2sin (example 3.3)

Average Error: 36.7 → 0.2

Time: 11.7s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

↓

(FPCore (x eps)
 :precision binary64
 (fma (sin eps) (cos x) (* (* (sin eps) (tan (* eps 0.5))) (- (sin x)))))

C

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(eps) * tan((eps * 0.5))) * -sin(x)));
}

Julia

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) * tan(Float64(eps * 0.5))) * Float64(-sin(x))))
end

Wolfram

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[eps], $MachinePrecision] * N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Target

Original	36.7
Target	15.0
Herbie	0.2

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

1. Initial program 36.7

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

2. Applied egg-rr 21.6

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

3. Simplified 0.4

   \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)} \]
   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))))): 4 points increase in error, 6 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)))): 121 points increase in error, 5 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-rr 0.3

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

5. Taylor expanded in x around inf 0.3

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

6. Simplified 0.2

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(-\sin x\right) \cdot \left(\sin \varepsilon \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)} \]
   Proof
   (*.f64 (neg.f64 (sin.f64 x)) (*.f64 (sin.f64 eps) (tan.f64 (/.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (neg.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (sin.f64 x) 1))) (*.f64 (sin.f64 eps) (tan.f64 (/.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 (sin.f64 x)) 1)) (*.f64 (sin.f64 eps) (tan.f64 (/.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (sin.f64 x)) 1) (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (sin.f64 eps) 1)) (tan.f64 (/.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (sin.f64 x)) 1) (*.f64 (/.f64 (sin.f64 eps) 1) (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 eps) (+.f64 1 (cos.f64 eps)))))): 23 points increase in error, 28 points decrease in error
   (*.f64 (/.f64 (neg.f64 (sin.f64 x)) 1) (*.f64 (/.f64 (sin.f64 eps) 1) (/.f64 (sin.f64 eps) (Rewrite<= +-commutative_binary64 (+.f64 (cos.f64 eps) 1))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (sin.f64 x)) 1) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sin.f64 eps) (sin.f64 eps)) (*.f64 1 (+.f64 (cos.f64 eps) 1))))): 25 points increase in error, 17 points decrease in error
   (*.f64 (/.f64 (neg.f64 (sin.f64 x)) 1) (/.f64 (*.f64 (sin.f64 eps) (sin.f64 eps)) (Rewrite=> *-lft-identity_binary64 (+.f64 (cos.f64 eps) 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (sin.f64 x)) 1) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (neg.f64 (sin.f64 x)) (pow.f64 (sin.f64 eps) 2)) (*.f64 1 (+.f64 (cos.f64 eps) 1)))): 19 points increase in error, 23 points decrease in error
   (/.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2)))) (*.f64 1 (+.f64 (cos.f64 eps) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2)))) (*.f64 1 (+.f64 (cos.f64 eps) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 -1 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2))) (Rewrite=> *-lft-identity_binary64 (+.f64 (cos.f64 eps) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 -1 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2))) (Rewrite=> +-commutative_binary64 (+.f64 1 (cos.f64 eps)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2)) (+.f64 1 (cos.f64 eps))))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 0.2

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

8. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	26176

\[\sin \varepsilon \cdot \left(\cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right) \]

Alternative 2
Error	14.3
Cost	26048

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

Alternative 3
Error	15.0
Cost	13632

\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

Alternative 4
Error	14.7
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000235:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.0
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Error	28.4
Cost	6464

\[\sin \varepsilon \]

Alternative 7
Error	61.3
Cost	64

\[0 \]

Alternative 8
Error	45.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022338 
(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)))