Average Error: 31.8 → 0.0
Time: 18.0s
Precision: binary64
Cost: 26697
\[\frac{x - \sin x}{x - \tan x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.1 \lor \neg \left(x \leq 0.09\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.009642857142857142, {x}^{4}, 0.00024107142857142857 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \end{array} \]
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.1) (not (<= x 0.09)))
   (/ (- x (sin x)) (- x (tan x)))
   (+
    (fma
     -0.009642857142857142
     (pow x 4.0)
     (* 0.00024107142857142857 (pow x 6.0)))
    (fma 0.225 (* x x) -0.5))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.1) || !(x <= 0.09)) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else {
		tmp = fma(-0.009642857142857142, pow(x, 4.0), (0.00024107142857142857 * pow(x, 6.0))) + fma(0.225, (x * x), -0.5);
	}
	return tmp;
}

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end
↓
function code(x)
	tmp = 0.0
	if ((x <= -0.1) || !(x <= 0.09))
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	else
		tmp = Float64(fma(-0.009642857142857142, (x ^ 4.0), Float64(0.00024107142857142857 * (x ^ 6.0))) + fma(0.225, Float64(x * x), -0.5));
	end
	return tmp
end

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := If[Or[LessEqual[x, -0.1], N[Not[LessEqual[x, 0.09]], $MachinePrecision]], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]

\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.1 \lor \neg \left(x \leq 0.09\right):\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-0.009642857142857142, {x}^{4}, 0.00024107142857142857 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if x < -0.10000000000000001 or 0.089999999999999997 < x

    1. Initial program 0.0

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

    if -0.10000000000000001 < x < 0.089999999999999997

    1. Initial program 63.4

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

    2. Simplified63.4

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

      [Start]63.4

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

      sub-neg [=>]63.4

      \[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]

      +-commutative [=>]63.4

      \[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]

      neg-sub0 [=>]63.4

      \[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      associate-+l- [=>]63.4

      \[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      sub0-neg [=>]63.4

      \[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]

      neg-mul-1 [=>]63.4

      \[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]

      sub-neg [=>]63.4

      \[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]

      +-commutative [=>]63.4

      \[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]

      neg-sub0 [=>]63.4

      \[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      associate-+l- [=>]63.4

      \[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      sub0-neg [=>]63.4

      \[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]

      neg-mul-1 [=>]63.4

      \[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]

      times-frac [=>]63.4

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

      metadata-eval [=>]63.4

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

      *-lft-identity [=>]63.4

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

    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. Simplified0.0

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

      [Start]0.0

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

      +-commutative [=>]0.0

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

      associate--l+ [=>]0.0

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

      fma-def [=>]0.0

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

      unpow2 [=>]0.0

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

      fma-neg [=>]0.0

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

      metadata-eval [=>]0.0

      \[ \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, 0.00024107142857142857 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.5}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.1 \lor \neg \left(x \leq 0.09\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.009642857142857142, {x}^{4}, 0.00024107142857142857 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.4
Cost13513
\[\begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 4.9\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + {x}^{4} \cdot \left(-0.009642857142857142 + 0.00024107142857142857 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
Alternative 2
Error0.0
Cost13513
\[\begin{array}{l} \mathbf{if}\;x \leq -0.1 \lor \neg \left(x \leq 0.09\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + {x}^{4} \cdot \left(-0.009642857142857142 + 0.00024107142857142857 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
Alternative 3
Error0.8
Cost7817
\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + {x}^{4} \cdot \left(-0.009642857142857142 + 0.00024107142857142857 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
Alternative 4
Error0.8
Cost7433
\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right)\\ \end{array} \]
Alternative 5
Error0.9
Cost7049
\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + x \cdot \left(x \cdot 0.225\right)\\ \end{array} \]
Alternative 6
Error0.9
Cost6984
\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3:\\ \;\;\;\;-0.5 + x \cdot \left(x \cdot 0.225\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sin x}{x}\\ \end{array} \]
Alternative 7
Error0.9
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;-0.5 + x \cdot \left(x \cdot 0.225\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 8
Error1.0
Cost328
\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 9
Error31.5
Cost64
\[-0.5 \]

Error

Reproduce

herbie shell --seed 2023017 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))