2sin (example 3.3)

Average Error: 36.6 → 0.4

Time: 13.8s

Precision: binary64

Cost: 45440

Math

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

↓

\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\sin x}{\frac{-1 - \cos \varepsilon}{{\sin \varepsilon}^{2}}}\right) \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (/ (sin x) (/ (- -1.0 (cos eps)) (pow (sin eps) 2.0)))))

C

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

↓

double code(double x, double eps) {
	return fma(sin(eps), cos(x), (sin(x) / ((-1.0 - cos(eps)) / pow(sin(eps), 2.0))));
}

Julia

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

↓

function code(x, eps)
	return fma(sin(eps), cos(x), Float64(sin(x) / Float64(Float64(-1.0 - cos(eps)) / (sin(eps) ^ 2.0))))
end

Wolfram

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

↓

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[(N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	36.6
Target	14.7
Herbie	0.4

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

Derivation

1. Initial program 36.6

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

2. Applied egg-rr 0.4

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

3. Taylor expanded in x around inf 21.8

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

4. Simplified 0.4

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

   [Start] 21.8	\[ \left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \]
   *-commutative [=>] 21.8	\[ \left(\cos x \cdot \sin \varepsilon + \color{blue}{\sin x \cdot \cos \varepsilon}\right) - \sin x \]
   associate--l+ [=>] 0.4	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon - \sin x\right)} \]
   sub-neg [=>] 0.4	\[ \cos x \cdot \sin \varepsilon + \color{blue}{\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   neg-mul-1 [=>] 0.4	\[ \cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon + \color{blue}{-1 \cdot \sin x}\right) \]
   metadata-eval [<=] 0.4	\[ \cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon + \color{blue}{\left(-1\right)} \cdot \sin x\right) \]
   *-commutative [=>] 0.4	\[ \cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon + \color{blue}{\sin x \cdot \left(-1\right)}\right) \]
   distribute-lft-in [<=] 0.4	\[ \cos x \cdot \sin \varepsilon + \color{blue}{\sin x \cdot \left(\cos \varepsilon + \left(-1\right)\right)} \]
   sub-neg [<=] 0.4	\[ \cos x \cdot \sin \varepsilon + \sin x \cdot \color{blue}{\left(\cos \varepsilon - 1\right)} \]
   *-commutative [=>] 0.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \sin x \cdot \left(\cos \varepsilon - 1\right) \]
   fma-udef [<=] 0.4	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon - 1\right)\right)} \]
   *-commutative [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \sin x}\right) \]
   sub-neg [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)} \cdot \sin x\right) \]
   metadata-eval [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\cos \varepsilon + \color{blue}{-1}\right) \cdot \sin x\right) \]

5. Applied egg-rr 0.4

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{\sin x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}}\right) \]

6. Simplified 0.4

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{\sin x}{\frac{-1 - \cos \varepsilon}{{\sin \varepsilon}^{2}}}}\right) \]
   Proof

   [Start] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\sin x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}\right) \]
   associate-/l* [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{\sin x}{\frac{-1 - \cos \varepsilon}{{\sin \varepsilon}^{2}}}}\right) \]

7. Final simplification 0.4

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\sin x}{\frac{-1 - \cos \varepsilon}{{\sin \varepsilon}^{2}}}\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	32448

\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(-1 + \cos \varepsilon\right)\right) \]

Alternative 2
Error	0.4
Cost	32448

\[\mathsf{fma}\left(-1 + \cos \varepsilon, \sin x, \sin \varepsilon \cdot \cos x\right) \]

Alternative 3
Error	0.4
Cost	26176

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

Alternative 4
Error	14.0
Cost	26048

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

Alternative 5
Error	14.7
Cost	13888

\[\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot \cos \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \]

Alternative 6
Error	14.4
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000215 \lor \neg \left(\varepsilon \leq 0.5\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 7
Error	14.8
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000215:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.5:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8
Error	28.2
Cost	6464

\[\sin \varepsilon \]

Alternative 9
Error	44.6
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022356 
(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)))