Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 11.0s

Alternatives: 8

Speedup: 1.0×

Specification

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, 0.1× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 + \frac{-2}{t + 1}\right)}^{2}}\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (exp (log1p (/ -1.0 (+ 2.0 (pow (+ 2.0 (/ -2.0 (+ t 1.0))) 2.0))))))

C

double code(double t) {
	return exp(log1p((-1.0 / (2.0 + pow((2.0 + (-2.0 / (t + 1.0))), 2.0)))));
}

Java

public static double code(double t) {
	return Math.exp(Math.log1p((-1.0 / (2.0 + Math.pow((2.0 + (-2.0 / (t + 1.0))), 2.0)))));
}

Python

def code(t):
	return math.exp(math.log1p((-1.0 / (2.0 + math.pow((2.0 + (-2.0 / (t + 1.0))), 2.0)))))

Julia

function code(t)
	return exp(log1p(Float64(-1.0 / Float64(2.0 + (Float64(2.0 + Float64(-2.0 / Float64(t + 1.0))) ^ 2.0)))))
end

Wolfram

code[t_] := N[Exp[N[Log[1 + N[(-1.0 / N[(2.0 + N[Power[N[(2.0 + N[(-2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $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. Step-by-step derivation

   1. add-exp-log 100.0%

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

   2. sub-neg 100.0%

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

   3. log1p-def 100.0%

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

   4. distribute-neg-frac 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. rgt-mult-inverse 100.0%

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

   4. *-rgt-identity 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 + \frac{-2}{t + 1}\right)}^{2}}\right)} \]

Alternative 2: 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. Final simplification 100.0%

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

Alternative 3: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.58) (not (<= t 0.7)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ 1.0 (/ -1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.7)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.58d0)) .or. (.not. (t <= 0.7d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.7)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.58) or not (t <= 0.7):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.58) || !(t <= 0.7))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.58) || ~((t <= 0.7)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.58], N[Not[LessEqual[t, 0.7]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 0.69999999999999996 < 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. Taylor expanded in t around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   4. Simplified 99.2%

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

   if -0.57999999999999996 < t < 0.69999999999999996

   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. Taylor expanded in t around 0 98.6%

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

   3. Taylor expanded in t around 0 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{8 - \frac{4}{t}}{-1 - t}}\\ \mathbf{elif}\;t \leq 0.7:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.52)
   (+ 1.0 (/ -1.0 (+ 6.0 (/ (- 8.0 (/ 4.0 t)) (- -1.0 t)))))
   (if (<= t 0.7)
     (+ 1.0 (/ -1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))));
	} else if (t <= 0.7) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.52d0)) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((8.0d0 - (4.0d0 / t)) / ((-1.0d0) - t))))
    else if (t <= 0.7d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))));
	} else if (t <= 0.7) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.52:
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))))
	elif t <= 0.7:
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(8.0 - Float64(4.0 / t)) / Float64(-1.0 - t)))));
	elseif (t <= 0.7)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.52)
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))));
	elseif (t <= 0.7)
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.52], N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(8.0 - N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.7], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. frac-2neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. neg-sub0 100.0%

         \[\leadsto 1 - \frac{1}{6 + \frac{\left(\frac{2}{t + 1} + -4\right) \cdot -2}{\left(-1\right) + \color{blue}{\left(0 - t\right)}}} \]

      8. associate-+r- 100.0%

         \[\leadsto 1 - \frac{1}{6 + \frac{\left(\frac{2}{t + 1} + -4\right) \cdot -2}{\color{blue}{\left(\left(-1\right) + 0\right) - t}}} \]

      9. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{6 + \frac{\left(\frac{2}{t + 1} + -4\right) \cdot -2}{\left(\color{blue}{-1} + 0\right) - t}} \]

      10. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 98.8%

      \[\leadsto 1 - \frac{1}{6 + \frac{\color{blue}{8 - 4 \cdot \frac{1}{t}}}{-1 - t}} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   10. Simplified 98.8%

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

   if -0.52000000000000002 < t < 0.69999999999999996

   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. Taylor expanded in t around 0 98.6%

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

   3. Taylor expanded in t around 0 98.5%

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

   if 0.69999999999999996 < 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. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{8 - \frac{4}{t}}{-1 - t}}\\ \mathbf{elif}\;t \leq 0.7:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{6 + \frac{8 + \frac{-4}{t + 1}}{-1 - t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ 6.0 (/ (+ 8.0 (/ -4.0 (+ t 1.0))) (- -1.0 t))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. frac-2neg 100.0%

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

   3. associate-*r/ 100.0%

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

   4. +-commutative 100.0%

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

   5. metadata-eval 100.0%

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

   6. distribute-neg-in 100.0%

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

   7. neg-sub0 100.0%

      \[\leadsto 1 - \frac{1}{6 + \frac{\left(\frac{2}{t + 1} + -4\right) \cdot -2}{\left(-1\right) + \color{blue}{\left(0 - t\right)}}} \]

   8. associate-+r- 100.0%

      \[\leadsto 1 - \frac{1}{6 + \frac{\left(\frac{2}{t + 1} + -4\right) \cdot -2}{\color{blue}{\left(\left(-1\right) + 0\right) - t}}} \]

   9. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{6 + \frac{\left(\frac{2}{t + 1} + -4\right) \cdot -2}{\left(\color{blue}{-1} + 0\right) - t}} \]

   10. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-*r/ 100.0%

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

   6. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 6: 99.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.65)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(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 <= (-0.49d0)) .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 <= -0.49) || !(t <= 0.65)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) 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 <= -0.49) || !(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 <= -0.49) || ~((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, -0.49], 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 < -0.48999999999999999 or 0.650000000000000022 < 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. Taylor expanded in t around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   4. Simplified 99.2%

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

   if -0.48999999999999999 < t < 0.650000000000000022

   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. Taylor expanded in t around 0 98.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 98.5% accurate, 5.7× 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%

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

   2. Taylor expanded in t around inf 98.6%

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

   if -0.330000000000000016 < t < 1

   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. Taylor expanded in t around 0 98.1%

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

4. Final simplification 98.4%

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

Alternative 8: 59.4% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in t around 0 61.4%

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

3. Final simplification 61.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 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)))))))))