math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.5%

Time: 10.6s

Precision: binary64

Cost: 46537

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

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := e^{-im} - e^{im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (- (exp (- im)) (exp im))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-9)))
     (* t_0 t_1)
     (*
      t_0
      (+
       (* im -2.0)
       (+
        (* -0.016666666666666666 (pow im 5.0))
        (* -0.3333333333333333 (pow im 3.0))))))))

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 = 0.5 * cos(re);
	double t_1 = exp(-im) - exp(im);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-9)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * pow(im, 5.0)) + (-0.3333333333333333 * pow(im, 3.0))));
	}
	return tmp;
}

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 = 0.5 * Math.cos(re);
	double t_1 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-9)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * Math.pow(im, 5.0)) + (-0.3333333333333333 * Math.pow(im, 3.0))));
	}
	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 = 0.5 * math.cos(re)
	t_1 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-9):
		tmp = t_0 * t_1
	else:
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * math.pow(im, 5.0)) + (-0.3333333333333333 * math.pow(im, 3.0))))
	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(0.5 * cos(re))
	t_1 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-9))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_0 * Float64(Float64(im * -2.0) + Float64(Float64(-0.016666666666666666 * (im ^ 5.0)) + Float64(-0.3333333333333333 * (im ^ 3.0)))));
	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 = 0.5 * cos(re);
	t_1 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-9)))
		tmp = t_0 * t_1;
	else
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * (im ^ 5.0)) + (-0.3333333333333333 * (im ^ 3.0))));
	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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-9]], $MachinePrecision]], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$0 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	54.0%
Target	99.8%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 2.00000000000000012e-9 < (-.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 -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.00000000000000012e-9

   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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	46537

\[\begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := e^{-im} - e^{im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3
Accuracy	96.8%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -1.16 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00075:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0013:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	97.0%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -1.16 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.092:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0034:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	86.9%
Cost	13580

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -0.00034:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0106:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \end{array} \]

Alternative 6
Accuracy	75.9%
Cost	13516

\[\begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -2.15 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+61}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(re \cdot 0.75\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \end{array} \]

Alternative 7
Accuracy	75.9%
Cost	7824

\[\begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 480:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+99}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \end{array} \]

Alternative 8
Accuracy	75.8%
Cost	7180

\[\begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -9.8 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 620:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+100}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	57.7%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+70}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot re\right) - im\\ \end{array} \]

Alternative 10
Accuracy	35.0%
Cost	972

\[\begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+69}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 520:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+180}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 11
Accuracy	35.0%
Cost	844

\[\begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+69}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 780:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+186}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	35.8%
Cost	840

\[\begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+70}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+61}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot re\right) - im\\ \end{array} \]

Alternative 13
Accuracy	35.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+69}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+61}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	34.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;im \leq -2.5 \cdot 10^{+70} \lor \neg \left(im \leq 6 \cdot 10^{+66}\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 15
Accuracy	29.7%
Cost	128

\[-im \]

Alternative 16
Accuracy	2.9%
Cost	64

\[-3 \]

Alternative 17
Accuracy	2.9%
Cost	64

\[-0.015625 \]

Alternative 18
Accuracy	2.9%
Cost	64

\[-3.814697265625 \cdot 10^{-6} \]

Alternative 19
Accuracy	3.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023229 
(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))))