sintan (problem 3.4.5)

Average Error: 31.3 → 0.1

Time: 17.6s

Precision: binary64

Cost: 13636

Math

\[\frac{x - \sin x}{x - \tan x} \]

↓

\[\begin{array}{l} t_0 := x - \sin x\\ \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;-1 + \left(1 - \frac{t_0}{\tan x - x}\right)\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x - \tan x}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (sin x))))
   (if (<= x -0.0052)
     (+ -1.0 (- 1.0 (/ t_0 (- (tan x) x))))
     (if (<= x 0.0055) (- (* 0.225 (pow x 2.0)) 0.5) (/ t_0 (- x (tan x)))))))

C

double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}

↓

double code(double x) {
	double t_0 = x - sin(x);
	double tmp;
	if (x <= -0.0052) {
		tmp = -1.0 + (1.0 - (t_0 / (tan(x) - x)));
	} else if (x <= 0.0055) {
		tmp = (0.225 * pow(x, 2.0)) - 0.5;
	} else {
		tmp = t_0 / (x - tan(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x - sin(x)) / (x - tan(x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sin(x)
    if (x <= (-0.0052d0)) then
        tmp = (-1.0d0) + (1.0d0 - (t_0 / (tan(x) - x)))
    else if (x <= 0.0055d0) then
        tmp = (0.225d0 * (x ** 2.0d0)) - 0.5d0
    else
        tmp = t_0 / (x - tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (x - Math.sin(x)) / (x - Math.tan(x));
}

↓

public static double code(double x) {
	double t_0 = x - Math.sin(x);
	double tmp;
	if (x <= -0.0052) {
		tmp = -1.0 + (1.0 - (t_0 / (Math.tan(x) - x)));
	} else if (x <= 0.0055) {
		tmp = (0.225 * Math.pow(x, 2.0)) - 0.5;
	} else {
		tmp = t_0 / (x - Math.tan(x));
	}
	return tmp;
}

Python

def code(x):
	return (x - math.sin(x)) / (x - math.tan(x))

↓

def code(x):
	t_0 = x - math.sin(x)
	tmp = 0
	if x <= -0.0052:
		tmp = -1.0 + (1.0 - (t_0 / (math.tan(x) - x)))
	elif x <= 0.0055:
		tmp = (0.225 * math.pow(x, 2.0)) - 0.5
	else:
		tmp = t_0 / (x - math.tan(x))
	return tmp

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end

↓

function code(x)
	t_0 = Float64(x - sin(x))
	tmp = 0.0
	if (x <= -0.0052)
		tmp = Float64(-1.0 + Float64(1.0 - Float64(t_0 / Float64(tan(x) - x))));
	elseif (x <= 0.0055)
		tmp = Float64(Float64(0.225 * (x ^ 2.0)) - 0.5);
	else
		tmp = Float64(t_0 / Float64(x - tan(x)));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (x - sin(x)) / (x - tan(x));
end

↓

function tmp_2 = code(x)
	t_0 = x - sin(x);
	tmp = 0.0;
	if (x <= -0.0052)
		tmp = -1.0 + (1.0 - (t_0 / (tan(x) - x)));
	elseif (x <= 0.0055)
		tmp = (0.225 * (x ^ 2.0)) - 0.5;
	else
		tmp = t_0 / (x - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0052], N[(-1.0 + N[(1.0 - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0055], N[(N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(t$95$0 / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0051999999999999998

   1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{-1 + \left(1 - \frac{x - \sin x}{\tan x - x}\right)} \]

   if -0.0051999999999999998 < x < 0.0054999999999999997

   1. Initial program 63.5

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   if 0.0054999999999999997 < x

   1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;-1 + \left(1 - \frac{x - \sin x}{\tan x - x}\right)\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	13512

\[\begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	0.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	0.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	32.0
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))