2sin (example 3.3)

Average Accuracy: 41.6% → 99.7%

Time: 16.6s

Precision: binary64

Cost: 39040

Math

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

↓

\[\mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x, -\sin \varepsilon, \sin \varepsilon \cdot \cos x\right) \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (fma (* (tan (* eps 0.5)) (sin x)) (- (sin 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((tan((eps * 0.5)) * sin(x)), -sin(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(Float64(tan(Float64(eps * 0.5)) * sin(x)), Float64(-sin(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[(N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	41.6%
Target	76.3%
Herbie	99.7%

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

Derivation

1. Initial program 40.3%

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

2. Applied egg-rr 66.9%

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

   [Start] 40.3	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sin-sum [=>] 66.9	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]
   associate--l+ [=>] 66.9	\[ \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

3. Simplified 99.4%

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

   [Start] 66.9	\[ \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \]
   +-commutative [=>] 66.9	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon} \]
   sub-neg [=>] 66.9	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} + \sin x \cdot \cos \varepsilon \]
   associate-+l+ [=>] 99.4	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} \]
   *-commutative [=>] 99.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right) \]
   neg-mul-1 [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \left(\color{blue}{-1 \cdot \sin x} + \sin x \cdot \cos \varepsilon\right) \]
   *-commutative [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \left(-1 \cdot \sin x + \color{blue}{\cos \varepsilon \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(-1 + \cos \varepsilon\right)} \]
   +-commutative [<=] 99.4	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} \]

4. Applied egg-rr 99.5%

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

   [Start] 99.4	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
   flip-+ [=>] 99.3	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1}{\cos \varepsilon - -1}} \]
   associate-*r/ [=>] 99.3	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\frac{\sin x \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1\right)}{\cos \varepsilon - -1}} \]
   metadata-eval [=>] 99.3	\[ \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - \color{blue}{1}\right)}{\cos \varepsilon - -1} \]
   sub-1-cos [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \color{blue}{\left(-\sin \varepsilon \cdot \sin \varepsilon\right)}}{\cos \varepsilon - -1} \]
   pow2 [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left(-\color{blue}{{\sin \varepsilon}^{2}}\right)}{\cos \varepsilon - -1} \]
   sub-neg [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left(-{\sin \varepsilon}^{2}\right)}{\color{blue}{\cos \varepsilon + \left(--1\right)}} \]
   metadata-eval [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left(-{\sin \varepsilon}^{2}\right)}{\cos \varepsilon + \color{blue}{1}} \]

5. Simplified 99.6%

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x} \]
   Proof

   [Start] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left(-{\sin \varepsilon}^{2}\right)}{\cos \varepsilon + 1} \]
   *-commutative [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{\left(-{\sin \varepsilon}^{2}\right) \cdot \sin x}}{\cos \varepsilon + 1} \]
   associate-/l* [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\frac{-{\sin \varepsilon}^{2}}{\frac{\cos \varepsilon + 1}{\sin x}}} \]
   associate-/r/ [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1} \cdot \sin x} \]
   *-lft-identity [<=] 99.5	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(1 \cdot \frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}\right)} \cdot \sin x \]
   metadata-eval [<=] 99.5	\[ \sin \varepsilon \cdot \cos x + \left(\color{blue}{\frac{-1}{-1}} \cdot \frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}\right) \cdot \sin x \]
   times-frac [<=] 99.5	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\frac{-1 \cdot \left(-{\sin \varepsilon}^{2}\right)}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \cdot \sin x \]
   neg-mul-1 [<=] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{-\left(-{\sin \varepsilon}^{2}\right)}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \cdot \sin x \]
   remove-double-neg [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{{\sin \varepsilon}^{2}}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \cdot \sin x \]
   unpow2 [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \cdot \sin x \]
   times-frac [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \cdot \sin x \]
   +-commutative [=>] 99.5	\[ \sin \varepsilon \cdot \cos x + \left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \cdot \sin x \]
   hang-0p-tan [=>] 99.6	\[ \sin \varepsilon \cdot \cos x + \left(\frac{\sin \varepsilon}{-1} \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sin x \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x, -\sin \varepsilon, \sin \varepsilon \cdot \cos x\right)} \]
   Proof

   [Start] 99.6	\[ \sin \varepsilon \cdot \cos x + \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x \]
   +-commutative [=>] 99.6	\[ \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x + \sin \varepsilon \cdot \cos x} \]
   associate-*l* [=>] 99.6	\[ \color{blue}{\frac{\sin \varepsilon}{-1} \cdot \left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)} + \sin \varepsilon \cdot \cos x \]
   *-commutative [=>] 99.6	\[ \color{blue}{\left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \frac{\sin \varepsilon}{-1}} + \sin \varepsilon \cdot \cos x \]
   fma-def [=>] 99.6	\[ \color{blue}{\mathsf{fma}\left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin x, \frac{\sin \varepsilon}{-1}, \sin \varepsilon \cdot \cos x\right)} \]
   div-inv [=>] 99.6	\[ \mathsf{fma}\left(\tan \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin x, \frac{\sin \varepsilon}{-1}, \sin \varepsilon \cdot \cos x\right) \]
   metadata-eval [=>] 99.6	\[ \mathsf{fma}\left(\tan \left(\varepsilon \cdot \color{blue}{0.5}\right) \cdot \sin x, \frac{\sin \varepsilon}{-1}, \sin \varepsilon \cdot \cos x\right) \]
   frac-2neg [=>] 99.6	\[ \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x, \color{blue}{\frac{-\sin \varepsilon}{--1}}, \sin \varepsilon \cdot \cos x\right) \]
   metadata-eval [=>] 99.6	\[ \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x, \frac{-\sin \varepsilon}{\color{blue}{1}}, \sin \varepsilon \cdot \cos x\right) \]
   /-rgt-identity [=>] 99.6	\[ \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x, \color{blue}{-\sin \varepsilon}, \sin \varepsilon \cdot \cos x\right) \]

7. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x, -\sin \varepsilon, \sin \varepsilon \cdot \cos x\right) \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	32832

\[\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \]

Alternative 2
Accuracy	99.4%
Cost	26176

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

Alternative 3
Accuracy	99.6%
Cost	26176

\[\sin \varepsilon \cdot \left(\cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right) \]

Alternative 4
Accuracy	77.7%
Cost	12992

\[\sin \varepsilon \cdot \cos x \]

Alternative 5
Accuracy	76.3%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.55:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Accuracy	54.3%
Cost	6464

\[\sin \varepsilon \]

Alternative 7
Accuracy	28.3%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023159 
(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)))