2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 14.0s

Precision: binary64

Cost: 32448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double eps) {
	return fma((cos(eps) + -1.0), sin(x), (cos(x) * sin(eps)));
}

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

↓

function code(x, eps)
	return fma(Float64(cos(eps) + -1.0), sin(x), Float64(cos(x) * sin(eps)))
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.1
Target	15.4
Herbie	0.4

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

Derivation

1. Initial program 37.1

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

2. Applied egg-rr 21.7

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

3. Taylor expanded in x around inf 21.7

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

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))))): 13 points increase in error, 7 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)))): 6 points increase in error, 2 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))): 111 points increase in error, 12 points decrease in error

5. Taylor expanded in eps around inf 0.4

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

6. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \cos x \cdot \sin \varepsilon\right)} \]
   Proof
   (fma.f64 (+.f64 (cos.f64 eps) -1) (sin.f64 x) (*.f64 (cos.f64 x) (sin.f64 eps))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (+.f64 (cos.f64 eps) (Rewrite<= metadata-eval (neg.f64 1))) (sin.f64 x) (*.f64 (cos.f64 x) (sin.f64 eps))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (Rewrite<= sub-neg_binary64 (-.f64 (cos.f64 eps) 1)) (sin.f64 x) (*.f64 (cos.f64 x) (sin.f64 eps))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 (cos.f64 eps) 1) (sin.f64 x)) (*.f64 (cos.f64 x) (sin.f64 eps)))): 14 points increase in error, 3 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (-.f64 (cos.f64 eps) 1))) (*.f64 (cos.f64 x) (sin.f64 eps))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	14.6
Cost	26440

\[\begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ t_1 := \sin x + \left(t_0 - \sin x\right)\\ \mathbf{if}\;\varepsilon \leq -24.337863054659543:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 1.0556431116860713 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right), \sin x, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.6
Cost	26440

\[\begin{array}{l} t_0 := \sin x + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{if}\;\varepsilon \leq -24.337863054659543:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.0556431116860713 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.4
Cost	26176

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

Alternative 4
Error	16.4
Cost	26048

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

Alternative 5
Error	14.8
Cost	13768

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot -0.5\right)\right) + \varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.8
Cost	13640

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	14.9
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.7
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.4478825126045238 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9
Error	28.8
Cost	6464

\[\sin \varepsilon \]

Alternative 10
Error	45.5
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022308 
(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)))