math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.7%

Time: 15.2s

Precision: binary64

Cost: 59529

• 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.

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + -0.0001984126984126984 \cdot {im}^{7}\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -5.0) (not (<= t_0 5e-11)))
     (* t_0 (* 0.5 (sin re)))
     (+
      (*
       (sin re)
       (+
        (* -0.008333333333333333 (pow im 5.0))
        (* -0.0001984126984126984 (pow im 7.0))))
      (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

↓

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 5e-11)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (sin(re) * ((-0.008333333333333333 * pow(im, 5.0)) + (-0.0001984126984126984 * pow(im, 7.0)))) + (sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-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 <= (-5.0d0)) .or. (.not. (t_0 <= 5d-11))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = (sin(re) * (((-0.008333333333333333d0) * (im ** 5.0d0)) + ((-0.0001984126984126984d0) * (im ** 7.0d0)))) + (sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-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 <= -5.0) || !(t_0 <= 5e-11)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = (Math.sin(re) * ((-0.008333333333333333 * Math.pow(im, 5.0)) + (-0.0001984126984126984 * Math.pow(im, 7.0)))) + (Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im));
	}
	return tmp;
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

↓

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 5e-11):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = (math.sin(re) * ((-0.008333333333333333 * math.pow(im, 5.0)) + (-0.0001984126984126984 * math.pow(im, 7.0)))) + (math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im))
	return tmp

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

↓

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 5e-11))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64(sin(re) * Float64(Float64(-0.008333333333333333 * (im ^ 5.0)) + Float64(-0.0001984126984126984 * (im ^ 7.0)))) + Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)));
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

↓

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 5e-11)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = (sin(re) * ((-0.008333333333333333 * (im ^ 5.0)) + (-0.0001984126984126984 * (im ^ 7.0)))) + (sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $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, -5.0], N[Not[LessEqual[t$95$0, 5e-11]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0001984126984126984 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	65.7%
Target	99.8%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5 or 5.00000000000000018e-11 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   if -5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5.00000000000000018e-11

   1. Initial program 31.4%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 99.8	\[ -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right) \]
      associate-+r+ [=>] 99.8	\[ \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]
      associate-+l+ [=>] 99.8	\[ \color{blue}{-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right)} \]
      associate-*r* [=>] 99.8	\[ \color{blue}{\left(-0.008333333333333333 \cdot \sin re\right) \cdot {im}^{5}} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]
      *-commutative [=>] 99.8	\[ \color{blue}{\left(\sin re \cdot -0.008333333333333333\right)} \cdot {im}^{5} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]
      associate-*l* [=>] 99.8	\[ \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]
      +-commutative [<=] 99.8	\[ \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]
      mul-1-neg [=>] 99.8	\[ \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \color{blue}{\left(-\sin re \cdot im\right)}\right) \]
      unsub-neg [=>] 99.8	\[ \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) - \sin re \cdot im\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) \cdot \sin re + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \sin re} \]
      Step-by-step derivation

      [Start] 99.8	\[ \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right) \]
      associate-+r+ [=>] 99.8	\[ \sin re \cdot \color{blue}{\left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]
      distribute-rgt-in [=>] 99.8	\[ \color{blue}{\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) \cdot \sin re + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \sin re} \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right)} \cdot \sin re + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \sin re \]

   5. Taylor expanded in re around inf 99.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + -0.0001984126984126984 \cdot {im}^{7}\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \sin re \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + -0.0001984126984126984 \cdot {im}^{7}\right) + \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	53001

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + -0.0001984126984126984 \cdot {im}^{7}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	46089

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - im \cdot \sin re\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	45961

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4
Accuracy	97.3%
Cost	13840

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -2.7 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	97.6%
Cost	13840

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -2.7 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	92.7%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;im \leq -4.2 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 7
Accuracy	77.4%
Cost	8072

\[\begin{array}{l} t_0 := im \cdot \left(im \cdot -0.3333333333333333\right)\\ \mathbf{if}\;im \leq -5.5 \cdot 10^{+128}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot re\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)\\ \mathbf{elif}\;im \leq -3.3 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot re\right) \cdot \frac{4 - {t_0}^{2}}{-2 - t_0}\right)\\ \mathbf{elif}\;im \leq 80000000000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot t_0\right) + re \cdot \left(im \cdot -2\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	77.1%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;im \leq -3.7 \cdot 10^{+27} \lor \neg \left(im \leq 2600000\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 9
Accuracy	52.6%
Cost	1088

\[0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right) \]

Alternative 10
Accuracy	49.5%
Cost	832

\[0.5 \cdot \left(\left(im \cdot re\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right) \]

Alternative 11
Accuracy	32.1%
Cost	256

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

Alternative 12
Accuracy	2.7%
Cost	64

\[-3 \]

Alternative 13
Accuracy	2.7%
Cost	64

\[-0.5 \]

Alternative 14
Accuracy	14.7%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))