2sin (example 3.3)

Average Error: 37.2 → 14.7

Time: 15.3s

Precision: binary64

Cost: 46600

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.105:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.19:\\ \;\;\;\;\varepsilon \cdot \cos x + \left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \left(-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + -0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.105)
   (sin eps)
   (if (<= eps 0.19)
     (+
      (* eps (cos x))
      (+
       (* 0.041666666666666664 (* (sin x) (pow eps 4.0)))
       (+
        (* -0.5 (* (sin x) (pow eps 2.0)))
        (* -0.16666666666666666 (* (cos x) (pow eps 3.0))))))
     (- (sin eps) (sin x)))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.105) {
		tmp = sin(eps);
	} else if (eps <= 0.19) {
		tmp = (eps * cos(x)) + ((0.041666666666666664 * (sin(x) * pow(eps, 4.0))) + ((-0.5 * (sin(x) * pow(eps, 2.0))) + (-0.16666666666666666 * (cos(x) * pow(eps, 3.0)))));
	} else {
		tmp = sin(eps) - 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) :: tmp
    if (eps <= (-0.105d0)) then
        tmp = sin(eps)
    else if (eps <= 0.19d0) then
        tmp = (eps * cos(x)) + ((0.041666666666666664d0 * (sin(x) * (eps ** 4.0d0))) + (((-0.5d0) * (sin(x) * (eps ** 2.0d0))) + ((-0.16666666666666666d0) * (cos(x) * (eps ** 3.0d0)))))
    else
        tmp = sin(eps) - 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 tmp;
	if (eps <= -0.105) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.19) {
		tmp = (eps * Math.cos(x)) + ((0.041666666666666664 * (Math.sin(x) * Math.pow(eps, 4.0))) + ((-0.5 * (Math.sin(x) * Math.pow(eps, 2.0))) + (-0.16666666666666666 * (Math.cos(x) * Math.pow(eps, 3.0)))));
	} else {
		tmp = Math.sin(eps) - Math.sin(x);
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	tmp = 0
	if eps <= -0.105:
		tmp = math.sin(eps)
	elif eps <= 0.19:
		tmp = (eps * math.cos(x)) + ((0.041666666666666664 * (math.sin(x) * math.pow(eps, 4.0))) + ((-0.5 * (math.sin(x) * math.pow(eps, 2.0))) + (-0.16666666666666666 * (math.cos(x) * math.pow(eps, 3.0)))))
	else:
		tmp = math.sin(eps) - math.sin(x)
	return tmp

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.105)
		tmp = sin(eps);
	elseif (eps <= 0.19)
		tmp = Float64(Float64(eps * cos(x)) + Float64(Float64(0.041666666666666664 * Float64(sin(x) * (eps ^ 4.0))) + Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(-0.16666666666666666 * Float64(cos(x) * (eps ^ 3.0))))));
	else
		tmp = Float64(sin(eps) - 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)
	tmp = 0.0;
	if (eps <= -0.105)
		tmp = sin(eps);
	elseif (eps <= 0.19)
		tmp = (eps * cos(x)) + ((0.041666666666666664 * (sin(x) * (eps ^ 4.0))) + ((-0.5 * (sin(x) * (eps ^ 2.0))) + (-0.16666666666666666 * (cos(x) * (eps ^ 3.0)))));
	else
		tmp = sin(eps) - 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_] := If[LessEqual[eps, -0.105], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.19], N[(N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Target

Original	37.2
Target	15.3
Herbie	14.7

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

Derivation

1. Split input into 3 regimes

2. if eps < -0.104999999999999996

   1. Initial program 30.9

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

   2. Taylor expanded in x around 0 32.5

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

   3. Simplified 32.5

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

      [Start] 32.5	\[ \left(\cos \varepsilon - 1\right) \cdot x + \sin \varepsilon \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 32.5	\[ \color{blue}{\sin \varepsilon + \left(\cos \varepsilon - 1\right) \cdot x} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 32.5	\[ \sin \varepsilon + \color{blue}{x \cdot \left(\cos \varepsilon - 1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 32.5	\[ \sin \varepsilon + \color{blue}{\left(\cos \varepsilon \cdot x - x \cdot 1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 32.5	\[ \sin \varepsilon + \left(\cos \varepsilon \cdot x - \color{blue}{x}\right) \]

   4. Taylor expanded in x around 0 30.3

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

   if -0.104999999999999996 < eps < 0.19

   1. Initial program 44.3

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

   2. Taylor expanded in eps around 0 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(\cos x \cdot \varepsilon + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.3	\[ \color{blue}{\cos x \cdot \varepsilon + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \color{blue}{\varepsilon \cdot \cos x} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \varepsilon \cdot \cos x + \left(0.041666666666666664 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{4}\right)} + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ \varepsilon \cdot \cos x + \left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \varepsilon \cdot \cos x + \left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \left(-0.5 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{2}\right)} + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \varepsilon \cdot \cos x + \left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \left(-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + -0.16666666666666666 \cdot \color{blue}{\left(\cos x \cdot {\varepsilon}^{3}\right)}\right)\right) \]

   if 0.19 < eps

   1. Initial program 29.3

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

   2. Taylor expanded in x around 0 27.8

      \[\leadsto \color{blue}{\sin \varepsilon} - \sin x \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.105:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.19:\\ \;\;\;\;\varepsilon \cdot \cos x + \left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \left(-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + -0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	14.8
Cost	33352

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.014:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;\cos x \cdot \varepsilon + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 2
Error	14.9
Cost	20104

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00082:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \varepsilon + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 3
Error	15.0
Cost	13576

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.004:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.012:\\ \;\;\;\;\cos x \cdot \varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 4
Error	14.9
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -230000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.3
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -230000:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Error	28.8
Cost	6464

\[\sin \varepsilon \]

Alternative 7
Error	45.2
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023090 
(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)))