sintan (problem 3.4.5)

Average Error: 31.8 → 0.0

Time: 1.5min

Precision: binary64

Cost: 20036

Math

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

↓

\[\begin{array}{l} t_0 := \tan x - x\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\frac{\sin x}{t_0} - \frac{x}{t_0}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (tan x) x)))
   (if (<= x -0.034)
     (- (/ (sin x) t_0) (/ x t_0))
     (if (<= x 0.028)
       (- (+ (* 0.225 (pow x 2.0)) (* -0.009642857142857142 (pow x 4.0))) 0.5)
       (/ (- x (sin x)) (- x (tan x)))))))

C

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

↓

double code(double x) {
	double t_0 = tan(x) - x;
	double tmp;
	if (x <= -0.034) {
		tmp = (sin(x) / t_0) - (x / t_0);
	} else if (x <= 0.028) {
		tmp = ((0.225 * pow(x, 2.0)) + (-0.009642857142857142 * pow(x, 4.0))) - 0.5;
	} else {
		tmp = (x - sin(x)) / (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 = tan(x) - x
    if (x <= (-0.034d0)) then
        tmp = (sin(x) / t_0) - (x / t_0)
    else if (x <= 0.028d0) then
        tmp = ((0.225d0 * (x ** 2.0d0)) + ((-0.009642857142857142d0) * (x ** 4.0d0))) - 0.5d0
    else
        tmp = (x - sin(x)) / (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 = Math.tan(x) - x;
	double tmp;
	if (x <= -0.034) {
		tmp = (Math.sin(x) / t_0) - (x / t_0);
	} else if (x <= 0.028) {
		tmp = ((0.225 * Math.pow(x, 2.0)) + (-0.009642857142857142 * Math.pow(x, 4.0))) - 0.5;
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.tan(x) - x
	tmp = 0
	if x <= -0.034:
		tmp = (math.sin(x) / t_0) - (x / t_0)
	elif x <= 0.028:
		tmp = ((0.225 * math.pow(x, 2.0)) + (-0.009642857142857142 * math.pow(x, 4.0))) - 0.5
	else:
		tmp = (x - math.sin(x)) / (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(tan(x) - x)
	tmp = 0.0
	if (x <= -0.034)
		tmp = Float64(Float64(sin(x) / t_0) - Float64(x / t_0));
	elseif (x <= 0.028)
		tmp = Float64(Float64(Float64(0.225 * (x ^ 2.0)) + Float64(-0.009642857142857142 * (x ^ 4.0))) - 0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / 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 = tan(x) - x;
	tmp = 0.0;
	if (x <= -0.034)
		tmp = (sin(x) / t_0) - (x / t_0);
	elseif (x <= 0.028)
		tmp = ((0.225 * (x ^ 2.0)) + (-0.009642857142857142 * (x ^ 4.0))) - 0.5;
	else
		tmp = (x - sin(x)) / (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[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -0.034], N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.028], N[(N[(N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.034000000000000002

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}} \]

   if -0.034000000000000002 < x < 0.0280000000000000006

   1. Initial program 63.0

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

   2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5} \]

   if 0.0280000000000000006 < x

   1. Initial program 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	13768

\[\begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \leq -0.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13512

\[\begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \leq -0.0037:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.005:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\left(x + x\right) - \tan x\right) - x}\\ \end{array} \]

Alternative 4
Error	0.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 5
Error	1.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	31.5
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))