2sin (example 3.3)

Average Error: 36.7 → 0.2

Time: 12.1s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (* (* (- (sin eps)) (tan (/ eps 2.0))) (sin x))))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

↓

double code(double x, double eps) {
	return fma(cos(x), sin(eps), ((-sin(eps) * tan((eps / 2.0))) * sin(x)));
}

Julia

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(Float64(-sin(eps)) * tan(Float64(eps / 2.0))) * 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[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[((-N[Sin[eps], $MachinePrecision]) * N[Tan[N[(eps / 2.0), $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

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

3. Taylor expanded in eps around inf 21.6

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

4. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)} \]
   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) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (*.f64 -1 (sin.f64 x))))): 15 points increase in error, 7 points decrease in error
   (fma.f64 (cos.f64 x) (sin.f64 eps) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (cos.f64 eps))) (*.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 (sin.f64 x) (cos.f64 eps)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 x) (sin.f64 eps) (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (-.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (sin.f64 x)))): 11 points increase in error, 2 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (-.f64 (Rewrite=> *-commutative_binary64 (*.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))): 109 points increase in error, 16 points decrease in error

5. Applied egg-rr 0.4

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

6. Simplified 0.2

   \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \color{blue}{\left(\left(-\sin \varepsilon\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x}\right) \]
   Proof
   (*.f64 (*.f64 (neg.f64 (sin.f64 eps)) (tan.f64 (/.f64 eps 2))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (neg.f64 (sin.f64 eps)) (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 eps) (+.f64 1 (cos.f64 eps))))) (sin.f64 x)): 34 points increase in error, 28 points decrease in error
   (*.f64 (*.f64 (neg.f64 (sin.f64 eps)) (/.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 (*.f64 (neg.f64 (sin.f64 eps)) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (sin.f64 eps) 1)) (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (neg.f64 (sin.f64 eps)) (Rewrite<= associate-*r/_binary64 (*.f64 (sin.f64 eps) (/.f64 1 (+.f64 (cos.f64 eps) 1))))) (sin.f64 x)): 17 points increase in error, 13 points decrease in error
   (*.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 eps) (*.f64 (sin.f64 eps) (/.f64 1 (+.f64 (cos.f64 eps) 1)))))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 eps) (sin.f64 eps)) (/.f64 1 (+.f64 (cos.f64 eps) 1))))) (sin.f64 x)): 33 points increase in error, 28 points decrease in error
   (*.f64 (neg.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 (Rewrite<= distribute-lft-neg-out_binary64 (*.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 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) 1) (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 14 points increase in error, 21 points decrease in error
   (*.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (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-*l/_binary64 (/.f64 (*.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (sin.f64 x)) (+.f64 (cos.f64 eps) 1))): 27 points increase in error, 20 points decrease in error

7. Final simplification 0.2

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

Alternatives

Alternative 1
Error	14.7
Cost	39624

\[\begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	26176

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

Alternative 3
Error	15.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0072:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \left(\varepsilon \cdot \sin x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 4
Error	15.1
Cost	13632

\[\cos \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

Alternative 5
Error	15.1
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.1 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0072:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Error	28.4
Cost	6464

\[\sin \varepsilon \]

Alternative 7
Error	44.8
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022335 
(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)))