2sin (example 3.3)

Average Error: 37.2 → 14.8

Time: 21.3s

Precision: binary64

Cost: 13256

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -8e-8) t_0 (if (<= eps 7e-6) (* (cos x) eps) 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 <= -8e-8) {
		tmp = t_0;
	} else if (eps <= 7e-6) {
		tmp = cos(x) * eps;
	} 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 <= (-8d-8)) then
        tmp = t_0
    else if (eps <= 7d-6) then
        tmp = cos(x) * eps
    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 <= -8e-8) {
		tmp = t_0;
	} else if (eps <= 7e-6) {
		tmp = Math.cos(x) * eps;
	} 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 <= -8e-8:
		tmp = t_0
	elif eps <= 7e-6:
		tmp = math.cos(x) * eps
	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 <= -8e-8)
		tmp = t_0;
	elseif (eps <= 7e-6)
		tmp = Float64(cos(x) * eps);
	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 <= -8e-8)
		tmp = t_0;
	elseif (eps <= 7e-6)
		tmp = cos(x) * eps;
	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, -8e-8], t$95$0, If[LessEqual[eps, 7e-6], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Target

Original	37.2
Target	15.2
Herbie	14.8

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

Derivation

1. Split input into 2 regimes

2. if eps < -8.0000000000000002e-8 or 6.99999999999999989e-6 < eps

   1. Initial program 30.1

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

   2. Taylor expanded in x around 0 28.6

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

   if -8.0000000000000002e-8 < eps < 6.99999999999999989e-6

   1. Initial program 44.5

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

   2. Taylor expanded in eps around 0 0.4

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	16.3
Cost	19776

\[\left(\left(\cos \varepsilon - 1\right) \cdot x + \sin \varepsilon\right) \cdot \cos x \]

Alternative 2
Error	15.3
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 3
Error	29.0
Cost	6464

\[\sin \varepsilon \]

Alternative 4
Error	45.4
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023088 
(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)))