math.sin on complex, imaginary part

Percentage Accurate: 54.6% → 99.7%

Time: 14.5s

Precision: binary64

Cost: 46281

• 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

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+14} \lor \neg \left(t_0 \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + \left({im}^{5} \cdot -0.008333333333333333 - im\right)\right)\\ \end{array} \]

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 -2e+14) (not (<= t_0 0.001)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (* (pow im 3.0) -0.16666666666666666)
       (- (* (pow im 5.0) -0.008333333333333333) 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 <= -2e+14) || !(t_0 <= 0.001)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) + ((pow(im, 5.0) * -0.008333333333333333) - 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 <= (-2d+14)) .or. (.not. (t_0 <= 0.001d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) + (((im ** 5.0d0) * (-0.008333333333333333d0)) - 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 <= -2e+14) || !(t_0 <= 0.001)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) + ((Math.pow(im, 5.0) * -0.008333333333333333) - 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 <= -2e+14) or not (t_0 <= 0.001):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) + ((math.pow(im, 5.0) * -0.008333333333333333) - 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 <= -2e+14) || !(t_0 <= 0.001))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) + Float64(Float64((im ^ 5.0) * -0.008333333333333333) - 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 <= -2e+14) || ~((t_0 <= 0.001)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) + (((im ^ 5.0) * -0.008333333333333333) - 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, -2e+14], N[Not[LessEqual[t$95$0, 0.001]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	54.6%
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)) < -2e14 or 1e-3 < (-.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 -2e14 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 9.6%

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

   2. Simplified 9.6%

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

      [Start] 9.6%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      sub0-neg [=>] 9.6%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Taylor expanded in im around 0 99.8%

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

   4. Simplified 99.8%

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

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

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+14} \lor \neg \left(e^{-im} - e^{im} \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + \left({im}^{5} \cdot -0.008333333333333333 - im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	46281

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+14} \lor \neg \left(t_0 \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + \left({im}^{5} \cdot -0.008333333333333333 - im\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+14} \lor \neg \left(t_0 \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3
Accuracy	97.1%
Cost	14096

\[\begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -1.05 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.082:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 33.5:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	91.1%
Cost	13712

\[\begin{array}{l} t_0 := \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00043:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 33.5:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	96.8%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00043:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0058:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	97.0%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.038:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.58:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	86.9%
Cost	7956

\[\begin{array}{l} t_0 := {im}^{5} \cdot -0.008333333333333333\\ t_1 := \frac{\cos re \cdot \left(3814697265625 - im \cdot im\right)}{im + 1953125}\\ t_2 := \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{if}\;im \leq -9.6 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.00043:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 33.5:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	86.2%
Cost	7764

\[\begin{array}{l} t_0 := {im}^{5} \cdot -0.008333333333333333\\ t_1 := \frac{\cos re \cdot \left(3814697265625 - im \cdot im\right)}{im + 1953125}\\ \mathbf{if}\;im \leq -9.6 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 33.5:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+58}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	79.9%
Cost	7180

\[\begin{array}{l} t_0 := {im}^{5} \cdot -0.008333333333333333\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 33.5:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+57}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \]

Alternative 10
Accuracy	39.2%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 11
Accuracy	79.9%
Cost	7052

\[\begin{array}{l} t_0 := {im}^{5} \cdot -0.008333333333333333\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 33.5:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	56.6%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;im \leq -4.4 \cdot 10^{+61} \lor \neg \left(im \leq 1.15 \cdot 10^{+58}\right):\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 13
Accuracy	35.1%
Cost	840

\[\begin{array}{l} t_0 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ \mathbf{if}\;im \leq -55:\\ \;\;\;\;t_0 \cdot 27\\ \mathbf{elif}\;im \leq 550:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -3\\ \end{array} \]

Alternative 14
Accuracy	31.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{+159}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 15
Accuracy	29.7%
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2023263 
(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))))