2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 14.8s

Precision: binary64

Cost: 39168

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) - ((sin(x) / (cos(eps) + 1.0d0)) * (sin(eps) ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

↓

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) - ((Math.sin(x) / (Math.cos(eps) + 1.0)) * Math.pow(Math.sin(eps), 2.0));
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

↓

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) - ((math.sin(x) / (math.cos(eps) + 1.0)) * math.pow(math.sin(eps), 2.0))

Julia

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

↓

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) - Float64(Float64(sin(x) / Float64(cos(eps) + 1.0)) * (sin(eps) ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

↓

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) - ((sin(x) / (cos(eps) + 1.0)) * (sin(eps) ^ 2.0));
end

Wolfram

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

↓

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.1
Target	14.8
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 22.2

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

3. Simplified 0.4

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

   [Start] 22.2	\[ \sin x \cdot \cos \varepsilon + \left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right) \]
   associate-+r+ [=>] 0.4	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon} \]
   +-commutative [<=] 0.4	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 0.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) \]
   fma-def [=>] 0.4	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\cos \varepsilon \cdot \sin x} + \left(-\sin x\right)\right) \]
   neg-mul-1 [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \cos \varepsilon \cdot \sin x + \color{blue}{-1 \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 0.4	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\sin x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

4. Applied egg-rr 0.4

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

5. Taylor expanded in eps around inf 0.4

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

6. Simplified 0.4

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

   [Start] 0.4	\[ \cos x \cdot \sin \varepsilon + -1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \]
   *-commutative [=>] 0.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + -1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \]
   mul-1-neg [=>] 0.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(-\frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right)} \]
   unsub-neg [=>] 0.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x - \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   associate-*l/ [<=] 0.4	\[ \sin \varepsilon \cdot \cos x - \color{blue}{\frac{\sin x}{1 + \cos \varepsilon} \cdot {\sin \varepsilon}^{2}} \]
   +-commutative [=>] 0.4	\[ \sin \varepsilon \cdot \cos x - \frac{\sin x}{\color{blue}{\cos \varepsilon + 1}} \cdot {\sin \varepsilon}^{2} \]

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	32448

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

Alternative 2
Error	0.4
Cost	32448

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

Alternative 3
Error	0.4
Cost	26432

\[\sin \varepsilon \cdot \cos x + \sin x \cdot \frac{1}{\frac{1}{\cos \varepsilon + -1}} \]

Alternative 4
Error	0.4
Cost	26176

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

Alternative 5
Error	14.8
Cost	13632

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

Alternative 6
Error	14.7
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00145:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Error	14.9
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00047:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8
Error	28.6
Cost	6464

\[\sin \varepsilon \]

Alternative 9
Error	45.4
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023057 
(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)))