Average Error: 36.7 → 0.4
Time: 14.6s
Precision: binary64
Cost: 45440
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} \cdot \sin x\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (* (/ (pow (sin eps) 2.0) (- -1.0 (cos eps))) (sin x))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(eps), cos(x), ((pow(sin(eps), 2.0) / (-1.0 - cos(eps))) * sin(x)));
}

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(eps) ^ 2.0) / Float64(-1.0 - cos(eps))) * sin(x)))
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[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original36.7
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 36.7

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

  2. Applied egg-rr21.3

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

  3. Taylor expanded in x around inf 21.3

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

  4. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\cos \varepsilon + -1\right) \cdot \sin x\right)} \]
    Proof
    (fma.f64 (sin.f64 eps) (cos.f64 x) (*.f64 (+.f64 (cos.f64 eps) -1) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (sin.f64 eps) (cos.f64 x) (*.f64 (+.f64 (cos.f64 eps) (Rewrite<= metadata-eval (neg.f64 1))) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (sin.f64 eps) (cos.f64 x) (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (cos.f64 eps) 1)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (sin.f64 eps) (cos.f64 x) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (-.f64 (cos.f64 eps) 1)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (sin.f64 eps) (cos.f64 x) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (*.f64 1 (sin.f64 x))))): 24 points increase in error, 17 points decrease in error
    (fma.f64 (sin.f64 eps) (cos.f64 x) (-.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (Rewrite=> *-lft-identity_binary64 (sin.f64 x)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (-.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (sin.f64 x)))): 7 points increase in error, 4 points decrease in error
    (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (sin.f64 eps))) (-.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (sin.f64 x))) (sin.f64 x))): 106 points increase in error, 14 points decrease in error

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost32704
\[\sin \varepsilon \cdot \left(\cos x - \frac{\sin x}{\frac{\cos \varepsilon + 1}{\sin \varepsilon}}\right) \]
Alternative 2
Error0.4
Cost32448
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(-1 + \cos \varepsilon\right)\right) \]
Alternative 3
Error14.5
Cost26312
\[\begin{array}{l} t_0 := \left(\sin x + \sin \varepsilon \cdot \cos x\right) - \sin x\\ \mathbf{if}\;\varepsilon \leq -12297687.609885052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0796388076311669:\\ \;\;\;\;\sin x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error0.4
Cost26176
\[\sin x \cdot \left(-1 + \cos \varepsilon\right) + \sin \varepsilon \cdot \cos x \]
Alternative 5
Error15.0
Cost13640
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -12297687.609885052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0796388076311669:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error15.6
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -12297687.609885052:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0796388076311669:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 7
Error61.3
Cost64
\[0 \]
Alternative 8
Error45.9
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2022289 
(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)))