math.sin on complex, imaginary part

Average Error: 57.9 → 0.6

Time: 9.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq 0.00030907239477484527:\\ \;\;\;\;-\mathsf{fma}\left(0.16666666666666666 \cdot {im}^{3}, \cos re, im \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\cos re \cdot 0.5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 0.00030907239477484527)
     (- (fma (* 0.16666666666666666 (pow im 3.0)) (cos re) (* im (cos re))))
     (* t_0 (* (cos re) 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 t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= 0.00030907239477484527) {
		tmp = -fma((0.16666666666666666 * pow(im, 3.0)), cos(re), (im * cos(re)));
	} else {
		tmp = t_0 * (cos(re) * 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)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if (t_0 <= 0.00030907239477484527)
		tmp = Float64(-fma(Float64(0.16666666666666666 * (im ^ 3.0)), cos(re), Float64(im * cos(re))));
	else
		tmp = Float64(t_0 * Float64(cos(re) * 0.5));
	end
	return 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, 0.00030907239477484527], (-N[(N[(0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision] + N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(t$95$0 * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus re

Bits error versus im

Target

Original	57.9
Target	0.3
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 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 3.09072394774845e-4

   1. Initial program 58.5

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

   2. Taylor expanded in im around 0 0.5

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

   3. Applied egg-rr 0.5

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

   if 3.09072394774845e-4 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 2.2

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

   2. Taylor expanded in re around inf 2.2

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

   3. Simplified 2.2

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq 0.00030907239477484527:\\ \;\;\;\;-\mathsf{fma}\left(0.16666666666666666 \cdot {im}^{3}, \cos re, im \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))))