math.sin on complex, imaginary part

Average Accuracy: 9.2% → 99.3%

Time: 14.0s

Precision: binary64

Cost: 40068

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot t_0 \leq -0.005:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]

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))))
   (if (<= (* (* (cos re) 0.5) t_0) -0.005)
     (* (cos re) (* 0.5 t_0))
     (*
      (cos re)
      (*
       0.5
       (+
        (* im -2.0)
        (+
         (* (pow im 5.0) -0.016666666666666666)
         (* (pow im 3.0) -0.3333333333333333))))))))

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 tmp;
	if (((cos(re) * 0.5) * t_0) <= -0.005) {
		tmp = cos(re) * (0.5 * t_0);
	} else {
		tmp = cos(re) * (0.5 * ((im * -2.0) + ((pow(im, 5.0) * -0.016666666666666666) + (pow(im, 3.0) * -0.3333333333333333))));
	}
	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) :: tmp
    t_0 = exp(-im) - exp(im)
    if (((cos(re) * 0.5d0) * t_0) <= (-0.005d0)) then
        tmp = cos(re) * (0.5d0 * t_0)
    else
        tmp = cos(re) * (0.5d0 * ((im * (-2.0d0)) + (((im ** 5.0d0) * (-0.016666666666666666d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0)))))
    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 tmp;
	if (((Math.cos(re) * 0.5) * t_0) <= -0.005) {
		tmp = Math.cos(re) * (0.5 * t_0);
	} else {
		tmp = Math.cos(re) * (0.5 * ((im * -2.0) + ((Math.pow(im, 5.0) * -0.016666666666666666) + (Math.pow(im, 3.0) * -0.3333333333333333))));
	}
	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)
	tmp = 0
	if ((math.cos(re) * 0.5) * t_0) <= -0.005:
		tmp = math.cos(re) * (0.5 * t_0)
	else:
		tmp = math.cos(re) * (0.5 * ((im * -2.0) + ((math.pow(im, 5.0) * -0.016666666666666666) + (math.pow(im, 3.0) * -0.3333333333333333))))
	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))
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * t_0) <= -0.005)
		tmp = Float64(cos(re) * Float64(0.5 * t_0));
	else
		tmp = Float64(cos(re) * Float64(0.5 * Float64(Float64(im * -2.0) + Float64(Float64((im ^ 5.0) * -0.016666666666666666) + Float64((im ^ 3.0) * -0.3333333333333333)))));
	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);
	tmp = 0.0;
	if (((cos(re) * 0.5) * t_0) <= -0.005)
		tmp = cos(re) * (0.5 * t_0);
	else
		tmp = cos(re) * (0.5 * ((im * -2.0) + (((im ^ 5.0) * -0.016666666666666666) + ((im ^ 3.0) * -0.3333333333333333))));
	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]}, If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], -0.005], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.2%
Target	99.6%
Herbie	99.3%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 1/2 (cos.f64 re)) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 98.3%

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

   2. Simplified 98.3%

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

      [Start] 98.3	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      *-commutative [=>] 98.3	\[ \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
      associate-*l* [=>] 98.3	\[ \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]
      sub-neg [=>] 98.3	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]
      sub-neg [<=] 98.3	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)}\right) \]
      sub0-neg [=>] 98.3	\[ \cos re \cdot \left(0.5 \cdot \left(e^{\color{blue}{-im}} - e^{im}\right)\right) \]

   if -0.0050000000000000001 < (*.f64 (*.f64 1/2 (cos.f64 re)) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)))

   1. Initial program 8.6%

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

   2. Simplified 8.6%

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

      [Start] 8.6	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      *-commutative [=>] 8.6	\[ \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
      associate-*l* [=>] 8.6	\[ \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]
      sub-neg [=>] 8.6	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]
      sub-neg [<=] 8.6	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)}\right) \]
      sub0-neg [=>] 8.6	\[ \cos re \cdot \left(0.5 \cdot \left(e^{\color{blue}{-im}} - e^{im}\right)\right) \]

   3. Taylor expanded in im around 0 99.4%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.005:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(e^{-im} - e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	39492

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot t_0 \leq -0.001:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	33280

\[\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) + \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) \]

Alternative 3
Accuracy	99.1%
Cost	27008

\[0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + \left({im}^{7} \cdot -0.0003968253968253968 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\right)\right) \]

Alternative 4
Accuracy	98.8%
Cost	7232

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

Alternative 5
Accuracy	98.2%
Cost	6656

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

Alternative 6
Accuracy	54.9%
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2023125 
(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))))