sintan (problem 3.4.5)

Average Error: 31.1 → 0.2

Time: 9.8s

Precision: binary64

Cost: 20104

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-\frac{\sin x - \frac{\sin x}{\cos x}}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.024)
   (/ (- x (sin x)) (- x (tan x)))
   (if (<= x 2.6)
     (- (+ (* 0.225 (pow x 2.0)) (* -0.009642857142857142 (pow x 4.0))) 0.5)
     (+ 1.0 (- (/ (- (sin x) (/ (sin x) (cos x))) x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.024) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else if (x <= 2.6) {
		tmp = ((0.225 * pow(x, 2.0)) + (-0.009642857142857142 * pow(x, 4.0))) - 0.5;
	} else {
		tmp = 1.0 + -((sin(x) - (sin(x) / cos(x))) / 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) :: tmp
    if (x <= (-0.024d0)) then
        tmp = (x - sin(x)) / (x - tan(x))
    else if (x <= 2.6d0) then
        tmp = ((0.225d0 * (x ** 2.0d0)) + ((-0.009642857142857142d0) * (x ** 4.0d0))) - 0.5d0
    else
        tmp = 1.0d0 + -((sin(x) - (sin(x) / cos(x))) / 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 tmp;
	if (x <= -0.024) {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	} else if (x <= 2.6) {
		tmp = ((0.225 * Math.pow(x, 2.0)) + (-0.009642857142857142 * Math.pow(x, 4.0))) - 0.5;
	} else {
		tmp = 1.0 + -((Math.sin(x) - (Math.sin(x) / Math.cos(x))) / x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.024:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	elif x <= 2.6:
		tmp = ((0.225 * math.pow(x, 2.0)) + (-0.009642857142857142 * math.pow(x, 4.0))) - 0.5
	else:
		tmp = 1.0 + -((math.sin(x) - (math.sin(x) / math.cos(x))) / x)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -0.024)
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	elseif (x <= 2.6)
		tmp = Float64(Float64(Float64(0.225 * (x ^ 2.0)) + Float64(-0.009642857142857142 * (x ^ 4.0))) - 0.5);
	else
		tmp = Float64(1.0 + Float64(-Float64(Float64(sin(x) - Float64(sin(x) / cos(x))) / x)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.024)
		tmp = (x - sin(x)) / (x - tan(x));
	elseif (x <= 2.6)
		tmp = ((0.225 * (x ^ 2.0)) + (-0.009642857142857142 * (x ^ 4.0))) - 0.5;
	else
		tmp = 1.0 + -((sin(x) - (sin(x) / cos(x))) / 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_] := If[LessEqual[x, -0.024], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6], 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[(1.0 + (-N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.024

   1. Initial program 0.1

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

   if -0.024 < x < 2.60000000000000009

   1. Initial program 62.9

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

   2. Taylor expanded in x around 0 0.2

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

   if 2.60000000000000009 < x

   1. Initial program 0.0

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

   2. Taylor expanded in x around -inf 0.6

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]

   3. Simplified 0.6

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

      [Start] 0.6	\[ 1 + -1 \cdot \frac{\sin x - \frac{\sin x}{\cos x}}{x} \]
      rational_best-simplify-3 [=>] 0.6	\[ 1 + \color{blue}{\frac{\sin x - \frac{\sin x}{\cos x}}{x} \cdot -1} \]
      rational_best-simplify-17 [=>] 0.6	\[ 1 + \color{blue}{\left(-\frac{\sin x - \frac{\sin x}{\cos x}}{x}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-\frac{\sin x - \frac{\sin x}{\cos x}}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	13768

\[\begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13512

\[\begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0042:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.8
Cost	7048

\[\begin{array}{l} t_0 := \frac{x}{x - \tan x}\\ \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	6984

\[\begin{array}{l} t_0 := \frac{x}{x - \tan x}\\ \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.1
Cost	328

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

Alternative 6
Error	32.2
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023104 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))