sintan (problem 3.4.5)

Average Error: 31.4 → 0.0

Time: 18.1s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_0 := \tan x - x\\ t_1 := \frac{\sin x}{t_0} - \frac{x}{t_0}\\ \mathbf{if}\;x \leq -0.105:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.098:\\ \;\;\;\;0.225 \cdot {x}^{2} + \left(\left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (tan x) x)) (t_1 (- (/ (sin x) t_0) (/ x t_0))))
   (if (<= x -0.105)
     t_1
     (if (<= x 0.098)
       (+
        (* 0.225 (pow x 2.0))
        (-
         (+
          (* -0.009642857142857142 (pow x 4.0))
          (* 0.00024107142857142857 (pow x 6.0)))
         0.5))
       t_1))))

C

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

↓

double code(double x) {
	double t_0 = tan(x) - x;
	double t_1 = (sin(x) / t_0) - (x / t_0);
	double tmp;
	if (x <= -0.105) {
		tmp = t_1;
	} else if (x <= 0.098) {
		tmp = (0.225 * pow(x, 2.0)) + (((-0.009642857142857142 * pow(x, 4.0)) + (0.00024107142857142857 * pow(x, 6.0))) - 0.5);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = tan(x) - x
    t_1 = (sin(x) / t_0) - (x / t_0)
    if (x <= (-0.105d0)) then
        tmp = t_1
    else if (x <= 0.098d0) then
        tmp = (0.225d0 * (x ** 2.0d0)) + ((((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.00024107142857142857d0 * (x ** 6.0d0))) - 0.5d0)
    else
        tmp = t_1
    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 t_1 = (Math.sin(x) / t_0) - (x / t_0);
	double tmp;
	if (x <= -0.105) {
		tmp = t_1;
	} else if (x <= 0.098) {
		tmp = (0.225 * Math.pow(x, 2.0)) + (((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.00024107142857142857 * Math.pow(x, 6.0))) - 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.tan(x) - x
	t_1 = (math.sin(x) / t_0) - (x / t_0)
	tmp = 0
	if x <= -0.105:
		tmp = t_1
	elif x <= 0.098:
		tmp = (0.225 * math.pow(x, 2.0)) + (((-0.009642857142857142 * math.pow(x, 4.0)) + (0.00024107142857142857 * math.pow(x, 6.0))) - 0.5)
	else:
		tmp = t_1
	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)
	t_1 = Float64(Float64(sin(x) / t_0) - Float64(x / t_0))
	tmp = 0.0
	if (x <= -0.105)
		tmp = t_1;
	elseif (x <= 0.098)
		tmp = Float64(Float64(0.225 * (x ^ 2.0)) + Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.00024107142857142857 * (x ^ 6.0))) - 0.5));
	else
		tmp = t_1;
	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;
	t_1 = (sin(x) / t_0) - (x / t_0);
	tmp = 0.0;
	if (x <= -0.105)
		tmp = t_1;
	elseif (x <= 0.098)
		tmp = (0.225 * (x ^ 2.0)) + (((-0.009642857142857142 * (x ^ 4.0)) + (0.00024107142857142857 * (x ^ 6.0))) - 0.5);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.105], t$95$1, If[LessEqual[x, 0.098], N[(N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.104999999999999996 or 0.098000000000000004 < x

   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 -0.104999999999999996 < x < 0.098000000000000004

   1. Initial program 63.2

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

   2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5} \]

   3. Simplified 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} + \left(\left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right) - 0.5\right)} \]
      Proof

      [Start] 0.0	\[ \left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      rational.json-simplify-5 [=>] 0.0	\[ \color{blue}{0.225 \cdot {x}^{2} + \left(\left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right) - 0.5\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.105:\\ \;\;\;\;\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.098:\\ \;\;\;\;0.225 \cdot {x}^{2} + \left(\left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	20168

\[\begin{array}{l} t_0 := \tan x - x\\ \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;\frac{x - \sin x}{\left(-x\right) + 1 \cdot \left(\left(x + x\right) - \tan x\right)}\\ \mathbf{elif}\;x \leq 0.005:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{t_0} - \frac{x}{t_0}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13960

\[\begin{array}{l} t_0 := \frac{x - \sin x}{\left(-x\right) + 1 \cdot \left(\left(x + x\right) - \tan x\right)}\\ \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	13512

\[\begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.9
Cost	7048

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

Alternative 5
Error	1.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	31.9
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))