2sin (example 3.3)

Average Error: 42.6% → 99.4%

Time: 15.5s

Precision: binary64

Cost: 38912.00

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (+ (fma (sin x) (cos eps) (- (sin x))) (* (cos x) (sin eps))))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

↓

double code(double x, double eps) {
	return fma(sin(x), cos(eps), -sin(x)) + (cos(x) * sin(eps));
}

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

↓

function code(x, eps)
	return Float64(fma(sin(x), cos(eps), Float64(-sin(x))) + Float64(cos(x) * sin(eps)))
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[Cos[eps], $MachinePrecision] + (-N[Sin[x], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	42.6%
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 42.6

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

2. Applied egg-rr 99.4

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

   [Start] 42.6	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sin-sum [=>] 66.5	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]
   +-commutative [=>] 66.5	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \cos \varepsilon\right)} - \sin x \]
   associate--l+ [=>] 99.4	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon - \sin x\right)} \]

3. Applied egg-rr 99.4

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

   [Start] 99.4	\[ \cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon - \sin x\right) \]
   fma-neg [=>] 99.4	\[ \cos x \cdot \sin \varepsilon + \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon, -\sin x\right)} \]

4. Final simplification 99.4

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

Alternatives

Alternative 1
Error	76.2%
Cost	39880.00

\[\begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	99.4%
Cost	32448.00

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

Alternative 3
Error	99.4%
Cost	32448.00

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

Alternative 4
Error	99.4%
Cost	26176.00

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

Alternative 5
Error	77.5%
Cost	25920.00

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

Alternative 6
Error	76.2%
Cost	13632.00

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

Alternative 7
Error	75.9%
Cost	13257.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000195 \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-33}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]

Alternative 8
Error	75.0%
Cost	6856.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -18000000:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9
Error	55.2%
Cost	6464.00

\[\sin \varepsilon \]

Alternative 10
Error	30.8%
Cost	836.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 63:\\ \;\;\;\;2 \cdot \frac{1}{\frac{2}{\varepsilon} + \varepsilon \cdot 0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 11
Error	30.6%
Cost	196.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 2:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 12
Error	28.9%
Cost	64.00

\[\varepsilon \]

Reproduce

herbie shell --seed 2023121 
(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)))