Average Error: 39.8 → 0.7

Time: 17.4s

Precision: binary64

Cost: 33218

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.028077178329225884:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0013137094133468092:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.028077178329225884:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \leq 0.0013137094133468092:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\end{array}

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.028077178329225884)
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
   (if (<= eps 0.0013137094133468092)
     (* -2.0 (* (sin (/ eps 2.0)) (sin (/ (+ x (+ eps x)) 2.0))))
     (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps)))))))

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.028077178329225884) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else if (eps <= 0.0013137094133468092) {
		tmp = -2.0 * (sin(eps / 2.0) * sin((x + (eps + x)) / 2.0));
	} else {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	16.1
Cost	92480

\[-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}}\right)\right)\right)\]

Alternative 2
Error	16.0
Cost	86080

\[-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}}\right)\right)\]

Alternative 3
Error	16.0
Cost	59968

\[-2 \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right)\]

Alternative 4
Error	39.8
Cost	58944

\[\frac{{\cos \left(\varepsilon + x\right)}^{3} - {\cos x}^{3}}{\cos x \cdot \cos x + \cos \left(\varepsilon + x\right) \cdot \left(\cos x + \cos \left(\varepsilon + x\right)\right)}\]

Alternative 5
Error	40.0
Cost	58688

\[\sqrt[3]{\cos \left(\varepsilon + x\right) - \cos x} \cdot \left(\sqrt[3]{\cos \left(\varepsilon + x\right) - \cos x} \cdot \sqrt[3]{\cos \left(\varepsilon + x\right) - \cos x}\right)\]

Alternative 6
Error	45.5
Cost	52800

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

Alternative 7
Error	49.2
Cost	51904

\[\left(\sqrt{\cos \left(\varepsilon + x\right)} + \sqrt{\cos x}\right) \cdot \left(\sqrt{\cos \left(\varepsilon + x\right)} - \sqrt{\cos x}\right)\]

Alternative 8
Error	15.9
Cost	46656

\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right)\right)\]

Alternative 9
Error	15.9
Cost	46656

\[-2 \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right)\right)\]

Alternative 10
Error	15.9
Cost	46528

Alternative 11
Error	15.9
Cost	46528

\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(x + \varepsilon \cdot 0.5\right)}\right)\right)\]

Alternative 12
Error	36.2
Cost	46272

Alternative 13
Error	36.2
Cost	46272

Alternative 14
Error	16.0
Cost	46144

\[-2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)\right)\]

Alternative 15
Error	40.1
Cost	45632

\[\sqrt[3]{\cos \left(\varepsilon + x\right)} \cdot \left(\sqrt[3]{\cos \left(\varepsilon + x\right)} \cdot \sqrt[3]{\cos \left(\varepsilon + x\right)}\right) - \cos x\]

Alternative 16
Error	48.1
Cost	40128

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

Alternative 17
Error	26.0
Cost	40000

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

Alternative 18
Error	39.9
Cost	39488

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

Alternative 19
Error	39.3
Cost	33344

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

Alternative 20
Error	39.2
Cost	33088

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

Alternative 21
Error	24.5
Cost	32576

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

Alternative 22
Error	48.9
Cost	32576

\[\sqrt{\cos \left(\varepsilon + x\right)} \cdot \sqrt{\cos \left(\varepsilon + x\right)} - \cos x\]

Alternative 23
Error	24.5
Cost	32576

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

Alternative 24
Error	41.5
Cost	26880

\[\left(\cos \varepsilon \cdot \left(-0.5 \cdot \left(x \cdot x\right) + 1\right) + \sin \varepsilon \cdot \left(0.16666666666666666 \cdot {x}^{3} - x\right)\right) - \cos x\]

Alternative 25
Error	43.5
Cost	26496

\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\sqrt[3]{{\left(x + \left(\varepsilon + x\right)\right)}^{3}}}{2}\right)\right)\]

Alternative 26
Error	17.3
Cost	26496

\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}^{3}}\right)\]

Alternative 27
Error	39.6
Cost	26432

\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot e^{\log \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\]

Alternative 28
Error	24.0
Cost	26432

\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right)\]

Alternative 29
Error	27.3
Cost	26432

\[-2 \cdot e^{\log \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]

Alternative 30
Error	39.8
Cost	25984

\[\sqrt[3]{{\left(\cos \left(\varepsilon + x\right) - \cos x\right)}^{3}}\]

Alternative 31
Error	39.9
Cost	25984

\[\sqrt[3]{{\cos \left(\varepsilon + x\right)}^{3}} - \cos x\]

Alternative 32
Error	39.8
Cost	25920

\[\log \left(e^{\cos \left(\varepsilon + x\right) - \cos x}\right)\]

Alternative 33
Error	48.9
Cost	25920

\[e^{\log \cos \left(\varepsilon + x\right)} - \cos x\]

Alternative 34
Error	39.9
Cost	25920

\[\log \left(e^{\cos \left(\varepsilon + x\right)}\right) - \cos x\]

Alternative 35
Error	40.9
Cost	19648

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

Alternative 36
Error	15.2
Cost	13632

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

Alternative 37
Error	31.8
Cost	13376

\[\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\]

Alternative 38
Error	34.2
Cost	13376

\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\]

Alternative 39
Error	40.6
Cost	13248

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

Alternative 40
Error	34.2
Cost	13184

\[-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\]

Alternative 41
Error	39.8
Cost	13120

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

Alternative 42
Error	39.4
Cost	12992

\[\cos \varepsilon - \cos x\]

Alternative 43
Error	37.2
Cost	6656

\[-\varepsilon \cdot \sin x\]

Alternative 44
Error	39.5
Cost	6592

\[-1 + \cos \varepsilon\]

Alternative 45
Error	61.0
Cost	64

\[1\]

Alternative 46
Error	56.0
Cost	64

\[0\]

Alternative 47
Error	58.0
Cost	64

\[-1\]

Error

Derivation

Split input into 2 regimes
if eps < -0.028077178329225884 or 0.0013137094133468092 < eps
1. Initial program 30.3
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum_binary64_12040.8
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Applied associate--l-_binary64_10080.8
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
5. Simplified0.8
  \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin x \cdot \sin \varepsilon\right)}\]
6. Simplified0.8
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)}\]
if -0.028077178329225884 < eps < 0.0013137094133468092
1. Initial program 49.4
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied diff-cos_binary64_122137.9
  \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Simplified0.6
  \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
5. Simplified0.6
  \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.028077178329225884:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0013137094133468092:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))

Error

Try it out

Alternatives

Error

Derivation

`if eps < -0.028077178329225884 or 0.0013137094133468092 < eps`

`if -0.028077178329225884 < eps < 0.0013137094133468092`

Reproduce