Average Error: 36.6 → 0.4
Time: 14.7s
Precision: binary64
Cost: 45504
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\sin x \cdot \left(-{\sin \varepsilon}^{2}\right)}{\cos \varepsilon + 1}\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (/ (* (sin x) (- (pow (sin eps) 2.0))) (+ (cos eps) 1.0))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(eps), cos(x), ((sin(x) * -pow(sin(eps), 2.0)) / (cos(eps) + 1.0)));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return fma(sin(eps), cos(x), Float64(Float64(sin(x) * Float64(-(sin(eps) ^ 2.0))) / Float64(cos(eps) + 1.0)))
end

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * (-N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original36.6
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 36.6

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

  2. Applied egg-rr21.4

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

  3. Applied egg-rr0.4

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

  4. Taylor expanded in x around inf 0.4

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

  5. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(\cos \varepsilon + -1\right) \cdot \sin x}\right) \]
    Proof
    (*.f64 (+.f64 (cos.f64 eps) -1) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (+.f64 (cos.f64 eps) (Rewrite<= metadata-eval (neg.f64 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (cos.f64 eps) 1)) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (-.f64 (cos.f64 eps) 1))): 0 points increase in error, 0 points decrease in error
    (*.f64 (sin.f64 x) (Rewrite=> sub-neg_binary64 (+.f64 (cos.f64 eps) (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (*.f64 (neg.f64 1) (sin.f64 x)))): 39 points increase in error, 35 points decrease in error
    (+.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (*.f64 (Rewrite=> metadata-eval -1) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (sin.f64 x)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error

  6. Applied egg-rr0.4

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

  7. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost45440
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}\right) \]
Alternative 2
Error0.4
Cost38848
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon - \sin x\right) \]
Alternative 3
Error0.4
Cost32448
\[\mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \sin \varepsilon \cdot \cos x\right) \]
Alternative 4
Error0.4
Cost32448
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
Alternative 5
Error14.0
Cost26440
\[\begin{array}{l} t_0 := \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{if}\;\varepsilon \leq -3091.2144165842037:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2260.4293333508335:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error14.4
Cost26312
\[\begin{array}{l} t_0 := \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{if}\;\varepsilon \leq -3091.2144165842037:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2260.4293333508335:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error0.4
Cost26176
\[\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
Alternative 8
Error14.7
Cost13640
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -3091.2144165842037:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0016076085439205817:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error14.8
Cost13256
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -3091.2144165842037:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0016076085439205817:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error15.3
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3091.2144165842037:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0016076085439205817:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 11
Error28.6
Cost6464
\[\sin \varepsilon \]
Alternative 12
Error61.3
Cost64
\[0 \]
Alternative 13
Error45.4
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2022297 
(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)))