math.sin on complex, imaginary part

Percentage Accurate: 53.8% → 99.7%

Time: 18.8s

Precision: binary64

Cost: 46537

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

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

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

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

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

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

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

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

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

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -4000 \lor \neg \left(t_0 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

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 -4000.0) (not (<= t_0 5e-7)))
     (* 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 <= -4000.0) || !(t_0 <= 5e-7)) {
		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 <= (-4000.0d0)) .or. (.not. (t_0 <= 5d-7))) 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 <= -4000.0) || !(t_0 <= 5e-7)) {
		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 <= -4000.0) or not (t_0 <= 5e-7):
		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 <= -4000.0) || !(t_0 <= 5e-7))
		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 <= -4000.0) || ~((t_0 <= 5e-7)))
		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, -4000.0], N[Not[LessEqual[t$95$0, 5e-7]], $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	53.8%
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)) < -4e3 or 4.99999999999999977e-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 -4e3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 4.99999999999999977e-7

   1. Initial program 8.8%

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

   2. Simplified 8.8%

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

      [Start] 8.8	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      sub0-neg [=>] 8.8	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -4000 \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.004 \lor \neg \left(t_0 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2
Accuracy	96.3%
Cost	13832

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -7.6 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.082:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 8.5:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	92.4%
Cost	13712

\[\begin{array}{l} t_0 := \frac{\cos re}{im} \cdot \left(im \cdot \left(-im\right)\right)\\ t_1 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.0036:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 8.5:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+151}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	96.9%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00081:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 8.5:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	97.2%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.082:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 8.5:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	85.2%
Cost	8084

\[\begin{array}{l} t_0 := \frac{\cos re}{im} \cdot \left(im \cdot \left(-im\right)\right)\\ t_1 := {im}^{5} \cdot -0.008333333333333333 - im\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+27}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+151}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	85.2%
Cost	7704

\[\begin{array}{l} t_0 := \frac{\cos re}{im} \cdot \left(im \cdot \left(-im\right)\right)\\ t_1 := {im}^{5} \cdot -0.008333333333333333 - im\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+27}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;im \cdot \left(0.08333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+151}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	76.1%
Cost	7445

\[\begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -7.8 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -840:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102} \lor \neg \left(im \leq 1.9 \cdot 10^{+278}\right):\\ \;\;\;\;im \cdot \left(0.08333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	78.1%
Cost	7445

\[\begin{array}{l} t_0 := {im}^{5} \cdot -0.008333333333333333 - im\\ \mathbf{if}\;im \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+102} \lor \neg \left(im \leq 1.9 \cdot 10^{+278}\right):\\ \;\;\;\;im \cdot \left(0.08333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 10
Accuracy	60.7%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;im \leq -500:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+20}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.08333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	38.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+20}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.08333333333333333 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	38.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{-24} \lor \neg \left(im \leq 4 \cdot 10^{+20}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 13
Accuracy	38.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+20}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot re\right)\right) - im\\ \end{array} \]

Alternative 14
Accuracy	33.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+158} \lor \neg \left(re \leq 2.2 \cdot 10^{+219}\right):\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 15
Accuracy	37.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;im \leq -660 \lor \neg \left(im \leq 4 \cdot 10^{+20}\right):\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 16
Accuracy	37.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;im \leq -290 \lor \neg \left(im \leq 4 \cdot 10^{+20}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 17
Accuracy	34.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;im \leq -620:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot 27\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+20}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot -3\right)\\ \end{array} \]

Alternative 18
Accuracy	30.1%
Cost	128

\[-im \]

Alternative 19
Accuracy	2.8%
Cost	64

\[-3 \]

Alternative 20
Accuracy	2.9%
Cost	64

\[-0.015625 \]

Alternative 21
Accuracy	2.9%
Cost	64

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

Alternative 22
Accuracy	3.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023160 
(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))))