Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 8

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: 100.0% 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

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 - (2.0 / (t + 1.0));
	double t_2 = 2.0 + (-2.0 / (t + 1.0));
	return (1.0 + (t_1 * t_1)) / (2.0 + (t_2 * 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))
    t_2 = 2.0d0 + ((-2.0d0) / (t + 1.0d0))
    code = (1.0d0 + (t_1 * t_1)) / (2.0d0 + (t_2 * t_2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\left(2 - \frac{2}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right) \cdot \left(2 - \frac{2}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right) + 1}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. rgt-mult-inverse N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 100.0

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

   6. lift-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. rgt-mult-inverse N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 100.0

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

8. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. rgt-mult-inverse N/A

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

   7. lift-+.f64 100.0

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lift-/.f64 N/A

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

   13. rgt-mult-inverse N/A

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

   14. lift-+.f64 100.0

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.2:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.2)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))
   (fma t (fma -2.0 (* t t) t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.2) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = fma(t, fma(-2.0, (t * t), t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.2)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	else
		tmp = fma(t, fma(-2.0, Float64(t * t), t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.2], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(t * N[(-2.0 * N[(t * t), $MachinePrecision] + t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.20000000000000001

   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

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{t}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{5}{6} + \frac{\color{blue}{\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}}{t} \]

      5. +-commutative N/A

         \[\leadsto \frac{5}{6} + \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \frac{2}{9}\right)}\right)}{t} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{5}{6} + \frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right) + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}}{t} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{5}{6} + \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{5}{6} + \frac{\color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{5}{6} + \frac{\color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}}{t} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{5}{6} + \frac{\color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{5}{6} + \frac{\frac{\color{blue}{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{5}{6} + \frac{\frac{\frac{1}{27} + \color{blue}{\frac{\frac{4}{81} \cdot 1}{t}}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t} \]

      13. metadata-eval N/A

          \[\leadsto \frac{5}{6} + \frac{\frac{\frac{1}{27} + \frac{\color{blue}{\frac{4}{81}}}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{5}{6} + \frac{\frac{\frac{1}{27} + \color{blue}{\frac{\frac{4}{81}}{t}}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t} \]

      15. metadata-eval 99.6

          \[\leadsto 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + \color{blue}{-0.2222222222222222}}{t} \]

   8. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}} \]

   if 0.20000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.3

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(-2, \color{blue}{t \cdot t}, t\right), 0.5\right) \]

   8. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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))))))))