math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 99.9%

Time: 14.3s

Precision: binary64

Cost: 46537

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

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -0.01 \lor \neg \left(t_0 \leq 0.02\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (cos re))))
   (if (or (<= t_0 -0.01) (not (<= t_0 0.02)))
     (* t_1 t_0)
     (*
      t_1
      (+
       (* 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 = exp(-im) - exp(im);
	double t_1 = 0.5 * cos(re);
	double tmp;
	if ((t_0 <= -0.01) || !(t_0 <= 0.02)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.016666666666666666 * pow(im, 5.0)) + (-0.3333333333333333 * pow(im, 3.0))));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = 0.5d0 * cos(re)
    if ((t_0 <= (-0.01d0)) .or. (.not. (t_0 <= 0.02d0))) then
        tmp = t_1 * t_0
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((-0.016666666666666666d0) * (im ** 5.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0))))
    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 t_1 = 0.5 * Math.cos(re);
	double tmp;
	if ((t_0 <= -0.01) || !(t_0 <= 0.02)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((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 = math.exp(-im) - math.exp(im)
	t_1 = 0.5 * math.cos(re)
	tmp = 0
	if (t_0 <= -0.01) or not (t_0 <= 0.02):
		tmp = t_1 * t_0
	else:
		tmp = t_1 * ((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(exp(Float64(-im)) - exp(im))
	t_1 = Float64(0.5 * cos(re))
	tmp = 0.0
	if ((t_0 <= -0.01) || !(t_0 <= 0.02))
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_1 * 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 = exp(-im) - exp(im);
	t_1 = 0.5 * cos(re);
	tmp = 0.0;
	if ((t_0 <= -0.01) || ~((t_0 <= 0.02)))
		tmp = t_1 * t_0;
	else
		tmp = t_1 * ((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[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.01], N[Not[LessEqual[t$95$0, 0.02]], $MachinePrecision]], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$1 * 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	54.7%
Target	99.8%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0100000000000000002 or 0.0200000000000000004 < (-.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 -0.0100000000000000002 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0200000000000000004

   1. Initial program 8.2%

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

   2. Simplified 8.2%

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

      [Start] 8.2%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      sub0-neg [=>] 8.2%	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.01 \lor \neg \left(e^{-im} - e^{im} \leq 0.02\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	46537

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -0.01 \lor \neg \left(t_0 \leq 0.02\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	46089

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.01 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.01 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4
Accuracy	95.0%
Cost	13842

\[\begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+84} \lor \neg \left(im \leq -0.0069 \lor \neg \left(im \leq 0.0085\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 5
Accuracy	87.1%
Cost	13581

\[\begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+265}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \mathbf{elif}\;im \leq -0.0013 \lor \neg \left(im \leq 0.00043\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 6
Accuracy	78.1%
Cost	7825

\[\begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := t_0 \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \mathbf{if}\;im \leq -2 \cdot 10^{+266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.5 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.0014 \lor \neg \left(im \leq 0.0035\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 7
Accuracy	64.1%
Cost	7184

\[\begin{array}{l} t_0 := \frac{im \cdot \left(-im\right)}{im + 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)}\\ t_1 := im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)\\ \mathbf{if}\;im \leq -2 \cdot 10^{+273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.9 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -700:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+156}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	74.3%
Cost	7181

\[\begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+273}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)\\ \mathbf{elif}\;im \leq -0.00034 \lor \neg \left(im \leq 0.0042\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 9
Accuracy	43.7%
Cost	1556

\[\begin{array}{l} t_0 := \frac{im \cdot \left(-im\right)}{im + 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)}\\ t_1 := im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)\\ \mathbf{if}\;im \leq -2.6 \cdot 10^{+273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2.45 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.0085:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	36.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;im \leq -480 \lor \neg \left(im \leq 700\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 11
Accuracy	35.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;re \leq -3.05 \cdot 10^{+189}:\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)\\ \end{array} \]

Alternative 12
Accuracy	35.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+193}:\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot re\right) - im\\ \end{array} \]

Alternative 13
Accuracy	29.2%
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2023269 
(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))))