2cos (problem 3.3.5)

Average Error: 39.7 → 0.6

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.28755678883086966:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 1.4910460004356991 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\sin x, \varepsilon \cdot \left(\varepsilon \cdot -0.125\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(-0.020833333333333332, {\varepsilon}^{3}, \varepsilon \cdot 0.5\right), \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, 0\right) - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.28755678883086966)
   (fma
    (cos x)
    (cos eps)
    (- (- (cos x)) (expm1 (log1p (* (sin x) (sin eps))))))
   (if (<= eps 1.4910460004356991e-5)
     (*
      (* -2.0 (sin (* (+ eps (- x x)) 0.5)))
      (fma
       (sin x)
       (* eps (* eps -0.125))
       (fma
        (cos x)
        (fma -0.020833333333333332 (pow eps 3.0) (* eps 0.5))
        (sin x))))
     (- (fma (cos x) (cos eps) 0.0) (fma (sin x) (sin eps) (cos x))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.28755678883086966) {
		tmp = fma(cos(x), cos(eps), (-cos(x) - expm1(log1p((sin(x) * sin(eps))))));
	} else if (eps <= 1.4910460004356991e-5) {
		tmp = (-2.0 * sin(((eps + (x - x)) * 0.5))) * fma(sin(x), (eps * (eps * -0.125)), fma(cos(x), fma(-0.020833333333333332, pow(eps, 3.0), (eps * 0.5)), sin(x)));
	} else {
		tmp = fma(cos(x), cos(eps), 0.0) - fma(sin(x), sin(eps), cos(x));
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.28755678883086966)
		tmp = fma(cos(x), cos(eps), Float64(Float64(-cos(x)) - expm1(log1p(Float64(sin(x) * sin(eps))))));
	elseif (eps <= 1.4910460004356991e-5)
		tmp = Float64(Float64(-2.0 * sin(Float64(Float64(eps + Float64(x - x)) * 0.5))) * fma(sin(x), Float64(eps * Float64(eps * -0.125)), fma(cos(x), fma(-0.020833333333333332, (eps ^ 3.0), Float64(eps * 0.5)), sin(x))));
	else
		tmp = Float64(fma(cos(x), cos(eps), 0.0) - fma(sin(x), sin(eps), cos(x)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, eps_] := If[LessEqual[eps, -0.28755678883086966], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Cos[x], $MachinePrecision]) - N[(Exp[N[Log[1 + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.4910460004356991e-5], N[(N[(-2.0 * N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(eps * -0.125), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision] + N[(eps * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + 0.0), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -0.28755678883086966

   1. Initial program 31.7

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.8

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)} - \left(-\cos x\right)\right)\right) \]

   if -0.28755678883086966 < eps < 1.4910460004356991e-5

   1. Initial program 49.0

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

   2. Applied egg-rr 0.6

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)} \]

   3. Taylor expanded in eps around 0 0.2

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

   4. Simplified 0.2

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

   if 1.4910460004356991e-5 < eps

   1. Initial program 29.9

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, 0\right) - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.28755678883086966:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 1.4910460004356991 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\sin x, \varepsilon \cdot \left(\varepsilon \cdot -0.125\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(-0.020833333333333332, {\varepsilon}^{3}, \varepsilon \cdot 0.5\right), \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, 0\right) - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022172 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))