Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 10

Speedup: 1.7×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 100.0%

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

   2. associate-/l/ 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. rgt-mult-inverse 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.425 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;1 + \frac{-1}{6 + \frac{8 + \frac{-4}{t}}{-1 - t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.425) (not (<= t 0.75)))
   (+ 1.0 (/ -1.0 (+ 6.0 (/ (+ 8.0 (/ -4.0 t)) (- -1.0 t)))))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.425) || !(t <= 0.75)) {
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / t)) / (-1.0 - t))));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.425d0)) .or. (.not. (t <= 0.75d0))) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((8.0d0 + ((-4.0d0) / t)) / ((-1.0d0) - t))))
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.425) || !(t <= 0.75)) {
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / t)) / (-1.0 - t))));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.425) or not (t <= 0.75):
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / t)) / (-1.0 - t))))
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.425) || !(t <= 0.75))
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(8.0 + Float64(-4.0 / t)) / Float64(-1.0 - t)))));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.425) || ~((t <= 0.75)))
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / t)) / (-1.0 - t))));
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.425], N[Not[LessEqual[t, 0.75]], $MachinePrecision]], 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], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.424999999999999989 or 0.75 < t

   1. Initial program 100.0%

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

   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. associate-*l/ 100.0%

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

      2. frac-2neg 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{-\left(-8 + 2 \cdot \frac{2}{1 + t}\right)}{-1 + \left(-t\right)}}} \]
   6. Step-by-step derivation

      1. distribute-neg-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

   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 98.5%

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

   if -0.424999999999999989 < t < 0.75

   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. Taylor expanded in t around 0 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.425 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;1 + \frac{-1}{6 + \frac{8 + \frac{-4}{t}}{-1 - t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ 1 + \frac{-1}{6 + t_1 \cdot \left(t_1 + -4\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))))
   (+ 1.0 (/ -1.0 (+ 6.0 (* t_1 (+ t_1 -4.0)))))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 / (1.0d0 + t)
    code = 1.0d0 + ((-1.0d0) / (6.0d0 + (t_1 * (t_1 + (-4.0d0)))))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))))

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	return Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(t_1 * Float64(t_1 + -4.0)))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	tmp = 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(6.0 + N[(t$95$1 * N[(t$95$1 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double t) {
	return 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / (1.0 + t))) / (-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) / (1.0d0 + t))) / ((-1.0d0) - t))))
end function

Java

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

Python

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

MATLAB

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

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(8.0 + N[(-4.0 / N[(1.0 + t), $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. associate-*l/ 100.0%

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

   2. frac-2neg 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. distribute-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{-\left(-8 + 2 \cdot \frac{2}{1 + t}\right)}{-1 + \left(-t\right)}}} \]
6. Step-by-step derivation

   1. distribute-neg-in 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-*r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. +-commutative 100.0%

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

   8. unsub-neg 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 5: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \left(1 + \left(-1 + \frac{-0.2222222222222222}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (+ 1.0 (+ (/ -0.2222222222222222 t) -0.16666666666666666))
   (if (<= t 0.68)
     0.5
     (+
      1.0
      (+ -0.16666666666666666 (+ 1.0 (+ -1.0 (/ -0.2222222222222222 t))))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 1.0d0 + (((-0.2222222222222222d0) / t) + (-0.16666666666666666d0))
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) + (1.0d0 + ((-1.0d0) + ((-0.2222222222222222d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(1.0 + Float64(Float64(-0.2222222222222222 / t) + -0.16666666666666666));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(1.0 + Float64(-1.0 + Float64(-0.2222222222222222 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(1.0 + N[(N[(-0.2222222222222222 / t), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(1.0 + N[(-0.16666666666666666 + N[(1.0 + N[(-1.0 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   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. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto 1 + \left(-\color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + 0.16666666666666666\right)}\right) \]

      2. distribute-neg-in 99.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(-0.16666666666666666\right)\right)} \]

      3. associate-*r/ 99.1%

         \[\leadsto 1 + \left(\left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) + \left(-0.16666666666666666\right)\right) \]

      4. metadata-eval 99.1%

         \[\leadsto 1 + \left(\left(-\frac{\color{blue}{0.2222222222222222}}{t}\right) + \left(-0.16666666666666666\right)\right) \]

      5. distribute-neg-frac 99.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-0.2222222222222222}{t}} + \left(-0.16666666666666666\right)\right) \]

      6. metadata-eval 99.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{-0.2222222222222222}}{t} + \left(-0.16666666666666666\right)\right) \]

      7. metadata-eval 99.1%

         \[\leadsto 1 + \left(\frac{-0.2222222222222222}{t} + \color{blue}{-0.16666666666666666}\right) \]

   6. Simplified 99.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} \]

   if -0.48999999999999999 < t < 0.680000000000000049

   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. Taylor expanded in t around 0 99.1%

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

   if 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 96.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 96.1%

         \[\leadsto 1 + \left(-\color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + 0.16666666666666666\right)}\right) \]

      2. distribute-neg-in 96.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(-0.16666666666666666\right)\right)} \]

      3. associate-*r/ 96.1%

         \[\leadsto 1 + \left(\left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) + \left(-0.16666666666666666\right)\right) \]

      4. metadata-eval 96.1%

         \[\leadsto 1 + \left(\left(-\frac{\color{blue}{0.2222222222222222}}{t}\right) + \left(-0.16666666666666666\right)\right) \]

      5. distribute-neg-frac 96.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-0.2222222222222222}{t}} + \left(-0.16666666666666666\right)\right) \]

      6. metadata-eval 96.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{-0.2222222222222222}}{t} + \left(-0.16666666666666666\right)\right) \]

      7. metadata-eval 96.1%

         \[\leadsto 1 + \left(\frac{-0.2222222222222222}{t} + \color{blue}{-0.16666666666666666}\right) \]

   6. Simplified 96.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 96.1%

         \[\leadsto 1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)\right)} + -0.16666666666666666\right) \]

      2. expm1-udef 96.1%

         \[\leadsto 1 + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)} - 1\right)} + -0.16666666666666666\right) \]

      3. log1p-udef 96.1%

         \[\leadsto 1 + \left(\left(e^{\color{blue}{\log \left(1 + \frac{-0.2222222222222222}{t}\right)}} - 1\right) + -0.16666666666666666\right) \]

      4. add-exp-log 96.1%

         \[\leadsto 1 + \left(\left(\color{blue}{\left(1 + \frac{-0.2222222222222222}{t}\right)} - 1\right) + -0.16666666666666666\right) \]

   8. Applied egg-rr 96.1%

      \[\leadsto 1 + \left(\color{blue}{\left(\left(1 + \frac{-0.2222222222222222}{t}\right) - 1\right)} + -0.16666666666666666\right) \]
   9. Step-by-step derivation

      1. associate--l+ 96.1%

         \[\leadsto 1 + \left(\color{blue}{\left(1 + \left(\frac{-0.2222222222222222}{t} - 1\right)\right)} + -0.16666666666666666\right) \]

   10. Simplified 96.1%

       \[\leadsto 1 + \left(\color{blue}{\left(1 + \left(\frac{-0.2222222222222222}{t} - 1\right)\right)} + -0.16666666666666666\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \left(1 + \left(-1 + \frac{-0.2222222222222222}{t}\right)\right)\right)\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(\left(-1 + \left(1 + \frac{-0.2222222222222222}{t}\right)\right) + -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \left(1 + \left(-1 + \frac{-0.2222222222222222}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (+ 1.0 (+ (+ -1.0 (+ 1.0 (/ -0.2222222222222222 t))) -0.16666666666666666))
   (if (<= t 0.68)
     0.5
     (+
      1.0
      (+ -0.16666666666666666 (+ 1.0 (+ -1.0 (/ -0.2222222222222222 t))))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + ((-1.0 + (1.0 + (-0.2222222222222222 / t))) + -0.16666666666666666);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 1.0d0 + (((-1.0d0) + (1.0d0 + ((-0.2222222222222222d0) / t))) + (-0.16666666666666666d0))
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) + (1.0d0 + ((-1.0d0) + ((-0.2222222222222222d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + ((-1.0 + (1.0 + (-0.2222222222222222 / t))) + -0.16666666666666666);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 + ((-1.0 + (1.0 + (-0.2222222222222222 / t))) + -0.16666666666666666)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(1.0 + Float64(-0.2222222222222222 / t))) + -0.16666666666666666));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(1.0 + Float64(-1.0 + Float64(-0.2222222222222222 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 1.0 + ((-1.0 + (1.0 + (-0.2222222222222222 / t))) + -0.16666666666666666);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 1.0 + (-0.16666666666666666 + (1.0 + (-1.0 + (-0.2222222222222222 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(1.0 + N[(N[(-1.0 + N[(1.0 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(1.0 + N[(-0.16666666666666666 + N[(1.0 + N[(-1.0 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   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. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto 1 + \left(-\color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + 0.16666666666666666\right)}\right) \]

      2. distribute-neg-in 99.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(-0.16666666666666666\right)\right)} \]

      3. associate-*r/ 99.1%

         \[\leadsto 1 + \left(\left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) + \left(-0.16666666666666666\right)\right) \]

      4. metadata-eval 99.1%

         \[\leadsto 1 + \left(\left(-\frac{\color{blue}{0.2222222222222222}}{t}\right) + \left(-0.16666666666666666\right)\right) \]

      5. distribute-neg-frac 99.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-0.2222222222222222}{t}} + \left(-0.16666666666666666\right)\right) \]

      6. metadata-eval 99.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{-0.2222222222222222}}{t} + \left(-0.16666666666666666\right)\right) \]

      7. metadata-eval 99.1%

         \[\leadsto 1 + \left(\frac{-0.2222222222222222}{t} + \color{blue}{-0.16666666666666666}\right) \]

   6. Simplified 99.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 99.1%

         \[\leadsto 1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)\right)} + -0.16666666666666666\right) \]

      2. expm1-udef 99.2%

         \[\leadsto 1 + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)} - 1\right)} + -0.16666666666666666\right) \]

      3. log1p-udef 99.2%

         \[\leadsto 1 + \left(\left(e^{\color{blue}{\log \left(1 + \frac{-0.2222222222222222}{t}\right)}} - 1\right) + -0.16666666666666666\right) \]

      4. add-exp-log 99.2%

         \[\leadsto 1 + \left(\left(\color{blue}{\left(1 + \frac{-0.2222222222222222}{t}\right)} - 1\right) + -0.16666666666666666\right) \]

   8. Applied egg-rr 99.2%

      \[\leadsto 1 + \left(\color{blue}{\left(\left(1 + \frac{-0.2222222222222222}{t}\right) - 1\right)} + -0.16666666666666666\right) \]

   if -0.48999999999999999 < t < 0.680000000000000049

   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. Taylor expanded in t around 0 99.1%

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

   if 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 96.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 96.1%

         \[\leadsto 1 + \left(-\color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + 0.16666666666666666\right)}\right) \]

      2. distribute-neg-in 96.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(-0.16666666666666666\right)\right)} \]

      3. associate-*r/ 96.1%

         \[\leadsto 1 + \left(\left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) + \left(-0.16666666666666666\right)\right) \]

      4. metadata-eval 96.1%

         \[\leadsto 1 + \left(\left(-\frac{\color{blue}{0.2222222222222222}}{t}\right) + \left(-0.16666666666666666\right)\right) \]

      5. distribute-neg-frac 96.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-0.2222222222222222}{t}} + \left(-0.16666666666666666\right)\right) \]

      6. metadata-eval 96.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{-0.2222222222222222}}{t} + \left(-0.16666666666666666\right)\right) \]

      7. metadata-eval 96.1%

         \[\leadsto 1 + \left(\frac{-0.2222222222222222}{t} + \color{blue}{-0.16666666666666666}\right) \]

   6. Simplified 96.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 96.1%

         \[\leadsto 1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)\right)} + -0.16666666666666666\right) \]

      2. expm1-udef 96.1%

         \[\leadsto 1 + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)} - 1\right)} + -0.16666666666666666\right) \]

      3. log1p-udef 96.1%

         \[\leadsto 1 + \left(\left(e^{\color{blue}{\log \left(1 + \frac{-0.2222222222222222}{t}\right)}} - 1\right) + -0.16666666666666666\right) \]

      4. add-exp-log 96.1%

         \[\leadsto 1 + \left(\left(\color{blue}{\left(1 + \frac{-0.2222222222222222}{t}\right)} - 1\right) + -0.16666666666666666\right) \]

   8. Applied egg-rr 96.1%

      \[\leadsto 1 + \left(\color{blue}{\left(\left(1 + \frac{-0.2222222222222222}{t}\right) - 1\right)} + -0.16666666666666666\right) \]
   9. Step-by-step derivation

      1. associate--l+ 96.1%

         \[\leadsto 1 + \left(\color{blue}{\left(1 + \left(\frac{-0.2222222222222222}{t} - 1\right)\right)} + -0.16666666666666666\right) \]

   10. Simplified 96.1%

       \[\leadsto 1 + \left(\color{blue}{\left(1 + \left(\frac{-0.2222222222222222}{t} - 1\right)\right)} + -0.16666666666666666\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(\left(-1 + \left(1 + \frac{-0.2222222222222222}{t}\right)\right) + -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \left(1 + \left(-1 + \frac{-0.2222222222222222}{t}\right)\right)\right)\\ \end{array} \]

Alternative 7: 99.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		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 <= 0.68d0))) 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 <= 0.68)) {
		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 <= 0.68):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.68))
		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 <= 0.68)))
		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, 0.68]], $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 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   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. Taylor expanded in t around 0 99.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 99.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (+ 1.0 (+ (/ -0.2222222222222222 t) -0.16666666666666666))
   (if (<= t 0.68) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} 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 <= (-0.49d0)) then
        tmp = 1.0d0 + (((-0.2222222222222222d0) / t) + (-0.16666666666666666d0))
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(1.0 + Float64(Float64(-0.2222222222222222 / t) + -0.16666666666666666));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 1.0 + ((-0.2222222222222222 / t) + -0.16666666666666666);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(1.0 + N[(N[(-0.2222222222222222 / t), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   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. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto 1 + \left(-\color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + 0.16666666666666666\right)}\right) \]

      2. distribute-neg-in 99.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(-0.16666666666666666\right)\right)} \]

      3. associate-*r/ 99.1%

         \[\leadsto 1 + \left(\left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) + \left(-0.16666666666666666\right)\right) \]

      4. metadata-eval 99.1%

         \[\leadsto 1 + \left(\left(-\frac{\color{blue}{0.2222222222222222}}{t}\right) + \left(-0.16666666666666666\right)\right) \]

      5. distribute-neg-frac 99.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-0.2222222222222222}{t}} + \left(-0.16666666666666666\right)\right) \]

      6. metadata-eval 99.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{-0.2222222222222222}}{t} + \left(-0.16666666666666666\right)\right) \]

      7. metadata-eval 99.1%

         \[\leadsto 1 + \left(\frac{-0.2222222222222222}{t} + \color{blue}{-0.16666666666666666}\right) \]

   6. Simplified 99.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} \]

   if -0.48999999999999999 < t < 0.680000000000000049

   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. Taylor expanded in t around 0 99.1%

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

   if 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 96.1%

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

      1. associate-*r/ 96.1%

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

      2. metadata-eval 96.1%

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

   6. Simplified 96.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 9: 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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 95.4%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.1%

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

4. Final simplification 97.3%

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

Alternative 10: 58.6% 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%

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

3. Simplified 100.0%

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

4. Taylor expanded in t around 0 60.0%

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

5. Final simplification 60.0%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023333 
(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)))))))))