Average Error: 0.0 → 0.0
Time: 3.3s
Precision: binary64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
↓
\[\begin{array}{l} t_1 := {\left(\frac{t}{1 + t}\right)}^{2} \cdot 4\\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + t_1}{2 + t_1}\right)\right) \end{array} \]
(FPCore (t)
 :precision binary64
 (/
  (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))
  (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))
↓
(FPCore (t)
 :precision binary64
 (let* ((t_1 (* (pow (/ t (+ 1.0 t)) 2.0) 4.0)))
   (log1p (expm1 (/ (+ 1.0 t_1) (+ 2.0 t_1))))))
double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}
↓
double code(double t) {
	double t_1 = pow((t / (1.0 + t)), 2.0) * 4.0;
	return log1p(expm1(((1.0 + t_1) / (2.0 + t_1))));
}

public static double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}
↓
public static double code(double t) {
	double t_1 = Math.pow((t / (1.0 + t)), 2.0) * 4.0;
	return Math.log1p(Math.expm1(((1.0 + t_1) / (2.0 + t_1))));
}

def code(t):
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))))
↓
def code(t):
	t_1 = math.pow((t / (1.0 + t)), 2.0) * 4.0
	return math.log1p(math.expm1(((1.0 + t_1) / (2.0 + t_1))))

function code(t)
	return Float64(Float64(1.0 + Float64(Float64(Float64(2.0 * t) / Float64(1.0 + t)) * Float64(Float64(2.0 * t) / Float64(1.0 + t)))) / Float64(2.0 + Float64(Float64(Float64(2.0 * t) / Float64(1.0 + t)) * Float64(Float64(2.0 * t) / Float64(1.0 + t)))))
end
↓
function code(t)
	t_1 = Float64((Float64(t / Float64(1.0 + t)) ^ 2.0) * 4.0)
	return log1p(expm1(Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1))))
end

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

\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\begin{array}{l}
t_1 := {\left(\frac{t}{1 + t}\right)}^{2} \cdot 4\\
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + t_1}{2 + t_1}\right)\right)
\end{array}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

  2. Applied egg-rr0.5

    \[\leadsto \color{blue}{1 \cdot \frac{{\left(\mathsf{hypot}\left(1, \frac{t}{\frac{t + 1}{2}}\right)\right)}^{2}}{2 + {\left(\frac{t}{\frac{t + 1}{2}}\right)}^{2}}} \]

  3. Applied egg-rr0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + {\left(\frac{t}{1 + t}\right)}^{2} \cdot 4}{2 + {\left(\frac{t}{1 + t}\right)}^{2} \cdot 4}\right)\right)} \]

  4. Final simplification0.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + {\left(\frac{t}{1 + t}\right)}^{2} \cdot 4}{2 + {\left(\frac{t}{1 + t}\right)}^{2} \cdot 4}\right)\right) \]

Reproduce

herbie shell --seed 2022170 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))