math.sin on complex, imaginary part

Percentage Accurate: 54.5% → 99.7%

Time: 15.4s

Precision: binary64

Cost: 53001

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(\left(-0.16666666666666666 \cdot {im}^{3} + -0.0001984126984126984 \cdot {im}^{7}\right) + -0.008333333333333333 \cdot {im}^{5}\right) - im\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -5.0) (not (<= t_0 5e-11)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (-
       (+
        (+
         (* -0.16666666666666666 (pow im 3.0))
         (* -0.0001984126984126984 (pow im 7.0)))
        (* -0.008333333333333333 (pow im 5.0)))
       im)))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

↓

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 5e-11)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((((-0.16666666666666666 * pow(im, 3.0)) + (-0.0001984126984126984 * pow(im, 7.0))) + (-0.008333333333333333 * pow(im, 5.0))) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 5d-11))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((((-0.16666666666666666d0) * (im ** 3.0d0)) + ((-0.0001984126984126984d0) * (im ** 7.0d0))) + ((-0.008333333333333333d0) * (im ** 5.0d0))) - im)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 5e-11)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((((-0.16666666666666666 * Math.pow(im, 3.0)) + (-0.0001984126984126984 * Math.pow(im, 7.0))) + (-0.008333333333333333 * Math.pow(im, 5.0))) - im);
	}
	return tmp;
}

Python

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

↓

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 5e-11):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((((-0.16666666666666666 * math.pow(im, 3.0)) + (-0.0001984126984126984 * math.pow(im, 7.0))) + (-0.008333333333333333 * math.pow(im, 5.0))) - im)
	return tmp

Julia

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

↓

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 5e-11))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) + Float64(-0.0001984126984126984 * (im ^ 7.0))) + Float64(-0.008333333333333333 * (im ^ 5.0))) - im));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 5e-11)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((-0.16666666666666666 * (im ^ 3.0)) + (-0.0001984126984126984 * (im ^ 7.0))) + (-0.008333333333333333 * (im ^ 5.0))) - im);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 5e-11]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0001984126984126984 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Target

Original	54.5%
Target	99.8%
Herbie	99.7%

\[\begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -5 or 5.00000000000000018e-11 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      sub0-neg [=>] 100.0	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   if -5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.00000000000000018e-11

   1. Initial program 9.1%

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

   2. Simplified 9.1%

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

      [Start] 9.1	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      sub0-neg [=>] 9.1	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)} \]
      Step-by-step derivation

      [Start] 99.8	\[ -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\right) \]
      associate-+r+ [=>] 99.8	\[ \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)} \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(-1 \cdot \left(\cos re \cdot im\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      mul-1-neg [=>] 99.8	\[ \left(\color{blue}{\left(-\cos re \cdot im\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      *-commutative [=>] 99.8	\[ \left(\left(-\color{blue}{im \cdot \cos re}\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      distribute-lft-neg-in [=>] 99.8	\[ \left(\color{blue}{\left(-im\right) \cdot \cos re} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      *-commutative [=>] 99.8	\[ \left(\left(-im\right) \cdot \cos re + -0.16666666666666666 \cdot \color{blue}{\left({im}^{3} \cdot \cos re\right)}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      associate-*r* [=>] 99.8	\[ \left(\left(-im\right) \cdot \cos re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      distribute-rgt-out [=>] 99.8	\[ \color{blue}{\cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      *-commutative [=>] 99.8	\[ \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      associate-*l* [=>] 99.8	\[ \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]
      *-commutative [=>] 99.8	\[ \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984}\right) \]
      associate-*l* [=>] 99.8	\[ \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\cos re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)}\right) \]
      distribute-lft-out [=>] 99.8	\[ \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)} \]

   5. Taylor expanded in re around inf 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} + -0.0001984126984126984 \cdot {im}^{7}\right)\right) - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(\left(-0.16666666666666666 \cdot {im}^{3} + -0.0001984126984126984 \cdot {im}^{7}\right) + -0.008333333333333333 \cdot {im}^{5}\right) - im\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	53001

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 3
Accuracy	94.7%
Cost	13842

\[\begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+132} \lor \neg \left(im \leq -0.55 \lor \neg \left(im \leq 2.3\right) \land im \leq 5.7 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4
Accuracy	38.6%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;\cos re \leq 0.94:\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 5
Accuracy	86.8%
Cost	13580

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.16666666666666666 \cdot {im}^{3} - im\\ \mathbf{if}\;im \leq -0.55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0155:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+193}:\\ \;\;\;\;t_1 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	77.6%
Cost	7824

\[\begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ t_1 := t_0 - im\\ t_2 := t_1 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.15:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	76.0%
Cost	7432

\[\begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ t_1 := t_0 - im\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2400:\\ \;\;\;\;t_1 \cdot \left(-0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	54.0%
Cost	7184

\[\begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 15500:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+73}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	75.9%
Cost	7180

\[\begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -7400:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \]

Alternative 10
Accuracy	76.0%
Cost	7052

\[\begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -780:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 2.2:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	33.4%
Cost	972

\[\begin{array}{l} t_0 := -3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{if}\;re \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+212}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	33.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;im \leq -780 \lor \neg \left(im \leq 5300\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 13
Accuracy	33.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;im \leq -1.15:\\ \;\;\;\;1.5 + \left(re \cdot re\right) \cdot -0.75\\ \mathbf{elif}\;im \leq 3600:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 14
Accuracy	28.8%
Cost	128

\[-im \]

Alternative 15
Accuracy	2.9%
Cost	64

\[-3 \]

Alternative 16
Accuracy	2.9%
Cost	64

\[-2 \]

Alternative 17
Accuracy	2.9%
Cost	64

\[-0.25 \]

Alternative 18
Accuracy	2.9%
Cost	64

\[-0.015625 \]

Alternative 19
Accuracy	3.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))