Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 2.1s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

Alternative 2: 60.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = (2.0 / t) / (1.0 + (1.0 / t));
	double t_2 = 2.0 - t_1;
	double tmp;
	if (t_1 <= 0.1) {
		tmp = 1.0 / (2.0 + (1.0 - t_2));
	} else {
		tmp = 1.0 / (2.0 + (t_2 * t_2));
	}
	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 = (2.0d0 / t) / (1.0d0 + (1.0d0 / t))
    t_2 = 2.0d0 - t_1
    if (t_1 <= 0.1d0) then
        tmp = 1.0d0 / (2.0d0 + (1.0d0 - t_2))
    else
        tmp = 1.0d0 / (2.0d0 + (t_2 * t_2))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{1}{2 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]

   8. Applied rewrites 21.2%

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

   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%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.2%

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

Alternative 3: 59.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (- 1.0 t_1)))
   (if (<= t_1 0.002) (/ 1.0 (+ 2.0 (- 1.0 t_2))) (/ 1.0 (+ 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 = 1.0 - t_1;
	double tmp;
	if (t_1 <= 0.002) {
		tmp = 1.0 / (2.0 + (1.0 - t_2));
	} else {
		tmp = 1.0 / (2.0 + t_2);
	}
	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 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = 1.0d0 - t_1
    if (t_1 <= 0.002d0) then
        tmp = 1.0d0 / (2.0d0 + (1.0d0 - t_2))
    else
        tmp = 1.0d0 / (2.0d0 + t_2)
    end if
    code = tmp
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 = 1.0 - t_1;
	double tmp;
	if (t_1 <= 0.002) {
		tmp = 1.0 / (2.0 + (1.0 - t_2));
	} else {
		tmp = 1.0 / (2.0 + t_2);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = 1.0 - t_1;
	tmp = 0.0;
	if (t_1 <= 0.002)
		tmp = 1.0 / (2.0 + (1.0 - t_2));
	else
		tmp = 1.0 / (2.0 + t_2);
	end
	tmp_2 = tmp;
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[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], N[(1.0 / N[(2.0 + N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

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%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{1}{2 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 96.7%

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

   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%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{1}{2 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]

   8. Applied rewrites 21.2%

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

Alternative 4: 19.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = 2.0 - ((2.0 / t) / t_1);
	double tmp;
	if (t_2 <= 0.002) {
		tmp = 1.0 - t_2;
	} else {
		tmp = 2.0 - 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 - ((2.0d0 / t) / t_1)
    if (t_2 <= 0.002d0) then
        tmp = 1.0d0 - t_2
    else
        tmp = 2.0d0 - 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 - ((2.0 / t) / t_1);
	double tmp;
	if (t_2 <= 0.002) {
		tmp = 1.0 - t_2;
	} else {
		tmp = 2.0 - t_1;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(t)
	t_1 = 1.0 + (1.0 / t);
	t_2 = 2.0 - ((2.0 / t) / t_1);
	tmp = 0.0;
	if (t_2 <= 0.002)
		tmp = 1.0 - t_2;
	else
		tmp = 2.0 - 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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.002], N[(1.0 - t$95$2), $MachinePrecision], N[(2.0 - t$95$1), $MachinePrecision]]]]

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%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 18.7%

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

   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%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. 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 16.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 18.1%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 21.2%

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

Alternative 5: 19.1% accurate, 1.2× speedup

Math

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

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 (- 2.0 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 = 2.0 - 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 = 2.0d0 - 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 = 2.0 - 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 = 2.0 - 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 = Float64(2.0 - 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 = 2.0 - 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, N[(2.0 - t$95$1), $MachinePrecision]]]]

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%

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

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. 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 16.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 18.1%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 21.2%

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

Alternative 6: 20.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + \left(1 - \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 56.9%

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

6. Taylor expanded in t around 0

   \[\leadsto \frac{1}{2 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]

8. Applied rewrites 20.5%

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

Alternative 7: 18.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ 2 + \left(1 - \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. 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 10.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 10.9%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 19.0%

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

Alternative 8: 12.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - \left(1 + \frac{1}{t}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 56.9%

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

6. Taylor expanded in t around 0

   \[\leadsto \frac{1}{2 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]

8. Applied rewrites 20.5%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 56.6%

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

12. Taylor expanded in t around 0

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

14. Applied rewrites 12.8%

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

Alternative 9: 12.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 56.9%

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

6. Taylor expanded in t around 0

   \[\leadsto \frac{1}{2 + {t}^{2} \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]

8. Applied rewrites 20.5%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 56.6%

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

12. Taylor expanded in t around 0

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

14. Applied rewrites 12.7%

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

Alternative 10: 12.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ 2 - \left(1 + \frac{1}{t}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. 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 10.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 10.9%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 12.5%

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

Alternative 11: 12.1% accurate, 6.7× 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%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. 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 10.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 10.4%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 12.4%

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

Alternative 12: 3.5% accurate, 8.4× 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%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. 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 10.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 10.9%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 3.6%

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

Alternative 13: 3.5% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t):
	return 1.0 / t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. 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 10.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 10.4%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 3.6%

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

Reproduce

herbie shell --seed 2024321 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  :pre (TRUE)
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))