2sin (example 3.3)

Average Error: 37.0 → 0.4

Time: 16.8s

Precision: binary64

Cost: 26176

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double eps) {
	return (sin(eps) * cos(x)) + ((cos(eps) + -1.0) * 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 = (sin(eps) * cos(x)) + ((cos(eps) + (-1.0d0)) * 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.sin(eps) * Math.cos(x)) + ((Math.cos(eps) + -1.0) * Math.sin(x));
}

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.cos(eps) + -1.0) * math.sin(x))

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) + ((cos(eps) + -1.0) * 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[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.0
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 37.0

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

2. Applied egg-rr 22.0

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

3. Taylor expanded in x around inf 22.0

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

4. Simplified 0.4

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

   [Start] 22.0	\[ \left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \]
   rational_best-simplify-1 [=>] 22.0	\[ \color{blue}{\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right)} - \sin x \]
   rational_best-simplify-2 [<=] 22.0	\[ \left(\color{blue}{\sin x \cdot \cos \varepsilon} + \cos x \cdot \sin \varepsilon\right) - \sin x \]
   rational_best-simplify-2 [=>] 22.0	\[ \left(\sin x \cdot \cos \varepsilon + \color{blue}{\sin \varepsilon \cdot \cos x}\right) - \sin x \]
   rational_best-simplify-6 [<=] 22.0	\[ \left(\sin x \cdot \cos \varepsilon + \sin \varepsilon \cdot \cos x\right) - \color{blue}{\left(\sin x - 0\right)} \]
   rational_best-simplify-45 [<=] 22.0	\[ \color{blue}{0 - \left(\sin x - \left(\sin x \cdot \cos \varepsilon + \sin \varepsilon \cdot \cos x\right)\right)} \]
   rational_best-simplify-46 [=>] 22.0	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \sin \varepsilon \cdot \cos x\right) + \left(0 - \sin x\right)} \]
   rational_best-simplify-11 [<=] 22.0	\[ \left(\sin x \cdot \cos \varepsilon + \sin \varepsilon \cdot \cos x\right) + \color{blue}{\left(-\sin x\right)} \]
   rational_best-simplify-1 [<=] 22.0	\[ \color{blue}{\left(-\sin x\right) + \left(\sin x \cdot \cos \varepsilon + \sin \varepsilon \cdot \cos x\right)} \]
   rational_best-simplify-43 [=>] 0.4	\[ \color{blue}{\sin \varepsilon \cdot \cos x + \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right)} \]
   rational_best-simplify-13 [=>] 0.4	\[ \sin \varepsilon \cdot \cos x + \left(\sin x \cdot \cos \varepsilon + \color{blue}{\sin x \cdot -1}\right) \]
   rational_best-simplify-47 [=>] 0.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(-1 + \cos \varepsilon\right)} \]
   rational_best-simplify-1 [<=] 0.4	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} \]
   rational_best-simplify-19 [<=] 0.4	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\cos \varepsilon - 1\right)} \]
   rational_best-simplify-2 [=>] 0.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \sin x} \]
   rational_best-simplify-19 [=>] 0.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(\cos \varepsilon + -1\right)} \cdot \sin x \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	14.0
Cost	26312

\[\begin{array}{l} t_0 := \left(\sin x + \sin \varepsilon \cdot \cos x\right) - \sin x\\ \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.68 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \varepsilon + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.5
Cost	20104

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0019:\\ \;\;\;\;\cos x \cdot \varepsilon + -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 3
Error	15.9
Cost	19776

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

Alternative 4
Error	14.6
Cost	13256

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 5
Error	14.8
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00082:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Error	28.6
Cost	6464

\[\sin \varepsilon \]

Alternative 7
Error	45.4
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023094 
(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)))