math.cos on complex, imaginary part

Average Accuracy: 16.5% → 52.2%

Time: 8.7s

Precision: binary64

Cost: 6656

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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
    code = sin(re) * -im
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) {
	return Math.sin(re) * -im;
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(re, im)
	tmp = sin(re) * -im;
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_] := N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision]

Target

Original	16.5%
Target	50.8%
Herbie	52.2%

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

Derivation

1. Initial program 16.5%

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

2. Taylor expanded in im around 0 52.2%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]

3. Taylor expanded in re around inf 52.2%

   \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]

4. Simplified 52.2%

   \[\leadsto \color{blue}{\sin re \cdot \left(-im\right)} \]
   Proof

   [Start] 52.2	\[ -1 \cdot \left(\sin re \cdot im\right) \]
   mul-1-neg [=>] 52.2	\[ \color{blue}{-\sin re \cdot im} \]
   distribute-rgt-neg-in [=>] 52.2	\[ \color{blue}{\sin re \cdot \left(-im\right)} \]

5. Final simplification 52.2%

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

Alternatives

Alternative 1
Accuracy	27.4%
Cost	256

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

Alternative 2
Accuracy	14.4%
Cost	192

\[re \cdot im \]

Reproduce

herbie shell --seed 2023153 
(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))))