Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 14.8s

Alternatives: 6

Speedup: 1.5×

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, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{2}{-1 - t}\\ t_2 := \frac{4}{-1 - t}\\ \frac{1 + \left(4 + \left(t\_2 + \frac{4 + t\_2}{-1 - t}\right)\right)}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ 2.0 (- -1.0 t)))) (t_2 (/ 4.0 (- -1.0 t))))
   (/ (+ 1.0 (+ 4.0 (+ t_2 (/ (+ 4.0 t_2) (- -1.0 t))))) (+ 2.0 (* t_1 t_1)))))

C

double code(double t) {
	double t_1 = 2.0 + (2.0 / (-1.0 - t));
	double t_2 = 4.0 / (-1.0 - t);
	return (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (2.0 + (t_1 * t_1));
}

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 / ((-1.0d0) - t))
    t_2 = 4.0d0 / ((-1.0d0) - t)
    code = (1.0d0 + (4.0d0 + (t_2 + ((4.0d0 + t_2) / ((-1.0d0) - t))))) / (2.0d0 + (t_1 * t_1))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 + (2.0 / (-1.0 - t));
	double t_2 = 4.0 / (-1.0 - t);
	return (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (2.0 + (t_1 * t_1));
}

Python

def code(t):
	t_1 = 2.0 + (2.0 / (-1.0 - t))
	t_2 = 4.0 / (-1.0 - t)
	return (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (2.0 + (t_1 * t_1))

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(2.0 / Float64(-1.0 - t)))
	t_2 = Float64(4.0 / Float64(-1.0 - t))
	return Float64(Float64(1.0 + Float64(4.0 + Float64(t_2 + Float64(Float64(4.0 + t_2) / Float64(-1.0 - t))))) / Float64(2.0 + Float64(t_1 * t_1)))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 + (2.0 / (-1.0 - t));
	t_2 = 4.0 / (-1.0 - t);
	tmp = (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (2.0 + (t_1 * t_1));
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-*r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r/ 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-lft-in 100.0%

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

   10. metadata-eval 100.0%

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

   11. associate-*r/ 100.0%

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

   12. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-+l+ 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{1 + \left(4 + \left(\frac{4}{-1 - t} + \frac{4 + \frac{4}{-1 - t}}{-1 - t}\right)\right)}{2 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(2 + \frac{2}{-1 - t}\right)} \]
10. Add Preprocessing

Alternative 2: 100.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ 2.0 (- -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 / (-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 / ((-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 / (-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 / (-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(2.0 / 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 / (-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[(2.0 / N[(-1.0 - t), $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]]]

Derivation

1. Initial program 99.6%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.55 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(t \cdot 2\right)}{2 + \left(t \cdot 2\right) \cdot \left(t \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.55) (not (<= t 0.75)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (/
    (+ 1.0 (* (+ 2.0 (/ 2.0 (- -1.0 t))) (* t 2.0)))
    (+ 2.0 (* (* t 2.0) (* t 2.0))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.55) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (t * 2.0))) / (2.0 + ((t * 2.0) * (t * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.55d0)) .or. (.not. (t <= 0.75d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (1.0d0 + ((2.0d0 + (2.0d0 / ((-1.0d0) - t))) * (t * 2.0d0))) / (2.0d0 + ((t * 2.0d0) * (t * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.55) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (t * 2.0))) / (2.0 + ((t * 2.0) * (t * 2.0)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.55) or not (t <= 0.75):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (t * 2.0))) / (2.0 + ((t * 2.0) * (t * 2.0)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.55) || !(t <= 0.75))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(2.0 + Float64(2.0 / Float64(-1.0 - t))) * Float64(t * 2.0))) / Float64(2.0 + Float64(Float64(t * 2.0) * Float64(t * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.55) || ~((t <= 0.75)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (t * 2.0))) / (2.0 + ((t * 2.0) * (t * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.55], N[Not[LessEqual[t, 0.75]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(t * 2.0), $MachinePrecision] * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.55000000000000004 or 0.75 < t

   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. Step-by-step derivation

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

   if -0.55000000000000004 < t < 0.75

   1. Initial program 99.1%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 99.7%

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

   6. Taylor expanded in t around 0 99.7%

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

   7. Taylor expanded in t around 0 99.9%

      \[\leadsto \frac{1 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(2 \cdot t\right)}{2 + \color{blue}{\left(2 \cdot t\right)} \cdot \left(2 \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.55 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(t \cdot 2\right)}{2 + \left(t \cdot 2\right) \cdot \left(t \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.48) (not (<= t 0.65)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.65)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.48d0)) .or. (.not. (t <= 0.65d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.65)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.48) or not (t <= 0.65):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.48) || !(t <= 0.65))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.48) || ~((t <= 0.65)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.48], N[Not[LessEqual[t, 0.65]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.47999999999999998 or 0.650000000000000022 < t

   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. Step-by-step derivation

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

   if -0.47999999999999998 < t < 0.650000000000000022

   1. Initial program 99.1%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 99.1%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   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. Step-by-step derivation

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 97.2%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 99.1%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 99.1%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% 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 99.6%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around 0 55.1%

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

6. Final simplification 55.1%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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))))))))