Average Error: 58.1 → 0.9
Time: 11.2s
Precision: binary64
Cost: 13568
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
↓
\[\cos re \cdot \left(im + \left(\left(-0.16666666666666666 \cdot {im}^{3} - im\right) - im\right)\right) \]
(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)) im) im))))
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)) - im) - im));
}

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)) - im) - im))
end function

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)) - im) - im));
}

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)) - im) - im))

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

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)) - im) - im));
end

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[Cos[re], $MachinePrecision] * N[(im + N[(N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\cos re \cdot \left(im + \left(\left(-0.16666666666666666 \cdot {im}^{3} - im\right) - im\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \]

Derivation

  1. Initial program 58.1

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

  2. Simplified58.1

    \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{-im} - e^{im}\right)\right)} \]
    Proof
    (*.f64 (cos.f64 re) (*.f64 1/2 (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (cos.f64 re) (*.f64 1/2 (-.f64 (exp.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 im))) (exp.f64 im)))): 8 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)) (neg.f64 (exp.f64 im)))))): 0 points increase in error, 8 points decrease in error
    (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (exp.f64 im)) (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 8 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, 8 points decrease in error
    (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= sub-neg_binary64 (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))))): 8 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)))): 8 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))): 8 points increase in error, 0 points decrease in error

  3. Taylor expanded in im around 0 0.9

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

  4. Simplified0.9

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

  5. Applied egg-rr0.9

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

  6. Simplified0.9

    \[\leadsto \color{blue}{\cos re \cdot \left(im + \left(\left(-0.16666666666666666 \cdot {im}^{3} - im\right) - im\right)\right)} \]
    Proof
    (*.f64 (cos.f64 re) (+.f64 im (-.f64 (-.f64 (*.f64 -1/6 (pow.f64 im 3)) im) im))): 0 points increase in error, 0 points decrease in error
    (*.f64 (cos.f64 re) (+.f64 im (-.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 im 3) -1/6)) im) im))): 0 points increase in error, 0 points decrease in error
    (*.f64 (cos.f64 re) (+.f64 im (Rewrite<= unsub-neg_binary64 (+.f64 (-.f64 (*.f64 (pow.f64 im 3) -1/6) im) (neg.f64 im))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (cos.f64 re) (+.f64 im (+.f64 (-.f64 (*.f64 (pow.f64 im 3) -1/6) im) (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 im) 1))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (cos.f64 re) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (-.f64 (*.f64 (pow.f64 im 3) -1/6) im) (*.f64 (neg.f64 im) 1)) im))): 0 points increase in error, 0 points decrease in error
    (*.f64 (cos.f64 re) (Rewrite<= associate-+r+_binary64 (+.f64 (-.f64 (*.f64 (pow.f64 im 3) -1/6) im) (+.f64 (*.f64 (neg.f64 im) 1) im)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (cos.f64 re) (+.f64 (-.f64 (*.f64 (pow.f64 im 3) -1/6) im) (Rewrite<= fma-udef_binary64 (fma.f64 (neg.f64 im) 1 im)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (cos.f64 re) (-.f64 (*.f64 (pow.f64 im 3) -1/6) im)) (*.f64 (cos.f64 re) (fma.f64 (neg.f64 im) 1 im)))): 0 points increase in error, 0 points decrease in error

  7. Final simplification0.9

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

Alternatives

Alternative 1
Error0.9
Cost13312
\[\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]
Alternative 2
Error1.3
Cost6656
\[\cos re \cdot \left(-im\right) \]
Alternative 3
Error29.0
Cost128
\[-im \]

Error

Reproduce

herbie shell --seed 2022343 
(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))))