Average Error: 30.8 → 0.2
Time: 16.2s
Precision: binary64
Cost: 19976
\[\frac{x - \sin x}{x - \tan x} \]
↓
\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -83.94563223606514:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00564586590815869:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.225, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- (sin x) x) (- (tan x) x))))
   (if (<= x -83.94563223606514)
     t_0
     (if (<= x 0.00564586590815869)
       (fma x (* x 0.225) (fma -0.009642857142857142 (pow x 4.0) -0.5))
       t_0))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double t_0 = (sin(x) - x) / (tan(x) - x);
	double tmp;
	if (x <= -83.94563223606514) {
		tmp = t_0;
	} else if (x <= 0.00564586590815869) {
		tmp = fma(x, (x * 0.225), fma(-0.009642857142857142, pow(x, 4.0), -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end
↓
function code(x)
	t_0 = Float64(Float64(sin(x) - x) / Float64(tan(x) - x))
	tmp = 0.0
	if (x <= -83.94563223606514)
		tmp = t_0;
	elseif (x <= 0.00564586590815869)
		tmp = fma(x, Float64(x * 0.225), fma(-0.009642857142857142, (x ^ 4.0), -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

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[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -83.94563223606514], t$95$0, If[LessEqual[x, 0.00564586590815869], N[(x * N[(x * 0.225), $MachinePrecision] + N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
t_0 := \frac{\sin x - x}{\tan x - x}\\
\mathbf{if}\;x \leq -83.94563223606514:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 0.00564586590815869:\\
\;\;\;\;\mathsf{fma}\left(x, x \cdot 0.225, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if x < -83.945632236065137 or 0.00564586590815868965 < x

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      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, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (sin.f64 x) x)) (*.f64 -1 (-.f64 (tan.f64 x) x)))): 0 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, 0 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))): 0 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, 0 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)))): 0 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, 0 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)): 0 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, 0 points decrease in error

    if -83.945632236065137 < x < 0.00564586590815868965

    1. Initial program 62.9

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

    2. Simplified62.9

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      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, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (sin.f64 x) x)) (*.f64 -1 (-.f64 (tan.f64 x) x)))): 0 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, 0 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))): 0 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, 0 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)))): 0 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, 0 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)): 0 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, 0 points decrease in error

    3. Taylor expanded in x around 0 0.3

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

    4. Simplified0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.225, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)} \]
      Proof
      (fma.f64 x (*.f64 x 9/40) (fma.f64 -27/2800 (pow.f64 x 4) -1/2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (*.f64 x 9/40) (fma.f64 -27/2800 (pow.f64 x 4) (Rewrite<= metadata-eval (neg.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (*.f64 x 9/40) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 -27/2800 (pow.f64 x 4)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 x 9/40)) (-.f64 (*.f64 -27/2800 (pow.f64 x 4)) 1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) 9/40)) (-.f64 (*.f64 -27/2800 (pow.f64 x 4)) 1/2)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) 9/40) (-.f64 (*.f64 -27/2800 (pow.f64 x 4)) 1/2)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 9/40 (pow.f64 x 2))) (-.f64 (*.f64 -27/2800 (pow.f64 x 4)) 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 9/40 (pow.f64 x 2)) (*.f64 -27/2800 (pow.f64 x 4))) 1/2)): 1 points increase in error, 1 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -83.94563223606514:\\ \;\;\;\;\frac{\sin x - x}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.00564586590815869:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.225, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x - x}{\tan x - x}\\ \end{array} \]

Alternatives

Alternative 1
Error0.8
Cost7432
\[\begin{array}{l} \mathbf{if}\;x \leq -83.94563223606514:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.309599646475398:\\ \;\;\;\;-0.5 + \left(x \cdot \left(x \cdot 0.225\right) + -0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 2
Error31.5
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022289 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))