Average Error: 37.1 → 0.2
Time: 11.8s
Precision: binary64
Cost: 32512
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\sin \varepsilon \cdot \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right), -\sin x, \cos x\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (* (sin eps) (fma (tan (* eps 0.5)) (- (sin x)) (cos x))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return sin(eps) * fma(tan((eps * 0.5)), -sin(x), cos(x));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return Float64(sin(eps) * fma(tan(Float64(eps * 0.5)), Float64(-sin(x)), cos(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[(N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right), -\sin x, \cos x\right)

Error

Target

Original37.1
Target15.3
Herbie0.2
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 37.1

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

  2. Applied egg-rr21.7

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

  3. Simplified0.4

    \[\leadsto \color{blue}{\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon - 1\right)} \]
    Proof
    (+.f64 (*.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)) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (*.f64 (sin.f64 x) 1)))): 10 points increase in error, 13 points decrease in error
    (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (-.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (Rewrite=> *-rgt-identity_binary64 (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 (sin.f64 x) (cos.f64 eps))) (sin.f64 x))): 113 points increase in error, 7 points decrease in error
    (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (*.f64 (cos.f64 x) (sin.f64 eps)))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r-_binary64 (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (sin.f64 x)))): 10 points increase in error, 9 points decrease in error

  4. Applied egg-rr0.3

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

  5. Simplified0.2

    \[\leadsto \cos x \cdot \sin \varepsilon + \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x} \]
    Proof
    (*.f64 (*.f64 (/.f64 (sin.f64 eps) -1) (tan.f64 (/.f64 eps 2))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (/.f64 (sin.f64 eps) -1) (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 eps) (+.f64 1 (cos.f64 eps))))) (sin.f64 x)): 39 points increase in error, 25 points decrease in error
    (*.f64 (*.f64 (/.f64 (sin.f64 eps) -1) (/.f64 (sin.f64 eps) (Rewrite<= +-commutative_binary64 (+.f64 (cos.f64 eps) 1)))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sin.f64 eps) (sin.f64 eps)) (*.f64 -1 (+.f64 (cos.f64 eps) 1)))) (sin.f64 x)): 23 points increase in error, 17 points decrease in error
    (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 eps) 2)) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)))) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (neg.f64 (pow.f64 (sin.f64 eps) 2)))) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1)))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite=> *-lft-identity_binary64 (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (/.f64 (+.f64 (cos.f64 eps) 1) (sin.f64 x)))): 25 points increase in error, 32 points decrease in error
    (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (sin.f64 x)) (+.f64 (cos.f64 eps) 1))): 29 points increase in error, 30 points decrease in error

  6. Applied egg-rr32.6

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(-\sin \varepsilon\right) \cdot \left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right)\right)\right)} - 1} \]

  7. Simplified0.2

    \[\leadsto \color{blue}{\sin \varepsilon \cdot \left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(-\sin x\right) + \cos x\right)} \]
    Proof
    (*.f64 (sin.f64 eps) (+.f64 (*.f64 (tan.f64 (*.f64 eps 1/2)) (neg.f64 (sin.f64 x))) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (sin.f64 eps) (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (sin.f64 eps) (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (sin.f64 eps) (*.f64 -1 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))) (*.f64 (sin.f64 eps) (cos.f64 x)))): 12 points increase in error, 8 points decrease in error
    (+.f64 (*.f64 (sin.f64 eps) (Rewrite=> mul-1-neg_binary64 (neg.f64 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))))) (*.f64 (sin.f64 eps) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (sin.f64 eps) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))))) (*.f64 (sin.f64 eps) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (sin.f64 eps)) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))) (*.f64 (sin.f64 eps) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (*.f64 (neg.f64 (sin.f64 eps)) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-udef_binary64 (fma.f64 (sin.f64 eps) (cos.f64 x) (*.f64 (neg.f64 (sin.f64 eps)) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))))): 7 points increase in error, 6 points decrease in error
    (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (fma.f64 (sin.f64 eps) (cos.f64 x) (*.f64 (neg.f64 (sin.f64 eps)) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))))))): 27 points increase in error, 2 points decrease in error
    (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (fma.f64 (sin.f64 eps) (cos.f64 x) (*.f64 (neg.f64 (sin.f64 eps)) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))))) 1)): 172 points increase in error, 4 points decrease in error

  8. Applied egg-rr0.2

    \[\leadsto \sin \varepsilon \cdot \color{blue}{\left(\cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right)} \]

  9. Simplified0.2

    \[\leadsto \sin \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right), -\sin x, \cos x\right)} \]
    Proof
    (fma.f64 (tan.f64 (*.f64 eps 1/2)) (neg.f64 (sin.f64 x)) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (tan.f64 (*.f64 eps 1/2)) (neg.f64 (sin.f64 x))) (cos.f64 x))): 11 points increase in error, 6 points decrease in error
    (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> +-commutative_binary64 (+.f64 (cos.f64 x) (*.f64 -1 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (cos.f64 x) (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= sub-neg_binary64 (-.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error

  10. Final simplification0.2

    \[\leadsto \sin \varepsilon \cdot \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right), -\sin x, \cos x\right) \]

Alternatives

Alternative 1
Error0.2
Cost26176
\[\sin \varepsilon \cdot \left(\cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right) \]
Alternative 2
Error15.3
Cost13632
\[\left(2 \cdot \cos \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) \]
Alternative 3
Error15.3
Cost13504
\[\cos \left(x + \varepsilon \cdot 0.5\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
Alternative 4
Error37.1
Cost13120
\[\sin \left(\varepsilon + x\right) - \sin x \]

Error

Reproduce

herbie shell --seed 2022334 
(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)))