Average Error: 36.9 → 0.4
Time: 11.5s
Precision: binary64
Cost: 32448
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \sin \varepsilon \cdot \cos x\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma (+ (cos eps) -1.0) (sin x) (* (sin eps) (cos x))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma((cos(eps) + -1.0), sin(x), (sin(eps) * cos(x)));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return fma(Float64(cos(eps) + -1.0), sin(x), Float64(sin(eps) * cos(x)))
end

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original36.9
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 36.9

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

  2. Applied egg-rr21.5

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

  3. Simplified0.4

    \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)} \]
    Proof
    (+.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (*.f64 (sin.f64 x) (+.f64 (cos.f64 eps) -1))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (sin.f64 eps))) (*.f64 (sin.f64 x) (+.f64 (cos.f64 eps) -1))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (sin.f64 x) (Rewrite=> +-commutative_binary64 (+.f64 -1 (cos.f64 eps))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 (sin.f64 x)) (*.f64 (cos.f64 eps) (sin.f64 x))))): 5 points increase in error, 10 points decrease in error
    (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sin.f64 x))) (*.f64 (cos.f64 eps) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (+.f64 (neg.f64 (sin.f64 x)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (cos.f64 eps))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (neg.f64 (sin.f64 x))) (*.f64 (sin.f64 x) (cos.f64 eps)))): 116 points increase in error, 3 points decrease in error
    (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (sin.f64 x))) (*.f64 (sin.f64 x) (cos.f64 eps))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost32448
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\cos \varepsilon + -1\right) \cdot \sin x\right) \]
Alternative 2
Error0.4
Cost26176
\[\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon + -1\right) \cdot \sin x \]
Alternative 3
Error14.8
Cost13640
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.015:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error15.3
Cost13632
\[\cos \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
Alternative 5
Error15.0
Cost13256
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error15.4
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 7
Error28.5
Cost6464
\[\sin \varepsilon \]
Alternative 8
Error44.9
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2022328 
(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)))