Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 14.7s

Alternatives: 5

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{8 + \frac{-4}{t + 1}}{t + 1}\\ \frac{t\_1 + -5}{t\_1 + -6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ 8.0 (/ -4.0 (+ t 1.0))) (+ t 1.0))))
   (/ (+ t_1 -5.0) (+ t_1 -6.0))))

C

double code(double t) {
	double t_1 = (8.0 + (-4.0 / (t + 1.0))) / (t + 1.0);
	return (t_1 + -5.0) / (t_1 + -6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = (8.0d0 + ((-4.0d0) / (t + 1.0d0))) / (t + 1.0d0)
    code = (t_1 + (-5.0d0)) / (t_1 + (-6.0d0))
end function

Java

public static double code(double t) {
	double t_1 = (8.0 + (-4.0 / (t + 1.0))) / (t + 1.0);
	return (t_1 + -5.0) / (t_1 + -6.0);
}

Python

def code(t):
	t_1 = (8.0 + (-4.0 / (t + 1.0))) / (t + 1.0)
	return (t_1 + -5.0) / (t_1 + -6.0)

Julia

function code(t)
	t_1 = Float64(Float64(8.0 + Float64(-4.0 / Float64(t + 1.0))) / Float64(t + 1.0))
	return Float64(Float64(t_1 + -5.0) / Float64(t_1 + -6.0))
end

MATLAB

function tmp = code(t)
	t_1 = (8.0 + (-4.0 / (t + 1.0))) / (t + 1.0);
	tmp = (t_1 + -5.0) / (t_1 + -6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(8.0 + N[(-4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 + -5.0), $MachinePrecision] / N[(t$95$1 + -6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 99.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)} \]

   2. expm1-udef 99.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)} - 1} \]

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 99.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)\right)} \]

   2. expm1-log1p 100.0%

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

   3. *-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. times-frac 100.0%

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

   6. neg-mul-1 100.0%

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

   7. neg-mul-1 100.0%

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

8. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -6}} \]

9. Final simplification 100.0%

   \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -6} \]
10. Add Preprocessing

Alternative 2: 57.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{5 + \frac{8 + \frac{-4}{t + 1}}{-1 - t}}{6 + \frac{2}{t \cdot -0.25 - 0.375}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+ 5.0 (/ (+ 8.0 (/ -4.0 (+ t 1.0))) (- -1.0 t)))
  (+ 6.0 (/ 2.0 (- (* t -0.25) 0.375)))))

C

double code(double t) {
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (2.0 / ((t * -0.25) - 0.375)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (5.0d0 + ((8.0d0 + ((-4.0d0) / (t + 1.0d0))) / ((-1.0d0) - t))) / (6.0d0 + (2.0d0 / ((t * (-0.25d0)) - 0.375d0)))
end function

Java

public static double code(double t) {
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (2.0 / ((t * -0.25) - 0.375)));
}

Python

def code(t):
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (2.0 / ((t * -0.25) - 0.375)))

Julia

function code(t)
	return Float64(Float64(5.0 + Float64(Float64(8.0 + Float64(-4.0 / Float64(t + 1.0))) / Float64(-1.0 - t))) / Float64(6.0 + Float64(2.0 / Float64(Float64(t * -0.25) - 0.375))))
end

MATLAB

function tmp = code(t)
	tmp = (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (2.0 / ((t * -0.25) - 0.375)));
end

Wolfram

code[t_] := N[(N[(5.0 + N[(N[(8.0 + N[(-4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(2.0 / N[(N[(t * -0.25), $MachinePrecision] - 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. associate-/l* 100.0%

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

   3. +-commutative 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in t around inf 58.9%

   \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{\color{blue}{-0.25 \cdot t - 0.375}}} \]
8. Step-by-step derivation

   1. associate-*l/ 58.9%

      \[\leadsto \frac{5 + \color{blue}{\frac{2 \cdot \left(\frac{2}{1 + t} - 4\right)}{1 + t}}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   2. frac-2neg 58.9%

      \[\leadsto \frac{5 + \color{blue}{\frac{-2 \cdot \left(\frac{2}{1 + t} - 4\right)}{-\left(1 + t\right)}}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   3. sub-neg 58.9%

      \[\leadsto \frac{5 + \frac{-2 \cdot \color{blue}{\left(\frac{2}{1 + t} + \left(-4\right)\right)}}{-\left(1 + t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   4. distribute-lft-in 58.9%

      \[\leadsto \frac{5 + \frac{-\color{blue}{\left(2 \cdot \frac{2}{1 + t} + 2 \cdot \left(-4\right)\right)}}{-\left(1 + t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   5. +-commutative 58.9%

      \[\leadsto \frac{5 + \frac{-\left(2 \cdot \frac{2}{\color{blue}{t + 1}} + 2 \cdot \left(-4\right)\right)}{-\left(1 + t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   6. metadata-eval 58.9%

      \[\leadsto \frac{5 + \frac{-\left(2 \cdot \frac{2}{t + 1} + 2 \cdot \color{blue}{-4}\right)}{-\left(1 + t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   7. metadata-eval 58.9%

      \[\leadsto \frac{5 + \frac{-\left(2 \cdot \frac{2}{t + 1} + \color{blue}{-8}\right)}{-\left(1 + t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   8. distribute-neg-in 58.9%

      \[\leadsto \frac{5 + \frac{-\left(2 \cdot \frac{2}{t + 1} + -8\right)}{\color{blue}{\left(-1\right) + \left(-t\right)}}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

   9. metadata-eval 58.9%

      \[\leadsto \frac{5 + \frac{-\left(2 \cdot \frac{2}{t + 1} + -8\right)}{\color{blue}{-1} + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

9. Applied egg-rr 58.9%

   \[\leadsto \frac{5 + \color{blue}{\frac{-\left(2 \cdot \frac{2}{t + 1} + -8\right)}{-1 + \left(-t\right)}}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]
10. Step-by-step derivation

    1. +-commutative 58.9%

       \[\leadsto \frac{5 + \frac{-\color{blue}{\left(-8 + 2 \cdot \frac{2}{t + 1}\right)}}{-1 + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

    2. distribute-neg-in 58.9%

       \[\leadsto \frac{5 + \frac{\color{blue}{\left(--8\right) + \left(-2 \cdot \frac{2}{t + 1}\right)}}{-1 + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

    3. metadata-eval 58.9%

       \[\leadsto \frac{5 + \frac{\color{blue}{8} + \left(-2 \cdot \frac{2}{t + 1}\right)}{-1 + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

    4. associate-*r/ 58.9%

       \[\leadsto \frac{5 + \frac{8 + \left(-\color{blue}{\frac{2 \cdot 2}{t + 1}}\right)}{-1 + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

    5. metadata-eval 58.9%

       \[\leadsto \frac{5 + \frac{8 + \left(-\frac{\color{blue}{4}}{t + 1}\right)}{-1 + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

    6. distribute-neg-frac 58.9%

       \[\leadsto \frac{5 + \frac{8 + \color{blue}{\frac{-4}{t + 1}}}{-1 + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

    7. metadata-eval 58.9%

       \[\leadsto \frac{5 + \frac{8 + \frac{\color{blue}{-4}}{t + 1}}{-1 + \left(-t\right)}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

    8. unsub-neg 58.9%

       \[\leadsto \frac{5 + \frac{8 + \frac{-4}{t + 1}}{\color{blue}{-1 - t}}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

11. Simplified 58.9%

    \[\leadsto \frac{5 + \color{blue}{\frac{8 + \frac{-4}{t + 1}}{-1 - t}}}{6 + \frac{2}{-0.25 \cdot t - 0.375}} \]

12. Final simplification 58.9%

    \[\leadsto \frac{5 + \frac{8 + \frac{-4}{t + 1}}{-1 - t}}{6 + \frac{2}{t \cdot -0.25 - 0.375}} \]
13. Add Preprocessing

Alternative 3: 49.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 - \frac{0.2222222222222222}{t} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (- 0.8333333333333334 (/ 0.2222222222222222 t)))

C

double code(double t) {
	return 0.8333333333333334 - (0.2222222222222222 / t);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
end function

Java

public static double code(double t) {
	return 0.8333333333333334 - (0.2222222222222222 / t);
}

Python

def code(t):
	return 0.8333333333333334 - (0.2222222222222222 / t)

Julia

function code(t)
	return Float64(0.8333333333333334 - Float64(0.2222222222222222 / t))
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334 - (0.2222222222222222 / t);
end

Wolfram

code[t_] := N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around inf 51.4%

   \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
4. Step-by-step derivation

   1. associate-*r/ 51.4%

      \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

   2. metadata-eval 51.4%

      \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

6. Final simplification 51.4%

   \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222}{t} \]
7. Add Preprocessing

Alternative 4: 60.3% accurate, 51.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0 59.2%

   \[\leadsto \color{blue}{0.5} \]

4. Final simplification 59.2%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Alternative 5: 57.8% accurate, 51.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0
end function

Java

public static double code(double t) {
	return 0.8333333333333334;
}

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334;
end

Wolfram

code[t_] := 0.8333333333333334

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around inf 59.0%

   \[\leadsto \color{blue}{0.8333333333333334} \]

4. Final simplification 59.0%

   \[\leadsto 0.8333333333333334 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(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))))))))