sintan (problem 3.4.5)

Average Error: 48.45% → 0.02%

Time: 18.5s

Precision: binary64

Cost: 13513

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.092 \lor \neg \left(x \leq 0.1\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857 + -0.009642857142857142\right)\right)\right) + -0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.092) (not (<= x 0.1)))
   (/ (- x (sin x)) (- x (tan x)))
   (+
    (+
     (* (* x x) 0.225)
     (*
      (* x x)
      (*
       (* x x)
       (+ (* (* x x) 0.00024107142857142857) -0.009642857142857142))))
    -0.5)))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -0.092) || !(x <= 0.1)) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else {
		tmp = (((x * x) * 0.225) + ((x * x) * ((x * x) * (((x * x) * 0.00024107142857142857) + -0.009642857142857142)))) + -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 <= (-0.092d0)) .or. (.not. (x <= 0.1d0))) then
        tmp = (x - sin(x)) / (x - tan(x))
    else
        tmp = (((x * x) * 0.225d0) + ((x * x) * ((x * x) * (((x * x) * 0.00024107142857142857d0) + (-0.009642857142857142d0))))) + (-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 <= -0.092) || !(x <= 0.1)) {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	} else {
		tmp = (((x * x) * 0.225) + ((x * x) * ((x * x) * (((x * x) * 0.00024107142857142857) + -0.009642857142857142)))) + -0.5;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (x <= -0.092) or not (x <= 0.1):
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	else:
		tmp = (((x * x) * 0.225) + ((x * x) * ((x * x) * (((x * x) * 0.00024107142857142857) + -0.009642857142857142)))) + -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 <= -0.092) || !(x <= 0.1))
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.225) + Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * 0.00024107142857142857) + -0.009642857142857142)))) + -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 <= -0.092) || ~((x <= 0.1)))
		tmp = (x - sin(x)) / (x - tan(x));
	else
		tmp = (((x * x) * 0.225) + ((x * x) * ((x * x) * (((x * x) * 0.00024107142857142857) + -0.009642857142857142)))) + -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, -0.092], N[Not[LessEqual[x, 0.1]], $MachinePrecision]], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.225), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.00024107142857142857), $MachinePrecision] + -0.009642857142857142), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.091999999999999998 or 0.10000000000000001 < x

   1. Initial program 0.04

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

   if -0.091999999999999998 < x < 0.10000000000000001

   1. Initial program 98.74

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

   2. Simplified 98.74

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

      [Start] 98.74	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 98.74	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 98.74	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 98.74	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 98.74	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 98.74	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 98.74	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 98.74	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 98.74	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 98.74	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 98.74	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 98.74	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 98.74	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 98.74	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 98.74	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 98.74	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 0.01

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

   4. Applied egg-rr 0.04

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.225 \cdot \left(x \cdot x\right)\right)} - 1\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   5. Simplified 0.01

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

      [Start] 0.04	\[ \left(\left(e^{\mathsf{log1p}\left(0.225 \cdot \left(x \cdot x\right)\right)} - 1\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      expm1-def [=>] 0.01	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.225 \cdot \left(x \cdot x\right)\right)\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      expm1-log1p [=>] 0.01	\[ \left(\color{blue}{0.225 \cdot \left(x \cdot x\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      *-commutative [=>] 0.01	\[ \left(\color{blue}{\left(x \cdot x\right) \cdot 0.225} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   6. Applied egg-rr 0.01

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

   7. Applied egg-rr 0.01

      \[\leadsto \left(\left(x \cdot x\right) \cdot 0.225 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857 + -0.009642857142857142\right)\right)}\right) - 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.02

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.092 \lor \neg \left(x \leq 0.1\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857 + -0.009642857142857142\right)\right)\right) + -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	1.03%
Cost	8073

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(x \cdot x\right) \cdot \left(0.00024107142857142857 \cdot {x}^{4} + \left(x \cdot x\right) \cdot -0.009642857142857142\right)\right) + -0.5\\ \end{array} \]

Alternative 2
Error	1.03%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857 + -0.009642857142857142\right)\right)\right) + -0.5\\ \end{array} \]

Alternative 3
Error	1.04%
Cost	1736

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.5:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857 + -0.009642857142857142\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	1.1%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -11:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.9:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.225 + \left(x \cdot x\right) \cdot -0.009642857142857142\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	1.21%
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	1.47%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -11:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	50.46%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))