Average Error: 39.7 → 0.5
Time: 12.8s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.003:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0027:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x - \left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.125\right)\right) - \cos x \cdot \left(0.5 \cdot \varepsilon - 0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (fma (cos x) (cos eps) (- (* (sin x) (sin eps)))) (cos x))))
   (if (<= eps -0.003)
     t_0
     (if (<= eps 0.0027)
       (*
        -2.0
        (*
         (sin (/ eps 2.0))
         (-
          (sin x)
          (-
           (* (sin x) (* eps (* eps 0.125)))
           (*
            (cos x)
            (- (* 0.5 eps) (* 0.020833333333333332 (pow eps 3.0))))))))
       t_0))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	double t_0 = fma(cos(x), cos(eps), -(sin(x) * sin(eps))) - cos(x);
	double tmp;
	if (eps <= -0.003) {
		tmp = t_0;
	} else if (eps <= 0.0027) {
		tmp = -2.0 * (sin((eps / 2.0)) * (sin(x) - ((sin(x) * (eps * (eps * 0.125))) - (cos(x) * ((0.5 * eps) - (0.020833333333333332 * pow(eps, 3.0)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end
↓
function code(x, eps)
	t_0 = Float64(fma(cos(x), cos(eps), Float64(-Float64(sin(x) * sin(eps)))) - cos(x))
	tmp = 0.0
	if (eps <= -0.003)
		tmp = t_0;
	elseif (eps <= 0.0027)
		tmp = Float64(-2.0 * Float64(sin(Float64(eps / 2.0)) * Float64(sin(x) - Float64(Float64(sin(x) * Float64(eps * Float64(eps * 0.125))) - Float64(cos(x) * Float64(Float64(0.5 * eps) - Float64(0.020833333333333332 * (eps ^ 3.0))))))));
	else
		tmp = t_0;
	end
	return tmp
end

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.003], t$95$0, If[LessEqual[eps, 0.0027], N[(-2.0 * N[(N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(eps * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] * N[(N[(0.5 * eps), $MachinePrecision] - N[(0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\
\mathbf{if}\;\varepsilon \leq -0.003:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\varepsilon \leq 0.0027:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x - \left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.125\right)\right) - \cos x \cdot \left(0.5 \cdot \varepsilon - 0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -0.0030000000000000001 or 0.0027000000000000001 < eps

    1. Initial program 29.9

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

    2. Applied egg-rr0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right)} - \cos x \]

    if -0.0030000000000000001 < eps < 0.0027000000000000001

    1. Initial program 49.9

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

    2. Applied egg-rr0.7

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)} \]

    3. Taylor expanded in eps around 0 0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right) - \left(0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)}\right) \]

    4. Simplified0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\sin x - \left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.125\right)\right) - \cos x \cdot \left(0.5 \cdot \varepsilon - 0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)\right)}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.003:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0027:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x - \left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.125\right)\right) - \cos x \cdot \left(0.5 \cdot \varepsilon - 0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))