2sin (example 3.3)

Average Error: 36.5 → 0.3

Time: 16.7s

Precision: binary64

Cost: 45504

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (/ (* (sin x) (- (pow (sin eps) 2.0))) (+ (cos eps) 1.0))))

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(x) * -pow(sin(eps), 2.0)) / (cos(eps) + 1.0)));
}

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

Target

Original	36.5
Target	15.1
Herbie	0.3

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

Derivation

1. Initial program 36.5

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

2. Applied egg-rr 21.3

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

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. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\cos \varepsilon + -1\right) \cdot \sin x\right)} \]
   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))))): 14 points increase in error, 6 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)))): 8 points increase in error, 5 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))): 103 points increase in error, 8 points decrease in error

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.5
Cost	39360

\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\sin x \cdot \left(0.5 \cdot \cos \left(\varepsilon + \varepsilon\right) + -0.5\right)}{\cos \varepsilon + 1}\right) \]

Alternative 2
Error	0.4
Cost	32448

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

Alternative 3
Error	0.4
Cost	32448

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

Alternative 4
Error	14.6
Cost	26312

\[\begin{array}{l} t_0 := \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9.96278517613511 \cdot 10^{-21}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.4
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - x\right)\right) \cdot \cos \left(0.5 \cdot \left(\varepsilon + x\right)\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 9.96278517613511 \cdot 10^{-21}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\varepsilon + x\right)\right)\right) - \sin x\\ \end{array} \]

Alternative 6
Error	0.4
Cost	26176

\[\sin x \cdot \left(\cos \varepsilon + -1\right) + \sin \varepsilon \cdot \cos x \]

Alternative 7
Error	15.4
Cost	13764

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - x\right)\right) \cdot \cos \left(0.5 \cdot \left(\varepsilon + x\right)\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 9.96278517613511 \cdot 10^{-21}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \]

Alternative 8
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 9.96278517613511 \cdot 10^{-21}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \]

Alternative 9
Error	14.9
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.68054905408696 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	15.4
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.96278517613511 \cdot 10^{-21}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 11
Error	28.5
Cost	6464

\[\sin \varepsilon \]

Alternative 12
Error	61.3
Cost	64

\[0 \]

Alternative 13
Error	45.4
Cost	64

\[\varepsilon \]

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)))