2sin (example 3.3)

Average Error: 36.8 → 14.3

Time: 18.5s

Precision: binary64

Cost: 33544

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(x + \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.06:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\sin \varepsilon} \cdot t_0\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \sin x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\cos \varepsilon \cdot x + \sin \varepsilon} \cdot t_0\right) - \sin x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (+ x eps))))
   (if (<= eps -0.06)
     (- (* t_0 (* (/ 1.0 (sin eps)) t_0)) (sin x))
     (if (<= eps 2.3e-12)
       (+
        (* (cos x) (+ (* -0.16666666666666666 (pow eps 3.0)) eps))
        (*
         (sin x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (pow eps 2.0)))))
       (- (* t_0 (* (/ 1.0 (+ (* (cos eps) x) (sin eps))) t_0)) (sin x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin((x + eps));
	double tmp;
	if (eps <= -0.06) {
		tmp = (t_0 * ((1.0 / sin(eps)) * t_0)) - sin(x);
	} else if (eps <= 2.3e-12) {
		tmp = (cos(x) * ((-0.16666666666666666 * pow(eps, 3.0)) + eps)) + (sin(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * pow(eps, 2.0))));
	} else {
		tmp = (t_0 * ((1.0 / ((cos(eps) * x) + sin(eps))) * t_0)) - sin(x);
	}
	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((x + eps))
    if (eps <= (-0.06d0)) then
        tmp = (t_0 * ((1.0d0 / sin(eps)) * t_0)) - sin(x)
    else if (eps <= 2.3d-12) then
        tmp = (cos(x) * (((-0.16666666666666666d0) * (eps ** 3.0d0)) + eps)) + (sin(x) * ((0.041666666666666664d0 * (eps ** 4.0d0)) + ((-0.5d0) * (eps ** 2.0d0))))
    else
        tmp = (t_0 * ((1.0d0 / ((cos(eps) * x) + sin(eps))) * t_0)) - sin(x)
    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((x + eps));
	double tmp;
	if (eps <= -0.06) {
		tmp = (t_0 * ((1.0 / Math.sin(eps)) * t_0)) - Math.sin(x);
	} else if (eps <= 2.3e-12) {
		tmp = (Math.cos(x) * ((-0.16666666666666666 * Math.pow(eps, 3.0)) + eps)) + (Math.sin(x) * ((0.041666666666666664 * Math.pow(eps, 4.0)) + (-0.5 * Math.pow(eps, 2.0))));
	} else {
		tmp = (t_0 * ((1.0 / ((Math.cos(eps) * x) + Math.sin(eps))) * t_0)) - Math.sin(x);
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = math.sin((x + eps))
	tmp = 0
	if eps <= -0.06:
		tmp = (t_0 * ((1.0 / math.sin(eps)) * t_0)) - math.sin(x)
	elif eps <= 2.3e-12:
		tmp = (math.cos(x) * ((-0.16666666666666666 * math.pow(eps, 3.0)) + eps)) + (math.sin(x) * ((0.041666666666666664 * math.pow(eps, 4.0)) + (-0.5 * math.pow(eps, 2.0))))
	else:
		tmp = (t_0 * ((1.0 / ((math.cos(eps) * x) + math.sin(eps))) * t_0)) - math.sin(x)
	return tmp

Julia

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

↓

function code(x, eps)
	t_0 = sin(Float64(x + eps))
	tmp = 0.0
	if (eps <= -0.06)
		tmp = Float64(Float64(t_0 * Float64(Float64(1.0 / sin(eps)) * t_0)) - sin(x));
	elseif (eps <= 2.3e-12)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.16666666666666666 * (eps ^ 3.0)) + eps)) + Float64(sin(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * (eps ^ 2.0)))));
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(1.0 / Float64(Float64(cos(eps) * x) + sin(eps))) * t_0)) - sin(x));
	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((x + eps));
	tmp = 0.0;
	if (eps <= -0.06)
		tmp = (t_0 * ((1.0 / sin(eps)) * t_0)) - sin(x);
	elseif (eps <= 2.3e-12)
		tmp = (cos(x) * ((-0.16666666666666666 * (eps ^ 3.0)) + eps)) + (sin(x) * ((0.041666666666666664 * (eps ^ 4.0)) + (-0.5 * (eps ^ 2.0))));
	else
		tmp = (t_0 * ((1.0 / ((cos(eps) * x) + sin(eps))) * t_0)) - sin(x);
	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[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps, -0.06], N[(N[(t$95$0 * N[(N[(1.0 / N[Sin[eps], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.3e-12], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(1.0 / N[(N[(N[Cos[eps], $MachinePrecision] * x), $MachinePrecision] + N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]]

Target

Original	36.8
Target	14.6
Herbie	14.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -0.059999999999999998

   1. Initial program 29.1

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

   2. Applied egg-rr 29.1

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) \cdot \left(\frac{1}{\sin \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right)\right)} - \sin x \]

   3. Taylor expanded in x around 0 28.3

      \[\leadsto \sin \left(x + \varepsilon\right) \cdot \left(\color{blue}{\frac{1}{\sin \varepsilon}} \cdot \sin \left(x + \varepsilon\right)\right) - \sin x \]

   if -0.059999999999999998 < eps < 2.29999999999999989e-12

   1. Initial program 44.7

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

   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. Simplified 0.2

      \[\leadsto \color{blue}{\cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \sin x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)} \]
      Proof

      [Start] 0.2	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(\cos x \cdot \varepsilon + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]
      rational_best-simplify-43 [=>] 0.2	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \cos x \cdot \varepsilon\right)\right)} \]
      rational_best-simplify-43 [=>] 0.2	\[ \color{blue}{\left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \cos x \cdot \varepsilon\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right)} \]
      rational_best-simplify-1 [=>] 0.2	\[ \color{blue}{\left(\cos x \cdot \varepsilon + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)} + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) \]
      rational_best-simplify-2 [=>] 0.2	\[ \left(\cos x \cdot \varepsilon + -0.16666666666666666 \cdot \color{blue}{\left(\cos x \cdot {\varepsilon}^{3}\right)}\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) \]
      rational_best-simplify-44 [=>] 0.2	\[ \left(\cos x \cdot \varepsilon + \color{blue}{\cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) \]
      rational_best-simplify-47 [=>] 0.2	\[ \color{blue}{\cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right)} + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) \]
      rational_best-simplify-2 [=>] 0.2	\[ \cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \left(-0.5 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{2}\right)} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) \]
      rational_best-simplify-44 [=>] 0.2	\[ \cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \left(\color{blue}{\sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) \]
      rational_best-simplify-2 [=>] 0.2	\[ \cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \left(\sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + 0.041666666666666664 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{4}\right)}\right) \]
      rational_best-simplify-44 [=>] 0.2	\[ \cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \left(\sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\sin x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4}\right)}\right) \]
      rational_best-simplify-47 [=>] 0.2	\[ \cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \color{blue}{\sin x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)} \]

   if 2.29999999999999989e-12 < eps

   1. Initial program 29.1

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

   2. Applied egg-rr 29.1

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) \cdot \left(\frac{1}{\sin \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right)\right)} - \sin x \]

   3. Taylor expanded in x around 0 27.5

      \[\leadsto \sin \left(x + \varepsilon\right) \cdot \left(\frac{1}{\color{blue}{\cos \varepsilon \cdot x + \sin \varepsilon}} \cdot \sin \left(x + \varepsilon\right)\right) - \sin x \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.06:\\ \;\;\;\;\sin \left(x + \varepsilon\right) \cdot \left(\frac{1}{\sin \varepsilon} \cdot \sin \left(x + \varepsilon\right)\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right) + \sin x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) \cdot \left(\frac{1}{\cos \varepsilon \cdot x + \sin \varepsilon} \cdot \sin \left(x + \varepsilon\right)\right) - \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	14.3
Cost	33352

\[\begin{array}{l} t_0 := \sin \left(x + \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.026:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\sin \varepsilon} \cdot t_0\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\cos \varepsilon \cdot x + \sin \varepsilon} \cdot t_0\right) - \sin x\\ \end{array} \]

Alternative 2
Error	14.4
Cost	26824

\[\begin{array}{l} t_0 := \sin \left(x + \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.057:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\sin \varepsilon} \cdot t_0\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3} + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 3
Error	14.4
Cost	26564

\[\begin{array}{l} t_0 := \sin \left(x + \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\sin \varepsilon} \cdot t_0\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\cos x \cdot \varepsilon + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 4
Error	14.9
Cost	20104

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\cos x \cdot \varepsilon + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.0
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.4
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Error	28.2
Cost	6464

\[\sin \varepsilon \]

Alternative 8
Error	61.3
Cost	64

\[0 \]

Alternative 9
Error	45.0
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023092 
(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)))