math.sin on complex, imaginary part

Percentage Accurate: 53.2% → 99.5%

Time: 15.0s

Precision: binary64

Cost: 45961

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

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

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 (- INFINITY)) (not (<= t_0 1e-6)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) 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 <= -((double) INFINITY)) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	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 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - 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 <= -math.inf) or not (t_0 <= 1e-6):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - 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 <= Float64(-Inf)) || !(t_0 <= 1e-6))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - 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 <= -Inf) || ~((t_0 <= 1e-6)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - 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, (-Infinity)], N[Not[LessEqual[t$95$0, 1e-6]], $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] - im), $MachinePrecision]), $MachinePrecision]]]

Target

Original	53.2%
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 9.99999999999999955e-7 < (-.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)) < 9.99999999999999955e-7

   1. Initial program 7.3%

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

   2. Simplified 7.3%

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

      [Start] 7.3%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      sub0-neg [=>] 7.3%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Taylor expanded in im around 0 99.8%

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

   4. Simplified 99.8%

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

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

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2
Accuracy	94.8%
Cost	14098

\[\begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+109} \lor \neg \left(im \leq -0.105 \lor \neg \left(im \leq 960000000\right) \land im \leq 5.8 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	94.1%
Cost	13842

\[\begin{array}{l} \mathbf{if}\;im \leq -2.9 \cdot 10^{+128} \lor \neg \left(im \leq -190000000000 \lor \neg \left(im \leq 0.125\right) \land im \leq 5.8 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4
Accuracy	86.1%
Cost	13580

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -190000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0068:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+186}:\\ \;\;\;\;\left(im - t_1\right) \cdot \left(-1 - -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - im\\ \end{array} \]

Alternative 5
Accuracy	77.8%
Cost	7692

\[\begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \left(im - t_0\right) \cdot \left(-1 - -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq -445:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+186}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \]

Alternative 6
Accuracy	74.3%
Cost	7445

\[\begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -3.8 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+167} \lor \neg \left(im \leq 1.32 \cdot 10^{+185}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 7
Accuracy	63.4%
Cost	7184

\[\begin{array}{l} t_0 := \left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(im \cdot -2\right)\right)\\ t_1 := \frac{im \cdot im - t_0 \cdot t_0}{\left(-im\right) - t_0}\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3 \cdot 10^{+30}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq -7 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.25 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+190}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	37.5%
Cost	1608

\[\begin{array}{l} t_0 := \left(re \cdot re\right) \cdot 0.75\\ \mathbf{if}\;re \leq -1.55 \cdot 10^{+147}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;re \leq -5.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{2.25 - t_0 \cdot t_0}{-1.5 - t_0}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 9
Accuracy	36.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{+179}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;re \leq 330000000000:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	37.5%
Cost	704

\[\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(im \cdot -2\right) \]

Alternative 11
Accuracy	34.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{+179} \lor \neg \left(re \leq 3.2 \cdot 10^{+122}\right):\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 12
Accuracy	37.5%
Cost	576

\[im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im \]

Alternative 13
Accuracy	30.5%
Cost	128

\[-im \]

Alternative 14
Accuracy	2.9%
Cost	64

\[-1.5 \]

Reproduce

herbie shell --seed 2023277 
(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))))