sintan (problem 3.4.5)

Average Error: 31.6 → 0.4

Time: 19.4s

Precision: binary64

Cost: 13513

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.001 \lor \neg \left(x \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.001) (not (<= x 2e-12)))
   (/ (- x (sin x)) (- x (tan x)))
   -0.5))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -0.001) || !(x <= 2e-12)) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else {
		tmp = -0.5;
	}
	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 <= (-0.001d0)) .or. (.not. (x <= 2d-12))) then
        tmp = (x - sin(x)) / (x - tan(x))
    else
        tmp = -0.5d0
    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 <= -0.001) || !(x <= 2e-12)) {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (x <= -0.001) or not (x <= 2e-12):
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	else:
		tmp = -0.5
	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 <= -0.001) || !(x <= 2e-12))
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	else
		tmp = -0.5;
	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 <= -0.001) || ~((x <= 2e-12)))
		tmp = (x - sin(x)) / (x - tan(x));
	else
		tmp = -0.5;
	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[Or[LessEqual[x, -0.001], N[Not[LessEqual[x, 2e-12]], $MachinePrecision]], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5]

Derivation

1. Split input into 2 regimes

2. if x < -1e-3 or 1.99999999999999996e-12 < x

   1. Initial program 0.6

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

   if -1e-3 < x < 1.99999999999999996e-12

   1. Initial program 63.7

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

   2. Simplified 63.7

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

      [Start] 63.7	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 63.7	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 63.7	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 63.7	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 63.7	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 63.7	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 63.7	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 63.7	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 63.7	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 63.7	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 63.7	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 63.7	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 63.7	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 63.7	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 63.7	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 63.7	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001 \lor \neg \left(x \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \lor \neg \left(x \leq 2.45\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{-0.5 + x \cdot \left(x \cdot 0.225\right)}}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	7108

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

Alternative 3
Error	0.8
Cost	6916

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

Alternative 4
Error	0.8
Cost	968

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

Alternative 5
Error	0.8
Cost	712

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

Alternative 6
Error	0.9
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 7
Error	31.7
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))