Kahan p13 Example 2

Percentage Accurate: 99.9% → 99.9%

Time: 2.0s

Alternatives: 8

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 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]]]

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. Add Preprocessing

Alternative 2: 60.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{1}{t}\\ t_2 := \frac{\frac{2}{t}}{t\_1}\\ \mathbf{if}\;t\_2 \leq 0.1:\\ \;\;\;\;1 + \left(2 - \left(2 - t\_2\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{2 + \left(2 - \frac{t\_2}{t\_1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ 1.0 t))) (t_2 (/ (/ 2.0 t) t_1)))
   (if (<= t_2 0.1)
     (+ 1.0 (- 2.0 (* (- 2.0 t_2) t_1)))
     (/ t_2 (+ 2.0 (- 2.0 (/ t_2 t_1)))))))

C

double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = (2.0 / t) / t_1;
	double tmp;
	if (t_2 <= 0.1) {
		tmp = 1.0 + (2.0 - ((2.0 - t_2) * t_1));
	} else {
		tmp = t_2 / (2.0 + (2.0 - (t_2 / t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 / t)
    t_2 = (2.0d0 / t) / t_1
    if (t_2 <= 0.1d0) then
        tmp = 1.0d0 + (2.0d0 - ((2.0d0 - t_2) * t_1))
    else
        tmp = t_2 / (2.0d0 + (2.0d0 - (t_2 / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = (2.0 / t) / t_1;
	double tmp;
	if (t_2 <= 0.1) {
		tmp = 1.0 + (2.0 - ((2.0 - t_2) * t_1));
	} else {
		tmp = t_2 / (2.0 + (2.0 - (t_2 / t_1)));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 1.0 + (1.0 / t)
	t_2 = (2.0 / t) / t_1
	tmp = 0
	if t_2 <= 0.1:
		tmp = 1.0 + (2.0 - ((2.0 - t_2) * t_1))
	else:
		tmp = t_2 / (2.0 + (2.0 - (t_2 / t_1)))
	return tmp

Julia

function code(t)
	t_1 = Float64(1.0 + Float64(1.0 / t))
	t_2 = Float64(Float64(2.0 / t) / t_1)
	tmp = 0.0
	if (t_2 <= 0.1)
		tmp = Float64(1.0 + Float64(2.0 - Float64(Float64(2.0 - t_2) * t_1)));
	else
		tmp = Float64(t_2 / Float64(2.0 + Float64(2.0 - Float64(t_2 / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 1.0 + (1.0 / t);
	t_2 = (2.0 / t) / t_1;
	tmp = 0.0;
	if (t_2 <= 0.1)
		tmp = 1.0 + (2.0 - ((2.0 - t_2) * t_1));
	else
		tmp = t_2 / (2.0 + (2.0 - (t_2 / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.1], N[(1.0 + N[(2.0 - N[(N[(2.0 - t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(2.0 + N[(2.0 - N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 0.10000000000000001

   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

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto 1 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(12 \cdot t - 8\right)\right)} \]

   8. Applied rewrites 17.4%

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

   9. Taylor expanded in t around 0

      \[\leadsto 1 + \left(2 - \left(2 + -2 \cdot t\right)\right) \]

   11. Applied rewrites 21.2%

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

   if 0.10000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 18.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{\frac{2}{t}}{1 + \frac{1}{t}}}{2 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]

   8. Applied rewrites 18.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 96.9%

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

Alternative 3: 19.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(N[(2.0 - t$95$1), $MachinePrecision] * t$95$1), $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 0

   \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]

5. Applied rewrites 17.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 20.0%

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

Alternative 4: 19.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{1}{t}\\ t_2 := \frac{\frac{2}{t}}{t\_1}\\ \mathbf{if}\;2 - t\_2 \leq 0.002:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ 1.0 t))) (t_2 (/ (/ 2.0 t) t_1)))
   (if (<= (- 2.0 t_2) 0.002) t_2 t_1)))

C

double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = (2.0 / t) / t_1;
	double tmp;
	if ((2.0 - t_2) <= 0.002) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 / t)
    t_2 = (2.0d0 / t) / t_1
    if ((2.0d0 - t_2) <= 0.002d0) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = (2.0 / t) / t_1;
	double tmp;
	if ((2.0 - t_2) <= 0.002) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 1.0 + (1.0 / t)
	t_2 = (2.0 / t) / t_1
	tmp = 0
	if (2.0 - t_2) <= 0.002:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(1.0 + Float64(1.0 / t))
	t_2 = Float64(Float64(2.0 / t) / t_1)
	tmp = 0.0
	if (Float64(2.0 - t_2) <= 0.002)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 1.0 + (1.0 / t);
	t_2 = (2.0 / t) / t_1;
	tmp = 0.0;
	if ((2.0 - t_2) <= 0.002)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(2.0 - t$95$2), $MachinePrecision], 0.002], t$95$2, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 2e-3

   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

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}}} \]

   if 2e-3 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 21.2%

      \[\leadsto 1 + \color{blue}{\frac{1}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 19.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(2.0 - N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $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. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]

5. Applied rewrites 17.7%

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

6. Taylor expanded in t around 0

   \[\leadsto 1 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(12 \cdot t - 8\right)\right)} \]

8. Applied rewrites 18.1%

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

9. Taylor expanded in t around 0

   \[\leadsto 1 + \left(2 - \left(2 + -2 \cdot t\right)\right) \]

11. Applied rewrites 19.0%

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

Alternative 6: 18.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \end{array} \]

FPCore

(FPCore (t) :precision binary64 (+ 1.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))

C

double code(double t) {
	return 1.0 + ((2.0 / t) / (1.0 + (1.0 / t)));
}

Fortran

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

Java

public static double code(double t) {
	return 1.0 + ((2.0 / t) / (1.0 + (1.0 / t)));
}

Python

def code(t):
	return 1.0 + ((2.0 / t) / (1.0 + (1.0 / t)))

Julia

function code(t)
	return Float64(1.0 + Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + ((2.0 / t) / (1.0 + (1.0 / t)));
end

Wolfram

code[t_] := N[(1.0 + N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]

5. Applied rewrites 17.7%

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

6. Taylor expanded in t around 0

   \[\leadsto 1 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(12 \cdot t - 8\right)\right)} \]

8. Applied rewrites 18.1%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 19.0%

    \[\leadsto 1 + \frac{\frac{2}{t}}{1 + \frac{1}{\color{blue}{t}}} \]
12. Add Preprocessing

Alternative 7: 12.1% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{t} \end{array} \]

FPCore

(FPCore (t) :precision binary64 (+ 1.0 (/ 1.0 t)))

C

double code(double t) {
	return 1.0 + (1.0 / t);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + (1.0d0 / t)
end function

Java

public static double code(double t) {
	return 1.0 + (1.0 / t);
}

Python

def code(t):
	return 1.0 + (1.0 / t)

Julia

function code(t)
	return Float64(1.0 + Float64(1.0 / t))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (1.0 / t);
end

Wolfram

code[t_] := N[(1.0 + N[(1.0 / 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 0

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

5. Applied rewrites 10.9%

   \[\leadsto \color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}} \]

6. Taylor expanded in t around 0

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

8. Applied rewrites 12.4%

   \[\leadsto 1 + \color{blue}{\frac{1}{t}} \]
9. Add Preprocessing

Alternative 8: 3.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{t} \end{array} \]

FPCore

(FPCore (t) :precision binary64 (/ 2.0 t))

C

double code(double t) {
	return 2.0 / t;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 2.0d0 / t
end function

Java

public static double code(double t) {
	return 2.0 / t;
}

Python

def code(t):
	return 2.0 / t

Julia

function code(t)
	return Float64(2.0 / t)
end

MATLAB

function tmp = code(t)
	tmp = 2.0 / t;
end

Wolfram

code[t_] := N[(2.0 / t), $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 0

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

5. Applied rewrites 10.4%

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

6. Taylor expanded in t around 0

   \[\leadsto {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)} \]

8. Applied rewrites 3.6%

   \[\leadsto \frac{2}{\color{blue}{t}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  :pre (TRUE)
  (/ (+ 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))))))))