Average Error: 36.5 → 0.3
Time: 15.3s
Precision: binary64
Cost: 45504
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{{\sin \varepsilon}^{2} \cdot \left(-\sin x\right)}{\cos \varepsilon + 1}\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  (sin eps)
  (/ (* (pow (sin eps) 2.0) (- (sin x))) (+ (cos eps) 1.0))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(cos(x), sin(eps), ((pow(sin(eps), 2.0) * -sin(x)) / (cos(eps) + 1.0)));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return fma(cos(x), sin(eps), Float64(Float64((sin(eps) ^ 2.0) * Float64(-sin(x))) / 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[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original36.5
Target15.1
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 36.5

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

  2. Applied egg-rr21.3

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

  3. Taylor expanded in eps around inf 21.3

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

  4. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)} \]
    Proof
    (fma.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
    (fma.f64 (cos.f64 x) (sin.f64 eps) (*.f64 (sin.f64 x) (+.f64 (cos.f64 eps) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (cos.f64 x) (sin.f64 eps) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (*.f64 (neg.f64 1) (sin.f64 x))))): 14 points increase in error, 6 points decrease in error
    (fma.f64 (cos.f64 x) (sin.f64 eps) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (*.f64 1 (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (cos.f64 x) (sin.f64 eps) (-.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 (cos.f64 x) (sin.f64 eps)) (-.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (sin.f64 x)))): 8 points increase in error, 5 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))): 103 points increase in error, 8 points decrease in error

  5. Applied egg-rr0.3

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

  6. Final simplification0.3

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

Alternatives

Alternative 1
Error0.5
Cost39360
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{\sin x \cdot \left(0.5 \cdot \cos \left(\varepsilon + \varepsilon\right) + -0.5\right)}{\cos \varepsilon + 1}\right) \]
Alternative 2
Error0.4
Cost32448
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
Alternative 3
Error0.4
Cost26176
\[\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
Alternative 4
Error15.1
Cost13632
\[2 \cdot \left(\cos \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
Alternative 5
Error15.3
Cost13384
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 9.96278517613511 \cdot 10^{-21}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \end{array} \]
Alternative 6
Error14.9
Cost13256
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.68054905408696 \cdot 10^{-15}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error15.4
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.96278517613511 \cdot 10^{-21}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 8
Error28.5
Cost6464
\[\sin \varepsilon \]
Alternative 9
Error45.4
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2022300 
(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)))