2sin (example 3.3)

Average Accuracy: 42.3% → 99.4%

Time: 16.9s

Precision: binary64

Cost: 45440

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (fma
  (pow (sin eps) 2.0)
  (/ (sin x) (- -1.0 (cos eps)))
  (* (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(pow(sin(eps), 2.0), (sin(x) / (-1.0 - cos(eps))), (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(eps) ^ 2.0), Float64(sin(x) / Float64(-1.0 - cos(eps))), 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[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	42.3%
Target	76.2%
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 42.3%

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

2. Applied egg-rr 66.3%

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

   [Start] 42.3	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sub-neg [=>] 42.3	\[ \color{blue}{\sin \left(x + \varepsilon\right) + \left(-\sin x\right)} \]
   sin-sum [=>] 66.3	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} + \left(-\sin x\right) \]
   associate-+l+ [=>] 66.3	\[ \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} \]
   +-commutative [=>] 66.3	\[ \sin x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right)} \]

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] 66.3	\[ \sin x \cdot \cos \varepsilon + \left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right) \]
   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 99.4%

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

   [Start] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
   *-commutative [=>] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(\cos \varepsilon + -1\right) \cdot \sin x}\right) \]
   +-commutative [=>] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(-1 + \cos \varepsilon\right)} \cdot \sin x\right) \]
   flip-+ [=>] 99.2	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{-1 \cdot -1 - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon}} \cdot \sin x\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
   sqr-cos-a [=>] 99.1	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{1 - \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)\right)}}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
   metadata-eval [<=] 99.1	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{1 - \left(\color{blue}{\frac{1}{2}} + 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)\right)}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
   associate--r+ [=>] 99.2	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\color{blue}{\left(1 - \frac{1}{2}\right) - 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)}}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\left(1 - \color{blue}{0.5}\right) - 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\color{blue}{0.5} - 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
   sqr-sin-a [<=] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
   associate-*l/ [=>] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin x}{-1 - \cos \varepsilon}}\right) \]
   pow2 [=>] 99.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\color{blue}{{\sin \varepsilon}^{2}} \cdot \sin x}{-1 - \cos \varepsilon}\right) \]

5. Taylor expanded in eps around inf 99.4%

   \[\leadsto \color{blue}{\cos x \cdot \sin \varepsilon + -1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]

6. Simplified 99.4%

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

   [Start] 99.4	\[ \cos x \cdot \sin \varepsilon + -1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \]
   +-commutative [=>] 99.4	\[ \color{blue}{-1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon} + \cos x \cdot \sin \varepsilon} \]
   +-commutative [<=] 99.4	\[ -1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{\color{blue}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon \]
   *-commutative [<=] 99.4	\[ -1 \cdot \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \sin x}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon \]
   *-commutative [=>] 99.4	\[ -1 \cdot \frac{{\sin \varepsilon}^{2} \cdot \sin x}{\cos \varepsilon + 1} + \color{blue}{\sin \varepsilon \cdot \cos x} \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	45440

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

Alternative 2
Accuracy	99.4%
Cost	39168

\[\sin \varepsilon \cdot \cos x + \frac{{\sin \varepsilon}^{2}}{\frac{-1 - \cos \varepsilon}{\sin x}} \]

Alternative 3
Accuracy	99.4%
Cost	39040

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

Alternative 4
Accuracy	99.4%
Cost	32448

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

Alternative 5
Accuracy	99.4%
Cost	26176

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

Alternative 6
Accuracy	76.2%
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 7
Accuracy	75.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \]

Alternative 8
Accuracy	76.7%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.00024\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9
Accuracy	76.1%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000215:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 10
Accuracy	55.4%
Cost	6464

\[\sin \varepsilon \]

Alternative 11
Accuracy	4.2%
Cost	64

\[0 \]

Alternative 12
Accuracy	29.2%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023133 
(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)))