   9. Applied rewrites 37.4%

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

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

   1. Initial program 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.8%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 21.6%

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

   10. Applied rewrites 21.6%

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

Alternative 7: 44.1% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq 0.002:\\ \;\;\;\;im \cdot \left(\left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot \left|re\right|\right) \cdot 0.16666666666666666 - \left|re\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - \left(1 + im\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) 0.002)
   (*
    im
    (-
     (* (* (* (fabs re) (fabs re)) (fabs re)) 0.16666666666666666)
     (fabs re)))
   (* (* 0.5 (fabs re)) (- 1.0 (+ 1.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 0.002) {
		tmp = im * ((((fabs(re) * fabs(re)) * fabs(re)) * 0.16666666666666666) - fabs(re));
	} else {
		tmp = (0.5 * fabs(re)) * (1.0 - (1.0 + 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))) <= 0.002) {
		tmp = im * ((((Math.abs(re) * Math.abs(re)) * Math.abs(re)) * 0.16666666666666666) - Math.abs(re));
	} else {
		tmp = (0.5 * Math.abs(re)) * (1.0 - (1.0 + im));
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= 0.002:
		tmp = im * ((((math.fabs(re) * math.fabs(re)) * math.fabs(re)) * 0.16666666666666666) - math.fabs(re))
	else:
		tmp = (0.5 * math.fabs(re)) * (1.0 - (1.0 + im))
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 0.002)
		tmp = Float64(im * Float64(Float64(Float64(Float64(abs(re) * abs(re)) * abs(re)) * 0.16666666666666666) - abs(re)));
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - Float64(1.0 + 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))) <= 0.002)
		tmp = im * ((((abs(re) * abs(re)) * abs(re)) * 0.16666666666666666) - abs(re));
	else
		tmp = (0.5 * abs(re)) * (1.0 - (1.0 + im));
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.002], N[(im * N[(N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

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

   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 37.4%

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

   7. Applied rewrites 37.4%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-neg.f64 37.4%

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

   9. Applied rewrites 37.4%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. sub-flip-reverse N/A

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

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

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-out-- N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 N/A

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

       12. associate-*r* N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 37.4%

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

   11. Applied rewrites 37.4%

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

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

   1. Initial program 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.8%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 21.6%

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

   10. Applied rewrites 21.6%

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

Alternative 8: 44.1% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq 0.002:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(0.16666666666666666 \cdot \left|re\right|, \left|re\right|, -1\right)\right) \cdot \left|re\right|\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - \left(1 + im\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) 0.002)
   (*
    (* im (fma (* 0.16666666666666666 (fabs re)) (fabs re) -1.0))
    (fabs re))
   (* (* 0.5 (fabs re)) (- 1.0 (+ 1.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 0.002) {
		tmp = (im * fma((0.16666666666666666 * fabs(re)), fabs(re), -1.0)) * fabs(re);
	} else {
		tmp = (0.5 * fabs(re)) * (1.0 - (1.0 + im));
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 0.002)
		tmp = Float64(Float64(im * fma(Float64(0.16666666666666666 * abs(re)), abs(re), -1.0)) * abs(re));
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - Float64(1.0 + im)));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(im * N[(N[(0.16666666666666666 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.0%

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

   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 51.4%

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

   4. Applied rewrites 51.4%

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

   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 37.4%

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

   7. Applied rewrites 37.4%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-neg.f64 37.4%

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

   9. Applied rewrites 37.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 37.4%

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

       4. lift-fma.f64 N/A

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

       5. lift-neg.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. mul-1-neg N/A

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

       9. distribute-rgt-out N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 37.4%

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

   11. Applied rewrites 37.4%

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

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

   1. Initial program 66.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.8%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 21.6%

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

   10. Applied rewrites 21.6%

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

Alternative 9: 32.7% accurate, 13.2× 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 66.0%

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

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 51.4%

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

4. Applied rewrites 51.4%

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

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 2025219 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))