2cos (problem 3.3.5)

Average Error: 39.6 → 0.4

Time: 17.5s

Precision: binary64

Cost: 33600

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := 0.5 \cdot \left(x \cdot 2\right)\\ \left(t_0 \cdot \left(t_0 \cdot \cos t_1 + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin t_1\right)\right) \cdot -2 \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))) (t_1 (* 0.5 (* x 2.0))))
   (* (* t_0 (+ (* t_0 (cos t_1)) (* (cos (* 0.5 eps)) (sin t_1)))) -2.0)))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	double t_1 = 0.5 * (x * 2.0);
	return (t_0 * ((t_0 * cos(t_1)) + (cos((0.5 * eps)) * sin(t_1)))) * -2.0;
}

Fortran

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

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin((0.5d0 * eps))
    t_1 = 0.5d0 * (x * 2.0d0)
    code = (t_0 * ((t_0 * cos(t_1)) + (cos((0.5d0 * eps)) * sin(t_1)))) * (-2.0d0)
end function

Java

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

↓

public static double code(double x, double eps) {
	double t_0 = Math.sin((0.5 * eps));
	double t_1 = 0.5 * (x * 2.0);
	return (t_0 * ((t_0 * Math.cos(t_1)) + (Math.cos((0.5 * eps)) * Math.sin(t_1)))) * -2.0;
}

Python

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

↓

def code(x, eps):
	t_0 = math.sin((0.5 * eps))
	t_1 = 0.5 * (x * 2.0)
	return (t_0 * ((t_0 * math.cos(t_1)) + (math.cos((0.5 * eps)) * math.sin(t_1)))) * -2.0

Julia

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

↓

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	t_1 = Float64(0.5 * Float64(x * 2.0))
	return Float64(Float64(t_0 * Float64(Float64(t_0 * cos(t_1)) + Float64(cos(Float64(0.5 * eps)) * sin(t_1)))) * -2.0)
end

MATLAB

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

↓

function tmp = code(x, eps)
	t_0 = sin((0.5 * eps));
	t_1 = 0.5 * (x * 2.0);
	tmp = (t_0 * ((t_0 * cos(t_1)) + (cos((0.5 * eps)) * sin(t_1)))) * -2.0;
end

Wolfram

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

↓

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * N[(N[(t$95$0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Initial program 39.6

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

2. Applied egg-rr 15.5

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \cdot -2} \]

3. Taylor expanded in x around -inf 15.5

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot -2 \]

4. Applied egg-rr 0.4

   \[\leadsto \left(\color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.9
Cost	32840

\[\begin{array}{l} t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.06558902832275153:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.0733017174529855 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.25\right) + \sin x \cdot \left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.06558902832275153:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 7.0733017174529855 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.25\right) + \sin x \cdot \left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \end{array} \]

Alternative 3
Error	15.5
Cost	13888

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

Alternative 4
Error	15.6
Cost	13640

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.06558902832275153:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.0733017174529855 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.5
Cost	13632

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

Alternative 6
Error	20.6
Cost	13512

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := -2 \cdot \left(t_0 \cdot \sin x\right)\\ \mathbf{if}\;x \leq -2.1785999829974862 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.3414793418423655 \cdot 10^{-66}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	21.9
Cost	13448

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;x \leq -2.1785999829974862 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3414793418423655 \cdot 10^{-66}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.0
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.06558902832275153:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.0733017174529855 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	33.2
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{if}\;\varepsilon \leq -1.7565918431791834 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.511609088854967 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 1.820551508707074 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 7.0733017174529855 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	21.4
Cost	7112

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.06558902832275153:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.0733017174529855 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	21.4
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.06558902832275153:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.0733017174529855 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	48.7
Cost	584

\[\begin{array}{l} t_0 := x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;x \leq -9.164731730283988 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.786131795453176 \cdot 10^{-126}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	52.6
Cost	256

\[x \cdot \left(-\varepsilon\right) \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))