2sin (example 3.3)

Percentage Accurate: 42.6% → 99.7%

Time: 12.8s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

Target

Original	42.6%
Target	75.9%
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 35.7%

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

2. Applied egg-rr 62.4%

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

   [Start] 35.7%	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sin-sum [=>] 62.4%	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]
   associate--l+ [=>] 62.4%	\[ \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

3. Simplified 99.3%

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

   [Start] 62.4%	\[ \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \]
   +-commutative [=>] 62.4%	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon} \]
   sub-neg [=>] 62.4%	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} + \sin x \cdot \cos \varepsilon \]
   associate-+l+ [=>] 99.3%	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} \]
   *-commutative [=>] 99.3%	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right) \]
   neg-mul-1 [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \left(\color{blue}{-1 \cdot \sin x} + \sin x \cdot \cos \varepsilon\right) \]
   *-commutative [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \left(-1 \cdot \sin x + \color{blue}{\cos \varepsilon \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(-1 + \cos \varepsilon\right)} \]
   +-commutative [<=] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} \]

4. Applied egg-rr 99.4%

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

   [Start] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
   flip-+ [=>] 99.0%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1}{\cos \varepsilon - -1}} \]
   frac-2neg [=>] 99.0%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{-\left(\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1\right)}{-\left(\cos \varepsilon - -1\right)}} \]
   metadata-eval [=>] 99.0%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(\cos \varepsilon \cdot \cos \varepsilon - \color{blue}{1}\right)}{-\left(\cos \varepsilon - -1\right)} \]
   sub-1-cos [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\color{blue}{\left(-\sin \varepsilon \cdot \sin \varepsilon\right)}}{-\left(\cos \varepsilon - -1\right)} \]
   pow2 [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-\color{blue}{{\sin \varepsilon}^{2}}\right)}{-\left(\cos \varepsilon - -1\right)} \]
   *-un-lft-identity [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\color{blue}{1 \cdot \cos \varepsilon} - -1\right)} \]
   fma-neg [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\color{blue}{\mathsf{fma}\left(1, \cos \varepsilon, --1\right)}} \]
   metadata-eval [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\mathsf{fma}\left(1, \cos \varepsilon, \color{blue}{1}\right)} \]
   fma-def [<=] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\color{blue}{\left(1 \cdot \cos \varepsilon + 1\right)}} \]
   *-un-lft-identity [<=] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\color{blue}{\cos \varepsilon} + 1\right)} \]

5. Simplified 99.4%

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

   [Start] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\cos \varepsilon + 1\right)} \]
   remove-double-neg [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{-\left(\cos \varepsilon + 1\right)} \]
   neg-sub0 [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{{\sin \varepsilon}^{2}}{\color{blue}{0 - \left(\cos \varepsilon + 1\right)}} \]
   +-commutative [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{{\sin \varepsilon}^{2}}{0 - \color{blue}{\left(1 + \cos \varepsilon\right)}} \]
   associate--r+ [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{{\sin \varepsilon}^{2}}{\color{blue}{\left(0 - 1\right) - \cos \varepsilon}} \]
   metadata-eval [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{{\sin \varepsilon}^{2}}{\color{blue}{-1} - \cos \varepsilon} \]

6. Taylor expanded in eps around inf 99.4%

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

7. Simplified 99.7%

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

   [Start] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right) \]
   metadata-eval [<=] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\color{blue}{\frac{1}{-1}} \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right) \]
   +-commutative [<=] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{1}{-1} \cdot \frac{{\sin \varepsilon}^{2}}{\color{blue}{\cos \varepsilon + 1}}\right) \]
   times-frac [<=] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{1 \cdot {\sin \varepsilon}^{2}}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \]
   *-lft-identity [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \]
   unpow2 [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \]
   times-frac [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \]
   +-commutative [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \]
   hang-0p-tan [=>] 99.7%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \]

8. Applied egg-rr 44.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x, \left(\sin \varepsilon \cdot -1\right) \cdot \tan \left(\varepsilon \cdot 0.5\right), \sin \varepsilon \cdot \cos x\right)\right)} - 1} \]
   Step-by-step derivation

   [Start] 99.7%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \]
   expm1-log1p-u [=>] 94.6%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)\right)} \]
   expm1-udef [=>] 44.8%	\[ \color{blue}{e^{\mathsf{log1p}\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)} - 1} \]
   +-commutative [=>] 44.8%	\[ e^{\mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) + \sin \varepsilon \cdot \cos x}\right)} - 1 \]
   fma-def [=>] 44.8%	\[ e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sin x, \frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right), \sin \varepsilon \cdot \cos x\right)}\right)} - 1 \]
   div-inv [=>] 44.8%	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x, \color{blue}{\left(\sin \varepsilon \cdot \frac{1}{-1}\right)} \cdot \tan \left(\frac{\varepsilon}{2}\right), \sin \varepsilon \cdot \cos x\right)\right)} - 1 \]
   metadata-eval [=>] 44.8%	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x, \left(\sin \varepsilon \cdot \color{blue}{-1}\right) \cdot \tan \left(\frac{\varepsilon}{2}\right), \sin \varepsilon \cdot \cos x\right)\right)} - 1 \]
   div-inv [=>] 44.8%	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x, \left(\sin \varepsilon \cdot -1\right) \cdot \tan \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)}, \sin \varepsilon \cdot \cos x\right)\right)} - 1 \]
   metadata-eval [=>] 44.8%	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x, \left(\sin \varepsilon \cdot -1\right) \cdot \tan \left(\varepsilon \cdot \color{blue}{0.5}\right), \sin \varepsilon \cdot \cos x\right)\right)} - 1 \]

9. Simplified 99.7%

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

   [Start] 44.8%	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x, \left(\sin \varepsilon \cdot -1\right) \cdot \tan \left(\varepsilon \cdot 0.5\right), \sin \varepsilon \cdot \cos x\right)\right)} - 1 \]
   expm1-def [=>] 94.5%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sin x, \left(\sin \varepsilon \cdot -1\right) \cdot \tan \left(\varepsilon \cdot 0.5\right), \sin \varepsilon \cdot \cos x\right)\right)\right)} \]
   expm1-log1p [=>] 99.7%	\[ \color{blue}{\mathsf{fma}\left(\sin x, \left(\sin \varepsilon \cdot -1\right) \cdot \tan \left(\varepsilon \cdot 0.5\right), \sin \varepsilon \cdot \cos x\right)} \]
   fma-udef [=>] 99.7%	\[ \color{blue}{\sin x \cdot \left(\left(\sin \varepsilon \cdot -1\right) \cdot \tan \left(\varepsilon \cdot 0.5\right)\right) + \sin \varepsilon \cdot \cos x} \]
   associate-*r* [=>] 99.6%	\[ \color{blue}{\left(\sin x \cdot \left(\sin \varepsilon \cdot -1\right)\right) \cdot \tan \left(\varepsilon \cdot 0.5\right)} + \sin \varepsilon \cdot \cos x \]
   fma-udef [<=] 99.7%	\[ \color{blue}{\mathsf{fma}\left(\sin x \cdot \left(\sin \varepsilon \cdot -1\right), \tan \left(\varepsilon \cdot 0.5\right), \sin \varepsilon \cdot \cos x\right)} \]
   *-commutative [=>] 99.7%	\[ \mathsf{fma}\left(\sin x \cdot \color{blue}{\left(-1 \cdot \sin \varepsilon\right)}, \tan \left(\varepsilon \cdot 0.5\right), \sin \varepsilon \cdot \cos x\right) \]
   neg-mul-1 [<=] 99.7%	\[ \mathsf{fma}\left(\sin x \cdot \color{blue}{\left(-\sin \varepsilon\right)}, \tan \left(\varepsilon \cdot 0.5\right), \sin \varepsilon \cdot \cos x\right) \]
   *-commutative [=>] 99.7%	\[ \mathsf{fma}\left(\sin x \cdot \left(-\sin \varepsilon\right), \tan \color{blue}{\left(0.5 \cdot \varepsilon\right)}, \sin \varepsilon \cdot \cos x\right) \]

10. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	39040

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

Alternative 2
Accuracy	99.6%
Cost	32704

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

Alternative 3
Accuracy	99.3%
Cost	26176

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

Alternative 4
Accuracy	77.2%
Cost	25920

\[\mathsf{fma}\left(0, \sin x, \sin \varepsilon \cdot \cos x\right) \]

Alternative 5
Accuracy	75.5%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -75000000000000:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0032:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Accuracy	75.9%
Cost	13632

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

Alternative 7
Accuracy	74.9%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{+20}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8
Accuracy	55.0%
Cost	6464

\[\sin \varepsilon \]

Alternative 9
Accuracy	28.9%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023167 
(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)))