2sin (example 3.3)

Average Error: 37.2 → 0.4

Time: 2.2min

Precision: binary64

Cost: 26176

Math

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

↓

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

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))))

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));
}

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))
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));
}

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))

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(sin(x) * Float64(cos(eps) - 1.0)))
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));
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[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.2
Target	15.1
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 37.2

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

2. Applied egg-rr 0.4

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

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	14.3
Cost	20040

\[\begin{array}{l} t_0 := \sin x \cdot \left(\cos \varepsilon - 1\right)\\ t_1 := \sin \varepsilon + t_0\\ \mathbf{if}\;\varepsilon \leq -0.00096:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 2.2:\\ \;\;\;\;\cos x \cdot \varepsilon + t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.4
Cost	19912

\[\begin{array}{l} t_0 := \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon - 1\right)\\ \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.2:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	14.5
Cost	19648

\[\sin \varepsilon \cdot \cos x + 0 \cdot \sin x \]

Alternative 4
Error	14.8
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00185:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.2:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.2
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.2:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Error	28.8
Cost	6464

\[\sin \varepsilon \]

Alternative 7
Error	45.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023099 
(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)))