2sin (example 3.3)

Average Error: 37.2 → 0.4

Time: 13.8s

Precision: binary64

Cost: 32448

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (fma (+ (cos eps) -1.0) (sin x) (* (cos x) (sin eps))))

C

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

↓

double code(double x, double eps) {
	return fma((cos(eps) + -1.0), sin(x), (cos(x) * sin(eps)));
}

Julia

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

↓

function code(x, eps)
	return fma(Float64(cos(eps) + -1.0), sin(x), Float64(cos(x) * sin(eps)))
end

Wolfram

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

↓

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

Target

Original	37.2
Target	15.3
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 37.2

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

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 0.4

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

4. Simplified 0.4

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

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

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	26176

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

Alternative 2
Error	14.6
Cost	26048

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

Alternative 3
Error	14.8
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00031 \lor \neg \left(\varepsilon \leq 0.00126\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4
Error	15.3
Cost	13632

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

Alternative 5
Error	14.9
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00026 \lor \neg \left(\varepsilon \leq 0.0013\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 6
Error	15.3
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0018:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Error	29.2
Cost	6464

\[\sin \varepsilon \]

Alternative 8
Error	61.3
Cost	64

\[0 \]

Alternative 9
Error	46.0
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023018 
(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)))