math.sin on complex, imaginary part

Average Error: 58.2 → 0.8

Time: 12.5s

Precision: binary64

Cost: 19904

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (+ (* (cos re) (- im)) (* -0.16666666666666666 (* (pow im 3.0) (cos re)))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

↓

double code(double re, double im) {
	return (cos(re) * -im) + (-0.16666666666666666 * (pow(im, 3.0) * cos(re)));
}

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
    code = (cos(re) * -im) + ((-0.16666666666666666d0) * ((im ** 3.0d0) * cos(re)))
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) {
	return (Math.cos(re) * -im) + (-0.16666666666666666 * (Math.pow(im, 3.0) * Math.cos(re)));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

↓

def code(re, im):
	return (math.cos(re) * -im) + (-0.16666666666666666 * (math.pow(im, 3.0) * math.cos(re)))

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)
	return Float64(Float64(cos(re) * Float64(-im)) + Float64(-0.16666666666666666 * Float64((im ^ 3.0) * cos(re))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

↓

function tmp = code(re, im)
	tmp = (cos(re) * -im) + (-0.16666666666666666 * ((im ^ 3.0) * cos(re)));
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_] := N[(N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[im, 3.0], $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	58.2
Target	0.2
Herbie	0.8

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

Derivation

1. Initial program 58.2

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

2. Simplified 58.2

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

   [Start] 58.2	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   rational.json-simplify-25 [=>] 58.2	\[ \color{blue}{\left(-0.5 \cdot \cos re\right) \cdot \left(e^{im} - e^{0 - im}\right)} \]
   rational.json-simplify-69 [=>] 58.2	\[ \color{blue}{\frac{0.5 \cdot \cos re}{-1}} \cdot \left(e^{im} - e^{0 - im}\right) \]
   rational.json-simplify-19 [=>] 58.2	\[ \color{blue}{\left(\cos re \cdot \frac{0.5}{-1}\right)} \cdot \left(e^{im} - e^{0 - im}\right) \]
   metadata-eval [=>] 58.2	\[ \left(\cos re \cdot \color{blue}{-0.5}\right) \cdot \left(e^{im} - e^{0 - im}\right) \]
   rational.json-simplify-68 [=>] 58.2	\[ \left(\cos re \cdot -0.5\right) \cdot \left(e^{im} - e^{\color{blue}{-im}}\right) \]

3. Taylor expanded in im around 0 0.8

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

4. Simplified 0.8

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

   [Start] 0.8	\[ -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right) \]
   rational.json-simplify-41 [=>] 0.8	\[ \color{blue}{-1 \cdot \left(\cos re \cdot im\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)} \]
   rational.json-simplify-3 [=>] 0.8	\[ \color{blue}{\cos re \cdot \left(-1 \cdot im\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) \]
   rational.json-simplify-39 [=>] 0.8	\[ \cos re \cdot \color{blue}{\left(im \cdot -1\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) \]
   rational.json-simplify-71 [<=] 0.8	\[ \cos re \cdot \color{blue}{\left(-im\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) \]
   rational.json-simplify-39 [=>] 0.8	\[ \cos re \cdot \left(-im\right) + -0.16666666666666666 \cdot \color{blue}{\left({im}^{3} \cdot \cos re\right)} \]

5. Final simplification 0.8

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

Alternatives

Alternative 1
Error	0.8
Cost	13568

\[\left(\cos re \cdot -0.5\right) \cdot \left(2 \cdot im + 0.3333333333333333 \cdot {im}^{3}\right) \]

Alternative 2
Error	0.8
Cost	13568

\[\frac{\cos re \cdot \left(im + \left(0.3333333333333333 \cdot {im}^{3} + im\right)\right)}{-2} \]

Alternative 3
Error	1.0
Cost	13312

\[-0.16666666666666666 \cdot {im}^{3} - im \cdot \cos re \]

Alternative 4
Error	1.1
Cost	6656

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

Alternative 5
Error	29.1
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2023066 
(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))))