2sin (example 3.3)

Average Error: 58.09% → 0.33%

Time: 13.3s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

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

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 = (cos(x) * sin(eps)) - ((sin(eps) * tan((eps / 2.0d0))) * sin(x))
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.cos(x) * Math.sin(eps)) - ((Math.sin(eps) * Math.tan((eps / 2.0))) * Math.sin(x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, eps)
	tmp = (cos(x) * sin(eps)) - ((sin(eps) * tan((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[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[eps], $MachinePrecision] * N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	58.09%
Target	23.06%
Herbie	0.33%

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

Derivation

1. Initial program 58.09

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

2. Applied egg-rr 34.74

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

3. Simplified 0.56

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

   [Start] 34.74	\[ \sin x \cdot \cos \varepsilon + \left(\left(-\sin x\right) + \cos x \cdot \sin \varepsilon\right) \]
   associate-+r+ [=>] 0.59	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon} \]
   +-commutative [<=] 0.59	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 0.59	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) \]
   fma-def [=>] 0.57	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   *-commutative [=>] 0.57	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\cos \varepsilon \cdot \sin x} + \left(-\sin x\right)\right) \]
   neg-mul-1 [=>] 0.57	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \cos \varepsilon \cdot \sin x + \color{blue}{-1 \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 0.56	\[ \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\sin x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

4. Applied egg-rr 0.53

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

5. Taylor expanded in eps around inf 0.55

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

6. Simplified 0.55

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

   [Start] 0.55	\[ \cos x \cdot \sin \varepsilon + -1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \]
   *-commutative [<=] 0.55	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + -1 \cdot \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \]
   mul-1-neg [=>] 0.55	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(-\frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right)} \]
   unsub-neg [=>] 0.55	\[ \color{blue}{\sin \varepsilon \cdot \cos x - \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   *-commutative [=>] 0.55	\[ \color{blue}{\cos x \cdot \sin \varepsilon} - \frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \]
   *-commutative [=>] 0.55	\[ \cos x \cdot \sin \varepsilon - \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \sin x}}{1 + \cos \varepsilon} \]
   associate-/l* [=>] 0.55	\[ \cos x \cdot \sin \varepsilon - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]
   associate-/r/ [=>] 0.55	\[ \cos x \cdot \sin \varepsilon - \color{blue}{\frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \cdot \sin x} \]
   +-commutative [=>] 0.55	\[ \cos x \cdot \sin \varepsilon - \frac{{\sin \varepsilon}^{2}}{\color{blue}{\cos \varepsilon + 1}} \cdot \sin x \]

7. Taylor expanded in eps around inf 0.55

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

8. Simplified 0.33

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

   [Start] 0.55	\[ \cos x \cdot \sin \varepsilon - \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon} \cdot \sin x \]
   unpow2 [=>] 0.55	\[ \cos x \cdot \sin \varepsilon - \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon} \cdot \sin x \]
   +-commutative [=>] 0.55	\[ \cos x \cdot \sin \varepsilon - \frac{\sin \varepsilon \cdot \sin \varepsilon}{\color{blue}{\cos \varepsilon + 1}} \cdot \sin x \]
   associate-*r/ [<=] 0.55	\[ \cos x \cdot \sin \varepsilon - \color{blue}{\left(\sin \varepsilon \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \cdot \sin x \]
   +-commutative [<=] 0.55	\[ \cos x \cdot \sin \varepsilon - \left(\sin \varepsilon \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \cdot \sin x \]
   hang-0p-tan [=>] 0.33	\[ \cos x \cdot \sin \varepsilon - \left(\sin \varepsilon \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sin x \]

9. Final simplification 0.33

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

Alternatives

Alternative 1
Error	0.56%
Cost	32448

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

Alternative 2
Error	0.58%
Cost	26176

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

Alternative 3
Error	22.06%
Cost	26048

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

Alternative 4
Error	23.12%
Cost	13888

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

Alternative 5
Error	22.69%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000115 \lor \neg \left(\varepsilon \leq 0.14\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]

Alternative 6
Error	23.37%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000115:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.14:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Error	45.3%
Cost	6464

\[\sin \varepsilon \]

Alternative 8
Error	95.73%
Cost	64

\[0 \]

Alternative 9
Error	70.64%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023102 
(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)))