math.sin on complex, imaginary part

Percentage Accurate: 54.8% → 99.8%

Time: 9.5s

Precision: binary64

Cost: 53001

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

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+23} \lor \neg \left(t_0 \leq 0.1\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)\right)\\ \end{array} \]

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 -4e+23) (not (<= t_0 0.1)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (- (* (pow im 3.0) -0.16666666666666666) im)
       (+
        (* (pow im 5.0) -0.008333333333333333)
        (* (pow im 7.0) -0.0001984126984126984)))))))

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 <= -4e+23) || !(t_0 <= 0.1)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 3.0) * -0.16666666666666666) - im) + ((pow(im, 5.0) * -0.008333333333333333) + (pow(im, 7.0) * -0.0001984126984126984)));
	}
	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 <= (-4d+23)) .or. (.not. (t_0 <= 0.1d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im ** 3.0d0) * (-0.16666666666666666d0)) - im) + (((im ** 5.0d0) * (-0.008333333333333333d0)) + ((im ** 7.0d0) * (-0.0001984126984126984d0))))
    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 <= -4e+23) || !(t_0 <= 0.1)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 3.0) * -0.16666666666666666) - im) + ((Math.pow(im, 5.0) * -0.008333333333333333) + (Math.pow(im, 7.0) * -0.0001984126984126984)));
	}
	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 <= -4e+23) or not (t_0 <= 0.1):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 3.0) * -0.16666666666666666) - im) + ((math.pow(im, 5.0) * -0.008333333333333333) + (math.pow(im, 7.0) * -0.0001984126984126984)))
	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 <= -4e+23) || !(t_0 <= 0.1))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) + Float64(Float64((im ^ 5.0) * -0.008333333333333333) + Float64((im ^ 7.0) * -0.0001984126984126984))));
	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 <= -4e+23) || ~((t_0 <= 0.1)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im ^ 3.0) * -0.16666666666666666) - im) + (((im ^ 5.0) * -0.008333333333333333) + ((im ^ 7.0) * -0.0001984126984126984)));
	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, -4e+23], N[Not[LessEqual[t$95$0, 0.1]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] + N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	54.8%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -3.9999999999999997e23 or 0.10000000000000001 < (-.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 -3.9999999999999997e23 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.10000000000000001

   1. Initial program 7.8%

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

   2. Simplified 7.8%

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

      [Start] 7.8%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      sub0-neg [=>] 7.8%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

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

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -4 \cdot 10^{+23} \lor \neg \left(e^{-im} - e^{im} \leq 0.1\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	53001

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+23} \lor \neg \left(t_0 \leq 0.1\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+23} \lor \neg \left(t_0 \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 3
Accuracy	96.9%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -4.9 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.043:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.016:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	97.2%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -4.9 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.076:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.025:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	93.2%
Cost	13580

\[\begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -4.9 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{3.936759889140842 \cdot 10^{-8} \cdot {im}^{14}}\\ \mathbf{elif}\;im \leq 4.2:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	82.3%
Cost	13188

\[\begin{array}{l} \mathbf{if}\;im \leq -2.7 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{3.936759889140842 \cdot 10^{-8} \cdot {im}^{14}}\\ \mathbf{elif}\;im \leq 54:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \end{array} \]

Alternative 7
Accuracy	59.4%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;im \leq -0.042 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 8
Accuracy	81.0%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;im \leq -4.6 \cdot 10^{+21} \lor \neg \left(im \leq 12.5\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 9
Accuracy	34.0%
Cost	840

\[\begin{array}{l} t_0 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ \mathbf{if}\;im \leq -2.05 \cdot 10^{+14}:\\ \;\;\;\;-3 \cdot t_0\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 27\\ \end{array} \]

Alternative 10
Accuracy	36.4%
Cost	840

\[\begin{array}{l} t_0 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ \mathbf{if}\;re \leq 2.95 \cdot 10^{+159}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+280}:\\ \;\;\;\;-3 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 27\\ \end{array} \]

Alternative 11
Accuracy	31.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;im \leq -2.05 \cdot 10^{+14}:\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 12
Accuracy	29.4%
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2023178 
(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))))