Average Error: 31.7 → 0.1
Time: 12.4s
Precision: binary64
Cost: 13513
\[\frac{x - \sin x}{x - \tan x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.005 \lor \neg \left(x \leq 0.0052\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \end{array} \]
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.005) (not (<= x 0.0052)))
   (/ (- x (sin x)) (- x (tan x)))
   (+ (* x (* x 0.225)) -0.5)))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.005) || !(x <= 0.0052)) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else {
		tmp = (x * (x * 0.225)) + -0.5;
	}
	return tmp;
}

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 <= (-0.005d0)) .or. (.not. (x <= 0.0052d0))) then
        tmp = (x - sin(x)) / (x - tan(x))
    else
        tmp = (x * (x * 0.225d0)) + (-0.5d0)
    end if
    code = tmp
end function

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 <= -0.005) || !(x <= 0.0052)) {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	} else {
		tmp = (x * (x * 0.225)) + -0.5;
	}
	return tmp;
}

def code(x):
	return (x - math.sin(x)) / (x - math.tan(x))
↓
def code(x):
	tmp = 0
	if (x <= -0.005) or not (x <= 0.0052):
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	else:
		tmp = (x * (x * 0.225)) + -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 <= -0.005) || !(x <= 0.0052))
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	else
		tmp = Float64(Float64(x * Float64(x * 0.225)) + -0.5);
	end
	return tmp
end

function tmp = code(x)
	tmp = (x - sin(x)) / (x - tan(x));
end
↓
function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.005) || ~((x <= 0.0052)))
		tmp = (x - sin(x)) / (x - tan(x));
	else
		tmp = (x * (x * 0.225)) + -0.5;
	end
	tmp_2 = 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, -0.005], N[Not[LessEqual[x, 0.0052]], $MachinePrecision]], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]]

\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.005 \lor \neg \left(x \leq 0.0052\right):\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0050000000000000001 or 0.0051999999999999998 < x

    1. Initial program 0.1

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

    if -0.0050000000000000001 < x < 0.0051999999999999998

    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
      (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x))): 0 points increase in error, 16 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (sin.f64 x) x)) (*.f64 -1 (-.f64 (tan.f64 x) x)))): 16 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 16 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (sin.f64 x)) x)) (*.f64 -1 (-.f64 (tan.f64 x) x))): 16 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sin.f64 x))) x) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 16 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (sin.f64 x)))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (sin.f64 x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (tan.f64 x) x)))): 16 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 16 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (tan.f64 x)) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (tan.f64 x))) x)): 16 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (tan.f64 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub-neg_binary64 (-.f64 x (tan.f64 x)))): 0 points increase in error, 16 points decrease in error

    3. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
      Proof
      (fma.f64 9/40 (*.f64 x x) -1/2): 0 points increase in error, 0 points decrease in error
      (fma.f64 9/40 (*.f64 x x) (Rewrite<= metadata-eval (neg.f64 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 9/40 (*.f64 x x)) 1/2)): 4 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 9/40 (Rewrite<= unpow2_binary64 (pow.f64 x 2))) 1/2): 0 points increase in error, 4 points decrease in error

    5. Applied egg-rr0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.005 \lor \neg \left(x \leq 0.0052\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error0.8
Cost6984
\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]
Alternative 2
Error0.8
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 3
Error1.0
Cost328
\[\begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 4
Error31.6
Cost64
\[-0.5 \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))