2sin (example 3.3)

Average Error: 37.3 → 0.4

Time: 15.7s

Precision: binary64

Cost: 45440

Math

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

↓

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

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

Target

Original	37.3
Target	15.0
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 37.3

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

2. Applied egg-rr 22.1

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

3. Simplified 0.4

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

   [Start] 22.1	\[ \sin x \cdot \cos \varepsilon + \left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right) \]
   associate-+r+ [=>] 0.4	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon} \]
   +-commutative [<=] 0.4	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 0.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) \]
   fma-def [=>] 0.4	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\cos \varepsilon \cdot \sin x} + \left(-\sin x\right)\right) \]
   neg-mul-1 [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \cos \varepsilon \cdot \sin x + \color{blue}{-1 \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\sin x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

4. Applied egg-rr 0.4

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

5. Simplified 0.4

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

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	32704

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

Alternative 2
Error	0.4
Cost	32448

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

Alternative 3
Error	0.4
Cost	32448

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

Alternative 4
Error	0.4
Cost	26176

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

Alternative 5
Error	15.0
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	14.6
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072 \lor \neg \left(\varepsilon \leq 0.0012\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7
Error	14.7
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 1.6 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 8
Error	15.1
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00012:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9
Error	29.1
Cost	6464

\[\sin \varepsilon \]

Alternative 10
Error	61.3
Cost	64

\[0 \]

Alternative 11
Error	45.6
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023046 
(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)))