Average Error: 0.0 → 0.0
Time: 8.9s
Precision: binary64
Cost: 8448
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
↓
\[\begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ \frac{1 + t_1 \cdot t_1}{2 + \left(\left(1 + {t_1}^{2}\right) + -1\right)} \end{array} \]
(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 (/ (* 2.0 t) (+ 1.0 t))))
   (/ (+ 1.0 (* t_1 t_1)) (+ 2.0 (+ (+ 1.0 (pow t_1 2.0)) -1.0)))))
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 = (2.0 * t) / (1.0 + t);
	return (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + pow(t_1, 2.0)) + -1.0));
}

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + (((2.0d0 * t) / (1.0d0 + t)) * ((2.0d0 * t) / (1.0d0 + t)))) / (2.0d0 + (((2.0d0 * t) / (1.0d0 + t)) * ((2.0d0 * t) / (1.0d0 + t))))
end function
↓
real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    code = (1.0d0 + (t_1 * t_1)) / (2.0d0 + ((1.0d0 + (t_1 ** 2.0d0)) + (-1.0d0)))
end function

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 = (2.0 * t) / (1.0 + t);
	return (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + Math.pow(t_1, 2.0)) + -1.0));
}

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 = (2.0 * t) / (1.0 + t)
	return (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + math.pow(t_1, 2.0)) + -1.0))

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(2.0 * t) / Float64(1.0 + t))
	return Float64(Float64(1.0 + Float64(t_1 * t_1)) / Float64(2.0 + Float64(Float64(1.0 + (t_1 ^ 2.0)) + -1.0)))
end

function tmp = code(t)
	tmp = (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))));
end
↓
function tmp = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + (t_1 ^ 2.0)) + -1.0));
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[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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 := \frac{2 \cdot t}{1 + t}\\
\frac{1 + t_1 \cdot t_1}{2 + \left(\left(1 + {t_1}^{2}\right) + -1\right)}
\end{array}

Error

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

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.8
Cost2248
\[\begin{array}{l} t_1 := \frac{\frac{\left(t \cdot t\right) \cdot 4}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -15267164.822466873:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 3.1710368597451677 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 2
Error0.0
Cost2240
\[\begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t_1 \cdot t_1\\ \frac{1 + t_2}{2 + t_2} \end{array} \]
Alternative 3
Error1.3
Cost1480
\[\begin{array}{l} \mathbf{if}\;t \leq -9966084.853204826:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 3.1710368597451677 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + \frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{2 + \left(t \cdot t\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 4
Error1.2
Cost1224
\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot 4\\ \mathbf{if}\;t \leq -9966084.853204826:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 3.1710368597451677 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 5
Error1.3
Cost836
\[\begin{array}{l} \mathbf{if}\;t \leq -9966084.853204826:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 3.1710368597451677 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 6
Error1.3
Cost452
\[\begin{array}{l} \mathbf{if}\;t \leq -9966084.853204826:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 3.1710368597451677 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 7
Error1.4
Cost328
\[\begin{array}{l} \mathbf{if}\;t \leq -9966084.853204826:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 3.1710368597451677 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 8
Error26.5
Cost64
\[0.8333333333333334 \]

Error

Reproduce

herbie shell --seed 2022294 
(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))))))