sintan (problem 3.4.5)

Average Error: 31.1 → 0.0

Time: 19.9s

Precision: binary64

Cost: 20164

Math

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

↓

\[\begin{array}{l} t_0 := \tan x - x\\ \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{1}{t_0} \cdot \sin x - \frac{x}{t_0}\\ \mathbf{elif}\;x \leq 0.086:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (tan x) x)))
   (if (<= x -0.095)
     (- (* (/ 1.0 t_0) (sin x)) (/ x t_0))
     (if (<= x 0.086)
       (+
        (+
         (* (* x x) 0.225)
         (+
          (* -0.009642857142857142 (pow x 4.0))
          (* 0.00024107142857142857 (pow x 6.0))))
        -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 <= -0.095) {
		tmp = ((1.0 / t_0) * sin(x)) - (x / t_0);
	} else if (x <= 0.086) {
		tmp = (((x * x) * 0.225) + ((-0.009642857142857142 * pow(x, 4.0)) + (0.00024107142857142857 * pow(x, 6.0)))) + -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 <= (-0.095d0)) then
        tmp = ((1.0d0 / t_0) * sin(x)) - (x / t_0)
    else if (x <= 0.086d0) then
        tmp = (((x * x) * 0.225d0) + (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.00024107142857142857d0 * (x ** 6.0d0)))) + (-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 <= -0.095) {
		tmp = ((1.0 / t_0) * Math.sin(x)) - (x / t_0);
	} else if (x <= 0.086) {
		tmp = (((x * x) * 0.225) + ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.00024107142857142857 * Math.pow(x, 6.0)))) + -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 <= -0.095:
		tmp = ((1.0 / t_0) * math.sin(x)) - (x / t_0)
	elif x <= 0.086:
		tmp = (((x * x) * 0.225) + ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.00024107142857142857 * math.pow(x, 6.0)))) + -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 <= -0.095)
		tmp = Float64(Float64(Float64(1.0 / t_0) * sin(x)) - Float64(x / t_0));
	elseif (x <= 0.086)
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.225) + Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.00024107142857142857 * (x ^ 6.0)))) + -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 <= -0.095)
		tmp = ((1.0 / t_0) * sin(x)) - (x / t_0);
	elseif (x <= 0.086)
		tmp = (((x * x) * 0.225) + ((-0.009642857142857142 * (x ^ 4.0)) + (0.00024107142857142857 * (x ^ 6.0)))) + -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, -0.095], N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.086], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.225), $MachinePrecision] + N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $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 < -0.095000000000000001

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 0.0	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 0.0	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 0.0	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 0.0	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 0.0	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 0.0	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 0.0	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 0.0	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Applied egg-rr 0.0

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

   4. Applied egg-rr 0.0

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

   if -0.095000000000000001 < x < 0.085999999999999993

   1. Initial program 63.0

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

   2. Simplified 63.0

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

      [Start] 63.0	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 63.0	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 63.0	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 63.0	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 63.0	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 63.0	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 63.0	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 63.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 63.0	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 63.0	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 0.0

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

   4. Applied egg-rr 0.0

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.225 \cdot {x}^{2}\right)} - 1\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   5. Simplified 0.0

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

      [Start] 0.0	\[ \left(\left(e^{\mathsf{log1p}\left(0.225 \cdot {x}^{2}\right)} - 1\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      expm1-def [=>] 0.0	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.225 \cdot {x}^{2}\right)\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      expm1-log1p [=>] 0.0	\[ \left(\color{blue}{0.225 \cdot {x}^{2}} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      *-commutative [=>] 0.0	\[ \left(\color{blue}{{x}^{2} \cdot 0.225} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      unpow2 [=>] 0.0	\[ \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.225 + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   if 0.085999999999999993 < x

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{1}{\tan x - x} \cdot \sin x - \frac{x}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.086:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	14152

\[\begin{array}{l} \mathbf{if}\;x \leq -0.096:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \mathbf{elif}\;x \leq 0.086:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -0.025 \lor \neg \left(x \leq 0.031\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \end{array} \]

Alternative 3
Error	0.0
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -0.025:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;x \leq -2.95:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.9:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	0.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.5:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	1.0
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	32.1
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))