math.sin on complex, imaginary part

Average Accuracy: 9.1% → 99.0%

Time: 14.8s

Precision: binary64

Cost: 32836

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) - im \cdot \cos re\\ \end{array} \]

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 (<= t_0 -2.0)
     (* (cos re) (* 0.5 t_0))
     (-
      (* (cos re) (* im (* -0.16666666666666666 (* im im))))
      (* im (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) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = cos(re) * (0.5 * t_0);
	} else {
		tmp = (cos(re) * (im * (-0.16666666666666666 * (im * im)))) - (im * cos(re));
	}
	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 (t_0 <= (-2.0d0)) then
        tmp = cos(re) * (0.5d0 * t_0)
    else
        tmp = (cos(re) * (im * ((-0.16666666666666666d0) * (im * im)))) - (im * cos(re))
    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 (t_0 <= -2.0) {
		tmp = Math.cos(re) * (0.5 * t_0);
	} else {
		tmp = (Math.cos(re) * (im * (-0.16666666666666666 * (im * im)))) - (im * Math.cos(re));
	}
	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 t_0 <= -2.0:
		tmp = math.cos(re) * (0.5 * t_0)
	else:
		tmp = (math.cos(re) * (im * (-0.16666666666666666 * (im * im)))) - (im * math.cos(re))
	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 (t_0 <= -2.0)
		tmp = Float64(cos(re) * Float64(0.5 * t_0));
	else
		tmp = Float64(Float64(cos(re) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im)))) - Float64(im * cos(re)));
	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 (t_0 <= -2.0)
		tmp = cos(re) * (0.5 * t_0);
	else
		tmp = (cos(re) * (im * (-0.16666666666666666 * (im * im)))) - (im * cos(re));
	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[t$95$0, -2.0], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.1%
Target	99.7%
Herbie	99.0%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -2

   1. Initial program 99.6%

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

   2. Simplified 99.6%

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

      [Start] 99.6	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      *-commutative [=>] 99.6	\[ \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
      associate-*l* [=>] 99.6	\[ \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]
      sub-neg [=>] 99.6	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]
      sub-neg [<=] 99.6	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)}\right) \]
      sub0-neg [=>] 99.6	\[ \cos re \cdot \left(0.5 \cdot \left(e^{\color{blue}{-im}} - e^{im}\right)\right) \]

   if -2 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 8.8%

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

   2. Simplified 8.8%

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

      [Start] 8.8	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
      *-commutative [=>] 8.8	\[ \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
      associate-*l* [=>] 8.8	\[ \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]
      sub-neg [=>] 8.8	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]
      sub-neg [<=] 8.8	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)}\right) \]
      sub0-neg [=>] 8.8	\[ \cos re \cdot \left(0.5 \cdot \left(e^{\color{blue}{-im}} - e^{im}\right)\right) \]

   3. Taylor expanded in im around 0 99.0%

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

   4. Simplified 99.0%

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

      [Start] 99.0	\[ -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right) \]
      mul-1-neg [=>] 99.0	\[ -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]
      unsub-neg [=>] 99.0	\[ \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]
      *-commutative [=>] 99.0	\[ \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]
      associate-*l* [=>] 99.0	\[ \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]
      distribute-lft-out-- [=>] 99.0	\[ \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   5. Applied egg-rr 99.0%

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

      [Start] 99.0	\[ \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]
      *-commutative [=>] 99.0	\[ \cos re \cdot \left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) \]
      unpow3 [=>] 99.0	\[ \cos re \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} - im\right) \]
      associate-*r* [=>] 99.0	\[ \cos re \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im} - im\right) \]
      *-un-lft-identity [=>] 99.0	\[ \cos re \cdot \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im - \color{blue}{1 \cdot im}\right) \]
      distribute-rgt-out-- [=>] 99.0	\[ \cos re \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) - 1\right)\right)} \]

   6. Applied egg-rr 99.0%

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

      [Start] 99.0	\[ \cos re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) - 1\right)\right) \]
      sub-neg [=>] 99.0	\[ \cos re \cdot \left(im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(im \cdot im\right) + \left(-1\right)\right)}\right) \]
      distribute-rgt-in [=>] 99.0	\[ \cos re \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im + \left(-1\right) \cdot im\right)} \]
      distribute-rgt-in [=>] 99.0	\[ \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \cos re + \left(\left(-1\right) \cdot im\right) \cdot \cos re} \]
      metadata-eval [=>] 99.0	\[ \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \cos re + \left(\color{blue}{-1} \cdot im\right) \cdot \cos re \]

   7. Taylor expanded in im around 0 99.0%

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

   8. Simplified 99.0%

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

      [Start] 99.0	\[ \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \cos re + -1 \cdot \left(\cos re \cdot im\right) \]
      *-commutative [=>] 99.0	\[ \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \cos re + -1 \cdot \color{blue}{\left(im \cdot \cos re\right)} \]
      mul-1-neg [=>] 99.0	\[ \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \cos re + \color{blue}{\left(-im \cdot \cos re\right)} \]
      *-commutative [<=] 99.0	\[ \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \cos re + \left(-\color{blue}{\cos re \cdot im}\right) \]
      distribute-rgt-neg-in [=>] 99.0	\[ \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \cos re + \color{blue}{\cos re \cdot \left(-im\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(e^{-im} - e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) - im \cdot \cos re\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.7%
Cost	13632

\[\cos re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) - im \cdot \cos re \]

Alternative 2
Accuracy	98.7%
Cost	7104

\[\cos re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right) \]

Alternative 3
Accuracy	98.7%
Cost	7104

\[\cos re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) - im\right) \]

Alternative 4
Accuracy	98.2%
Cost	6656

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

Alternative 5
Accuracy	54.7%
Cost	128

\[-im \]

Alternative 6
Accuracy	3.8%
Cost	64

\[-3 \]

Alternative 7
Accuracy	3.8%
Cost	64

\[-1.5 \]

Alternative 8
Accuracy	3.9%
Cost	64

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

Alternative 9
Accuracy	5.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023143 
(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))))