2sin (example 3.3)

Average Error: 37.7 → 14.9

Time: 28.2s

Precision: binary64

Cost: 46600

Math

\[\sin \left(x + \varepsilon\right) - \sin x \]

↓

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.058:\\ \;\;\;\;0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(\cos x \cdot \varepsilon + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin eps) (sin x))))
   (if (<= eps -0.0055)
     t_0
     (if (<= eps 0.058)
       (+
        (* 0.041666666666666664 (* (pow eps 4.0) (sin x)))
        (+
         (* (cos x) eps)
         (+
          (* -0.16666666666666666 (* (pow eps 3.0) (cos x)))
          (* -0.5 (* (pow eps 2.0) (sin x))))))
       t_0))))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

↓

double code(double x, double eps) {
	double t_0 = sin(eps) - sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = t_0;
	} else if (eps <= 0.058) {
		tmp = (0.041666666666666664 * (pow(eps, 4.0) * sin(x))) + ((cos(x) * eps) + ((-0.16666666666666666 * (pow(eps, 3.0) * cos(x))) + (-0.5 * (pow(eps, 2.0) * sin(x)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(eps) - sin(x)
    if (eps <= (-0.0055d0)) then
        tmp = t_0
    else if (eps <= 0.058d0) then
        tmp = (0.041666666666666664d0 * ((eps ** 4.0d0) * sin(x))) + ((cos(x) * eps) + (((-0.16666666666666666d0) * ((eps ** 3.0d0) * cos(x))) + ((-0.5d0) * ((eps ** 2.0d0) * sin(x)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

↓

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) - Math.sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = t_0;
	} else if (eps <= 0.058) {
		tmp = (0.041666666666666664 * (Math.pow(eps, 4.0) * Math.sin(x))) + ((Math.cos(x) * eps) + ((-0.16666666666666666 * (Math.pow(eps, 3.0) * Math.cos(x))) + (-0.5 * (Math.pow(eps, 2.0) * Math.sin(x)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

↓

def code(x, eps):
	t_0 = math.sin(eps) - math.sin(x)
	tmp = 0
	if eps <= -0.0055:
		tmp = t_0
	elif eps <= 0.058:
		tmp = (0.041666666666666664 * (math.pow(eps, 4.0) * math.sin(x))) + ((math.cos(x) * eps) + ((-0.16666666666666666 * (math.pow(eps, 3.0) * math.cos(x))) + (-0.5 * (math.pow(eps, 2.0) * math.sin(x)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

↓

function code(x, eps)
	t_0 = Float64(sin(eps) - sin(x))
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = t_0;
	elseif (eps <= 0.058)
		tmp = Float64(Float64(0.041666666666666664 * Float64((eps ^ 4.0) * sin(x))) + Float64(Float64(cos(x) * eps) + Float64(Float64(-0.16666666666666666 * Float64((eps ^ 3.0) * cos(x))) + Float64(-0.5 * Float64((eps ^ 2.0) * sin(x))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

↓

function tmp_2 = code(x, eps)
	t_0 = sin(eps) - sin(x);
	tmp = 0.0;
	if (eps <= -0.0055)
		tmp = t_0;
	elseif (eps <= 0.058)
		tmp = (0.041666666666666664 * ((eps ^ 4.0) * sin(x))) + ((cos(x) * eps) + ((-0.16666666666666666 * ((eps ^ 3.0) * cos(x))) + (-0.5 * ((eps ^ 2.0) * sin(x)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0055], t$95$0, If[LessEqual[eps, 0.058], N[(N[(0.041666666666666664 * N[(N[Power[eps, 4.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Power[eps, 3.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[eps, 2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	37.7
Target	15.6
Herbie	14.9

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0054999999999999997 or 0.0580000000000000029 < eps

   1. Initial program 30.6

      \[\sin \left(x + \varepsilon\right) - \sin x \]

   2. Taylor expanded in x around 0 29.3

      \[\leadsto \color{blue}{\sin \varepsilon} - \sin x \]

   if -0.0054999999999999997 < eps < 0.0580000000000000029

   1. Initial program 44.9

      \[\sin \left(x + \varepsilon\right) - \sin x \]

   2. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(\cos x \cdot \varepsilon + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.058:\\ \;\;\;\;0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(\cos x \cdot \varepsilon + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	14.9
Cost	40072

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.053:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(\cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.0
Cost	33352

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.047:\\ \;\;\;\;\cos x \cdot \varepsilon + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.0
Cost	26824

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.046:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.2
Cost	20360

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.075:\\ \;\;\;\;\cos x \cdot \varepsilon + \left(-1 + \left(1 - \left({\varepsilon}^{2} \cdot \sin x\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.1
Cost	20104

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.055:\\ \;\;\;\;\cos x \cdot \varepsilon + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.2
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.000106:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.045:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.7
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0002:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.045:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8
Error	29.6
Cost	6464

\[\sin \varepsilon \]

Alternative 9
Error	61.3
Cost	64

\[0 \]

Alternative 10
Error	45.7
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023075 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))