2sin (example 3.3)

Average Error: 56.92% → 0.56%

Time: 15.9s

Precision: binary64

Cost: 45440

Math

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

↓

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

FPCore

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

↓

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

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), (pow(sin(eps), 2.0) / ((-1.0 - cos(eps)) / sin(x))));
}

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(eps) ^ 2.0) / Float64(Float64(-1.0 - cos(eps)) / sin(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[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	56.92%
Target	23.4%
Herbie	0.56%

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

Derivation

1. Initial program 56.92

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

2. Applied egg-rr 33.37

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

3. Simplified 0.62

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

   [Start] 33.37	\[ \sin x \cdot \cos \varepsilon + \left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right) \]
   associate-+r+ [=>] 0.65	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon} \]
   +-commutative [=>] 0.65	\[ \color{blue}{\left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} + \cos x \cdot \sin \varepsilon \]
   +-commutative [=>] 0.65	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} \]
   *-commutative [=>] 0.65	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right) \]
   fma-def [=>] 0.63	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} \]
   neg-mul-1 [=>] 0.63	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{-1 \cdot \sin x} + \sin x \cdot \cos \varepsilon\right) \]
   *-commutative [=>] 0.63	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, -1 \cdot \sin x + \color{blue}{\cos \varepsilon \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 0.62	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\sin x \cdot \left(-1 + \cos \varepsilon\right)}\right) \]
   +-commutative [<=] 0.62	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)}\right) \]

4. Applied egg-rr 0.56

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

5. Applied egg-rr 0.57

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

6. Simplified 0.56

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

   [Start] 0.57	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{-1} \cdot \frac{\sin x}{\cos \varepsilon + 1}\right) \]
   associate-*l/ [=>] 0.57	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{{\sin \varepsilon}^{2} \cdot \frac{\sin x}{\cos \varepsilon + 1}}{-1}}\right) \]
   associate-/l* [=>] 0.58	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{-1}{\frac{\sin x}{\cos \varepsilon + 1}}}}\right) \]
   associate-/l* [<=] 0.56	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{\color{blue}{\frac{-1 \cdot \left(\cos \varepsilon + 1\right)}{\sin x}}}\right) \]
   mul-1-neg [=>] 0.56	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{\frac{\color{blue}{-\left(\cos \varepsilon + 1\right)}}{\sin x}}\right) \]
   distribute-neg-in [=>] 0.56	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{\frac{\color{blue}{\left(-\cos \varepsilon\right) + \left(-1\right)}}{\sin x}}\right) \]
   metadata-eval [=>] 0.56	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{\frac{\left(-\cos \varepsilon\right) + \color{blue}{-1}}{\sin x}}\right) \]
   +-commutative [=>] 0.56	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{\frac{\color{blue}{-1 + \left(-\cos \varepsilon\right)}}{\sin x}}\right) \]
   sub-neg [<=] 0.56	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{\frac{\color{blue}{-1 - \cos \varepsilon}}{\sin x}}\right) \]

7. Final simplification 0.56

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

Alternatives

Alternative 1
Error	0.57%
Cost	45440

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

Alternative 2
Error	23.01%
Cost	39625

\[\begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-6} \lor \neg \left(t_0 \leq 10^{-130}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3
Error	0.62%
Cost	32448

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

Alternative 4
Error	0.63%
Cost	26176

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

Alternative 5
Error	22.36%
Cost	26048

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

Alternative 6
Error	23.44%
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 7
Error	22.9%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0095 \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 8
Error	23.63%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0095:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00055:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9
Error	44.46%
Cost	6464

\[\sin \varepsilon \]

Alternative 10
Error	70.04%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023089 
(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)))