2sin (example 3.3)

Average Accuracy: 43.5% → 99.4%

Time: 15.2s

Precision: binary64

Cost: 32448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	43.5%
Target	76.9%
Herbie	99.4%

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

Derivation

1. Initial program 43.5%

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

2. Applied egg-rr 66.7%

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

   [Start] 43.5	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sub-neg [=>] 43.5	\[ \color{blue}{\sin \left(x + \varepsilon\right) + \left(-\sin x\right)} \]
   sin-sum [=>] 66.7	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} + \left(-\sin x\right) \]
   associate-+l+ [=>] 66.7	\[ \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} \]
   +-commutative [=>] 66.7	\[ \sin x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right)} \]

3. Simplified 99.4%

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

   [Start] 66.7	\[ \sin x \cdot \cos \varepsilon + \left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right) \]
   associate-+r+ [=>] 99.4	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon} \]
   +-commutative [<=] 99.4	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 99.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) \]
   fma-def [=>] 99.4	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\cos \varepsilon \cdot \sin x} + \left(-\sin x\right)\right) \]
   neg-mul-1 [=>] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \cos \varepsilon \cdot \sin x + \color{blue}{-1 \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\sin x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

4. Applied egg-rr 99.3%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon\right)\right)}, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
   Proof

   [Start] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
   expm1-log1p-u [=>] 99.3	\[ \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon\right)\right)}, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]

5. Taylor expanded in eps around inf 99.4%

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

6. Simplified 99.4%

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

   [Start] 99.4	\[ \cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon - 1\right) \]
   +-commutative [=>] 99.4	\[ \color{blue}{\sin x \cdot \left(\cos \varepsilon - 1\right) + \cos x \cdot \sin \varepsilon} \]
   *-commutative [<=] 99.4	\[ \sin x \cdot \left(\cos \varepsilon - 1\right) + \color{blue}{\sin \varepsilon \cdot \cos x} \]
   fma-def [=>] 99.4	\[ \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \sin \varepsilon \cdot \cos x\right)} \]
   sub-neg [=>] 99.4	\[ \mathsf{fma}\left(\sin x, \color{blue}{\cos \varepsilon + \left(-1\right)}, \sin \varepsilon \cdot \cos x\right) \]
   metadata-eval [=>] 99.4	\[ \mathsf{fma}\left(\sin x, \cos \varepsilon + \color{blue}{-1}, \sin \varepsilon \cdot \cos x\right) \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	32448

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

Alternative 2
Accuracy	99.4%
Cost	26176

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

Alternative 3
Accuracy	77.6%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006 \lor \neg \left(\varepsilon \leq 0.0045\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	76.9%
Cost	13632

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

Alternative 5
Accuracy	77.5%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.5 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.0029\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 6
Accuracy	76.8%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0029:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Accuracy	56.1%
Cost	6464

\[\sin \varepsilon \]

Alternative 8
Accuracy	29.2%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023137 
(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)))