?

Average Error: 0.0 → 0.0
Time: 10.4s
Precision: binary64
Cost: 14528

?

\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
↓
\[\begin{array}{l} t_1 := \frac{2}{1 + t}\\ t_2 := t_1 + -4\\ \frac{\mathsf{fma}\left(t_1, t_2, 5\right)}{\mathsf{fma}\left(t_1, t_2, 6\right)} \end{array} \]
(FPCore (t)
 :precision binary64
 (/
  (+
   1.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   2.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))
↓
(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))) (t_2 (+ t_1 -4.0)))
   (/ (fma t_1 t_2 5.0) (fma t_1 t_2 6.0))))
double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
}
↓
double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double t_2 = t_1 + -4.0;
	return fma(t_1, t_2, 5.0) / fma(t_1, t_2, 6.0);
}

function code(t)
	return Float64(Float64(1.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))))) / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))))))
end
↓
function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	t_2 = Float64(t_1 + -4.0)
	return Float64(fma(t_1, t_2, 5.0) / fma(t_1, t_2, 6.0))
end

code[t_] := N[(N[(1.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -4.0), $MachinePrecision]}, N[(N[(t$95$1 * t$95$2 + 5.0), $MachinePrecision] / N[(t$95$1 * t$95$2 + 6.0), $MachinePrecision]), $MachinePrecision]]]

\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\begin{array}{l}
t_1 := \frac{2}{1 + t}\\
t_2 := t_1 + -4\\
\frac{\mathsf{fma}\left(t_1, t_2, 5\right)}{\mathsf{fma}\left(t_1, t_2, 6\right)}
\end{array}

Error?

Derivation?

  1. Initial program 0.0

    \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 6\right)}} \]
    Proof

    [Start]0.0

    \[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

  3. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 6\right)} \]

Alternatives

Alternative 1
Error0.0
Cost1728
\[\begin{array}{l} t_1 := \frac{\frac{4}{1 + t} + -8}{1 + t}\\ \frac{5 + t_1}{6 + t_1} \end{array} \]
Alternative 2
Error0.5
Cost969
\[\begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]
Alternative 3
Error0.6
Cost713
\[\begin{array}{l} \mathbf{if}\;t \leq -0.85 \lor \neg \left(t \leq 0.39\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]
Alternative 4
Error0.6
Cost585
\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]
Alternative 5
Error0.9
Cost584
\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 6
Error1.0
Cost328
\[\begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 7
Error26.2
Cost64
\[0.5 \]

Error

Reproduce?

herbie shell --seed 2023018 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))