math.sin on complex, imaginary part

Average Error: 58.0 → 0.6

Time: 15.1s

Precision: binary64

Cost: 39556

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 10^{-6}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\frac{0.5}{e^{im}} + e^{im} \cdot -0.5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (- (exp (- im)) (exp im))) 1e-6)
   (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
   (* (cos re) (+ (/ 0.5 (exp im)) (* (exp im) -0.5)))))

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 tmp;
	if (((cos(re) * 0.5) * (exp(-im) - exp(im))) <= 1e-6) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = cos(re) * ((0.5 / exp(im)) + (exp(im) * -0.5));
	}
	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) :: tmp
    if (((cos(re) * 0.5d0) * (exp(-im) - exp(im))) <= 1d-6) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else
        tmp = cos(re) * ((0.5d0 / exp(im)) + (exp(im) * (-0.5d0)))
    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 tmp;
	if (((Math.cos(re) * 0.5) * (Math.exp(-im) - Math.exp(im))) <= 1e-6) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = Math.cos(re) * ((0.5 / Math.exp(im)) + (Math.exp(im) * -0.5));
	}
	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):
	tmp = 0
	if ((math.cos(re) * 0.5) * (math.exp(-im) - math.exp(im))) <= 1e-6:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = math.cos(re) * ((0.5 / math.exp(im)) + (math.exp(im) * -0.5))
	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)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) - exp(im))) <= 1e-6)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 / exp(im)) + Float64(exp(im) * -0.5)));
	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)
	tmp = 0.0;
	if (((cos(re) * 0.5) * (exp(-im) - exp(im))) <= 1e-6)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = cos(re) * ((0.5 / exp(im)) + (exp(im) * -0.5));
	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_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	58.0
Target	0.2
Herbie	0.6

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

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))) < 9.99999999999999955e-7

   1. Initial program 58.8

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

   2. Simplified 58.8

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]
      Proof
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) -1/2 (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (Rewrite<= metadata-eval (*.f64 1/2 -1)) (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (/.f64 (*.f64 1/2 (Rewrite<= exp-0_binary64 (exp.f64 0))) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (exp.f64 0) (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (*.f64 1/2 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 0 im)))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (cos.f64 re) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (exp.f64 im) (*.f64 1/2 -1)) (*.f64 1/2 (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 1/2 -1) (exp.f64 im))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 -1 (exp.f64 im)))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (+.f64 (*.f64 1/2 (Rewrite<= neg-mul-1_binary64 (neg.f64 (exp.f64 im)))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 (neg.f64 (exp.f64 im)) (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (-.f64 0 im)) (neg.f64 (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= sub-neg_binary64 (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 re) 1/2) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (cos.f64 re))) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in im around 0 0.5

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

   4. Simplified 0.5

      \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]
      Proof
      (*.f64 (cos.f64 re) (-.f64 (*.f64 -1/6 (pow.f64 im 3)) im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (*.f64 -1/6 (pow.f64 im 3)) (cos.f64 re)) (*.f64 im (cos.f64 re)))): 1 points increase in error, 1 points decrease in error
      (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/6 (*.f64 (pow.f64 im 3) (cos.f64 re)))) (*.f64 im (cos.f64 re))): 1 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1/6 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 re) (pow.f64 im 3)))) (*.f64 im (cos.f64 re))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1/6 (*.f64 (cos.f64 re) (pow.f64 im 3))) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 re) im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1/6 (*.f64 (cos.f64 re) (pow.f64 im 3))) (neg.f64 (*.f64 (cos.f64 re) im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/6 (*.f64 (cos.f64 re) (pow.f64 im 3))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (cos.f64 re) im)))): 0 points increase in error, 0 points decrease in error

   if 9.99999999999999955e-7 < (*.f64 (*.f64 1/2 (cos.f64 re)) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)))

   1. Initial program 5.1

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

   2. Simplified 5.3

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]
      Proof
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) -1/2 (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (Rewrite<= metadata-eval (*.f64 1/2 -1)) (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (/.f64 (*.f64 1/2 (Rewrite<= exp-0_binary64 (exp.f64 0))) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (exp.f64 0) (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (*.f64 1/2 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 0 im)))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (cos.f64 re) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (exp.f64 im) (*.f64 1/2 -1)) (*.f64 1/2 (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 1/2 -1) (exp.f64 im))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 -1 (exp.f64 im)))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (+.f64 (*.f64 1/2 (Rewrite<= neg-mul-1_binary64 (neg.f64 (exp.f64 im)))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 (neg.f64 (exp.f64 im)) (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (-.f64 0 im)) (neg.f64 (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= sub-neg_binary64 (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 re) 1/2) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (cos.f64 re))) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 5.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 10^{-6}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\frac{0.5}{e^{im}} + e^{im} \cdot -0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	26752

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

Alternative 2
Error	0.9
Cost	13312

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

Alternative 3
Error	1.1
Cost	6976

\[\frac{\cos re}{im \cdot 0.16666666666666666 + \frac{-1}{im}} \]

Alternative 4
Error	1.2
Cost	6656

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

Alternative 5
Error	28.9
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2022300 
(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))))