Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 20.1s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-2}{t + 1}\\ \frac{\mathsf{fma}\left(t_1, t_1 + 4, 5\right)}{6 + t_1 \cdot \left(4 - \frac{2}{t + 1}\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ -2.0 (+ t 1.0))))
   (/ (fma t_1 (+ t_1 4.0) 5.0) (+ 6.0 (* t_1 (- 4.0 (/ 2.0 (+ t 1.0))))))))

C

double code(double t) {
	double t_1 = -2.0 / (t + 1.0);
	return fma(t_1, (t_1 + 4.0), 5.0) / (6.0 + (t_1 * (4.0 - (2.0 / (t + 1.0)))));
}

Julia

function code(t)
	t_1 = Float64(-2.0 / Float64(t + 1.0))
	return Float64(fma(t_1, Float64(t_1 + 4.0), 5.0) / Float64(6.0 + Float64(t_1 * Float64(4.0 - Float64(2.0 / Float64(t + 1.0))))))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(-2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(t$95$1 + 4.0), $MachinePrecision] + 5.0), $MachinePrecision] / N[(6.0 + N[(t$95$1 * N[(4.0 - N[(2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

5. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate--l+ 100.0%

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

   4. associate-/r/ 100.0%

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

   5. metadata-eval 100.0%

      \[\leadsto \frac{\frac{-2}{1 + t} \cdot \left(\frac{-2}{1 + t} + 4\right) + \color{blue}{5}}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)} \]

   6. fma-udef 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-2}{1 + t}, \frac{-2}{1 + t} + 4, 5\right)}}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)} \]

   7. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{\color{blue}{t + 1}}, \frac{-2}{1 + t} + 4, 5\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)} \]

   8. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{t + 1}, \color{blue}{4 + \frac{-2}{1 + t}}, 5\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)} \]

   9. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{t + 1}, 4 + \frac{-2}{\color{blue}{t + 1}}, 5\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)} \]

7. Simplified 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-2}{t + 1}, 4 + \frac{-2}{t + 1}, 5\right)}}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)} \]

8. Final simplification 100.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{t + 1}, \frac{-2}{t + 1} + 4, 5\right)}{6 + \frac{-2}{t + 1} \cdot \left(4 - \frac{2}{t + 1}\right)} \]

Alternative 2: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25:\\ \;\;\;\;\frac{5 + \frac{8 - \frac{4}{t + 1}}{-1 - t}}{6 + \frac{8 - \frac{4}{t}}{-1 - t}}\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;\frac{5 + \left(t + 1\right) \cdot \frac{4}{-1 - t}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -1.25)
   (/
    (+ 5.0 (/ (- 8.0 (/ 4.0 (+ t 1.0))) (- -1.0 t)))
    (+ 6.0 (/ (- 8.0 (/ 4.0 t)) (- -1.0 t))))
   (if (<= t 0.65)
     (/ (+ 5.0 (* (+ t 1.0) (/ 4.0 (- -1.0 t)))) 2.0)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -1.25) {
		tmp = (5.0 + ((8.0 - (4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t)));
	} else if (t <= 0.65) {
		tmp = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0;
	} 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 <= (-1.25d0)) then
        tmp = (5.0d0 + ((8.0d0 - (4.0d0 / (t + 1.0d0))) / ((-1.0d0) - t))) / (6.0d0 + ((8.0d0 - (4.0d0 / t)) / ((-1.0d0) - t)))
    else if (t <= 0.65d0) then
        tmp = (5.0d0 + ((t + 1.0d0) * (4.0d0 / ((-1.0d0) - t)))) / 2.0d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -1.25) {
		tmp = (5.0 + ((8.0 - (4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t)));
	} else if (t <= 0.65) {
		tmp = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -1.25:
		tmp = (5.0 + ((8.0 - (4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t)))
	elif t <= 0.65:
		tmp = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -1.25)
		tmp = Float64(Float64(5.0 + Float64(Float64(8.0 - Float64(4.0 / Float64(t + 1.0))) / Float64(-1.0 - t))) / Float64(6.0 + Float64(Float64(8.0 - Float64(4.0 / t)) / Float64(-1.0 - t))));
	elseif (t <= 0.65)
		tmp = Float64(Float64(5.0 + Float64(Float64(t + 1.0) * Float64(4.0 / Float64(-1.0 - t)))) / 2.0);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -1.25)
		tmp = (5.0 + ((8.0 - (4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t)));
	elseif (t <= 0.65)
		tmp = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -1.25], N[(N[(5.0 + N[(N[(8.0 - N[(4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(N[(8.0 - N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.65], N[(N[(5.0 + N[(N[(t + 1.0), $MachinePrecision] * N[(4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{-2 \cdot \left(4 - \frac{2}{1 + t}\right)}{1 + t}}} \]

      2. frac-2neg 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{--2 \cdot \left(4 - \frac{2}{1 + t}\right)}{-\left(1 + t\right)}}} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{--2 \cdot \color{blue}{\left(4 + \left(-\frac{2}{1 + t}\right)\right)}}{-\left(1 + t\right)}} \]

      4. distribute-rgt-in 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(4 \cdot -2 + \left(-\frac{2}{1 + t}\right) \cdot -2\right)}}{-\left(1 + t\right)}} \]

      5. distribute-neg-frac 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \color{blue}{\frac{-2}{1 + t}} \cdot -2\right)}{-\left(1 + t\right)}} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \frac{\color{blue}{-2}}{1 + t} \cdot -2\right)}{-\left(1 + t\right)}} \]

      7. +-commutative 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(\frac{-2}{1 + t} \cdot -2 + 4 \cdot -2\right)}}{-\left(1 + t\right)}} \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + \color{blue}{-8}\right)}{-\left(1 + t\right)}} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{0 - \left(1 + t\right)}}} \]

      10. associate--r+ 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{\left(0 - 1\right) - t}}} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{-1} - t}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{-1 - t}}} \]
   6. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{0 - \left(\frac{-2}{1 + t} \cdot -2 + -8\right)}}{-1 - t}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{0 - \color{blue}{\left(-8 + \frac{-2}{1 + t} \cdot -2\right)}}{-1 - t}} \]

      3. associate--r+ 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{\left(0 - -8\right) - \frac{-2}{1 + t} \cdot -2}}{-1 - t}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{8} - \frac{-2}{1 + t} \cdot -2}{-1 - t}} \]

      5. associate-*l/ 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \color{blue}{\frac{-2 \cdot -2}{1 + t}}}{-1 - t}} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{\color{blue}{4}}{1 + t}}{-1 - t}} \]

      7. +-commutative 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{4}{\color{blue}{t + 1}}}{-1 - t}} \]

   7. Simplified 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{8 - \frac{4}{t + 1}}{-1 - t}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{-2 \cdot \left(4 - \frac{2}{1 + t}\right)}{1 + t}}} \]

      2. frac-2neg 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{--2 \cdot \left(4 - \frac{2}{1 + t}\right)}{-\left(1 + t\right)}}} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{--2 \cdot \color{blue}{\left(4 + \left(-\frac{2}{1 + t}\right)\right)}}{-\left(1 + t\right)}} \]

      4. distribute-rgt-in 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(4 \cdot -2 + \left(-\frac{2}{1 + t}\right) \cdot -2\right)}}{-\left(1 + t\right)}} \]

      5. distribute-neg-frac 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \color{blue}{\frac{-2}{1 + t}} \cdot -2\right)}{-\left(1 + t\right)}} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \frac{\color{blue}{-2}}{1 + t} \cdot -2\right)}{-\left(1 + t\right)}} \]

      7. +-commutative 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(\frac{-2}{1 + t} \cdot -2 + 4 \cdot -2\right)}}{-\left(1 + t\right)}} \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + \color{blue}{-8}\right)}{-\left(1 + t\right)}} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{0 - \left(1 + t\right)}}} \]

      10. associate--r+ 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{\left(0 - 1\right) - t}}} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{-1} - t}} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{5 + \color{blue}{\frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{-1 - t}}}{6 + \frac{8 - \frac{4}{t + 1}}{-1 - t}} \]
   10. Step-by-step derivation

       1. neg-sub0 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{0 - \left(\frac{-2}{1 + t} \cdot -2 + -8\right)}}{-1 - t}} \]

       2. +-commutative 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{0 - \color{blue}{\left(-8 + \frac{-2}{1 + t} \cdot -2\right)}}{-1 - t}} \]

       3. associate--r+ 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{\left(0 - -8\right) - \frac{-2}{1 + t} \cdot -2}}{-1 - t}} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{8} - \frac{-2}{1 + t} \cdot -2}{-1 - t}} \]

       5. associate-*l/ 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \color{blue}{\frac{-2 \cdot -2}{1 + t}}}{-1 - t}} \]

       6. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{\color{blue}{4}}{1 + t}}{-1 - t}} \]

       7. +-commutative 100.0%

          \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{4}{\color{blue}{t + 1}}}{-1 - t}} \]

   11. Simplified 100.0%

       \[\leadsto \frac{5 + \color{blue}{\frac{8 - \frac{4}{t + 1}}{-1 - t}}}{6 + \frac{8 - \frac{4}{t + 1}}{-1 - t}} \]

   12. Taylor expanded in t around inf 98.7%

       \[\leadsto \frac{5 + \frac{8 - \frac{4}{t + 1}}{-1 - t}}{6 + \frac{\color{blue}{8 - 4 \cdot \frac{1}{t}}}{-1 - t}} \]
   13. Step-by-step derivation

       1. associate-*r/ 98.7%

          \[\leadsto \frac{5 + \frac{8 - \frac{4}{t + 1}}{-1 - t}}{6 + \frac{8 - \color{blue}{\frac{4 \cdot 1}{t}}}{-1 - t}} \]

       2. metadata-eval 98.7%

          \[\leadsto \frac{5 + \frac{8 - \frac{4}{t + 1}}{-1 - t}}{6 + \frac{8 - \frac{\color{blue}{4}}{t}}{-1 - t}} \]

   14. Simplified 98.7%

       \[\leadsto \frac{5 + \frac{8 - \frac{4}{t + 1}}{-1 - t}}{6 + \frac{\color{blue}{8 - \frac{4}{t}}}{-1 - t}} \]

   if -1.25 < t < 0.650000000000000022

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. Taylor expanded in t around 0 98.1%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{-4}} \]

   5. Taylor expanded in t around 0 98.1%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \color{blue}{\left(2 + 2 \cdot t\right)}}{6 + -4} \]
   6. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(2 + \color{blue}{t \cdot 2}\right)}{6 + -4} \]

   7. Simplified 98.1%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \color{blue}{\left(2 + t \cdot 2\right)}}{6 + -4} \]
   8. Step-by-step derivation

      1. distribute-rgt-in 98.1%

         \[\leadsto \frac{5 + \color{blue}{\left(2 \cdot \frac{-2}{1 + t} + \left(t \cdot 2\right) \cdot \frac{-2}{1 + t}\right)}}{6 + -4} \]

      2. +-commutative 98.1%

         \[\leadsto \frac{5 + \color{blue}{\left(\left(t \cdot 2\right) \cdot \frac{-2}{1 + t} + 2 \cdot \frac{-2}{1 + t}\right)}}{6 + -4} \]

      3. associate-*l* 98.1%

         \[\leadsto \frac{5 + \left(\color{blue}{t \cdot \left(2 \cdot \frac{-2}{1 + t}\right)} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      4. frac-2neg 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \left(2 \cdot \color{blue}{\frac{--2}{-\left(1 + t\right)}}\right) + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      5. metadata-eval 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \left(2 \cdot \frac{\color{blue}{2}}{-\left(1 + t\right)}\right) + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      6. associate-*r/ 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \color{blue}{\frac{2 \cdot 2}{-\left(1 + t\right)}} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      7. metadata-eval 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \frac{\color{blue}{4}}{-\left(1 + t\right)} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      8. distribute-neg-in 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \frac{4}{\color{blue}{\left(-1\right) + \left(-t\right)}} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      9. metadata-eval 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \frac{4}{\color{blue}{-1} + \left(-t\right)} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      10. sub-neg 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{\color{blue}{-1 - t}} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      11. frac-2neg 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + 2 \cdot \color{blue}{\frac{--2}{-\left(1 + t\right)}}\right)}{6 + -4} \]

      12. metadata-eval 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + 2 \cdot \frac{\color{blue}{2}}{-\left(1 + t\right)}\right)}{6 + -4} \]

      13. associate-*r/ 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \color{blue}{\frac{2 \cdot 2}{-\left(1 + t\right)}}\right)}{6 + -4} \]

      14. metadata-eval 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{\color{blue}{4}}{-\left(1 + t\right)}\right)}{6 + -4} \]

      15. distribute-neg-in 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{4}{\color{blue}{\left(-1\right) + \left(-t\right)}}\right)}{6 + -4} \]

      16. metadata-eval 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{4}{\color{blue}{-1} + \left(-t\right)}\right)}{6 + -4} \]

      17. sub-neg 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{4}{\color{blue}{-1 - t}}\right)}{6 + -4} \]

   9. Applied egg-rr 98.1%

      \[\leadsto \frac{5 + \color{blue}{\left(t \cdot \frac{4}{-1 - t} + \frac{4}{-1 - t}\right)}}{6 + -4} \]
   10. Step-by-step derivation

       1. distribute-lft1-in 98.1%

          \[\leadsto \frac{5 + \color{blue}{\left(t + 1\right) \cdot \frac{4}{-1 - t}}}{6 + -4} \]

       2. *-commutative 98.1%

          \[\leadsto \frac{5 + \color{blue}{\frac{4}{-1 - t} \cdot \left(t + 1\right)}}{6 + -4} \]

   11. Simplified 98.1%

       \[\leadsto \frac{5 + \color{blue}{\frac{4}{-1 - t} \cdot \left(t + 1\right)}}{6 + -4} \]

   if 0.650000000000000022 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   5. 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%

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

   6. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25:\\ \;\;\;\;\frac{5 + \frac{8 - \frac{4}{t + 1}}{-1 - t}}{6 + \frac{8 - \frac{4}{t}}{-1 - t}}\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;\frac{5 + \left(t + 1\right) \cdot \frac{4}{-1 - t}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{8 - \frac{4}{t + 1}}{-1 - t}\\ \frac{5 + t_1}{6 + t_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (- 8.0 (/ 4.0 (+ t 1.0))) (- -1.0 t))))
   (/ (+ 5.0 t_1) (+ 6.0 t_1))))

C

double code(double t) {
	double t_1 = (8.0 - (4.0 / (t + 1.0))) / (-1.0 - t);
	return (5.0 + t_1) / (6.0 + t_1);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = (8.0d0 - (4.0d0 / (t + 1.0d0))) / ((-1.0d0) - t)
    code = (5.0d0 + t_1) / (6.0d0 + t_1)
end function

Java

public static double code(double t) {
	double t_1 = (8.0 - (4.0 / (t + 1.0))) / (-1.0 - t);
	return (5.0 + t_1) / (6.0 + t_1);
}

Python

def code(t):
	t_1 = (8.0 - (4.0 / (t + 1.0))) / (-1.0 - t)
	return (5.0 + t_1) / (6.0 + t_1)

Julia

function code(t)
	t_1 = Float64(Float64(8.0 - Float64(4.0 / Float64(t + 1.0))) / Float64(-1.0 - t))
	return Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1))
end

MATLAB

function tmp = code(t)
	t_1 = (8.0 - (4.0 / (t + 1.0))) / (-1.0 - t);
	tmp = (5.0 + t_1) / (6.0 + t_1);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(8.0 - N[(4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]
4. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{-2 \cdot \left(4 - \frac{2}{1 + t}\right)}{1 + t}}} \]

   2. frac-2neg 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{--2 \cdot \left(4 - \frac{2}{1 + t}\right)}{-\left(1 + t\right)}}} \]

   3. sub-neg 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{--2 \cdot \color{blue}{\left(4 + \left(-\frac{2}{1 + t}\right)\right)}}{-\left(1 + t\right)}} \]

   4. distribute-rgt-in 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(4 \cdot -2 + \left(-\frac{2}{1 + t}\right) \cdot -2\right)}}{-\left(1 + t\right)}} \]

   5. distribute-neg-frac 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \color{blue}{\frac{-2}{1 + t}} \cdot -2\right)}{-\left(1 + t\right)}} \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \frac{\color{blue}{-2}}{1 + t} \cdot -2\right)}{-\left(1 + t\right)}} \]

   7. +-commutative 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(\frac{-2}{1 + t} \cdot -2 + 4 \cdot -2\right)}}{-\left(1 + t\right)}} \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + \color{blue}{-8}\right)}{-\left(1 + t\right)}} \]

   9. neg-sub0 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{0 - \left(1 + t\right)}}} \]

   10. associate--r+ 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{\left(0 - 1\right) - t}}} \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{-1} - t}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{-1 - t}}} \]
6. Step-by-step derivation

   1. neg-sub0 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{0 - \left(\frac{-2}{1 + t} \cdot -2 + -8\right)}}{-1 - t}} \]

   2. +-commutative 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{0 - \color{blue}{\left(-8 + \frac{-2}{1 + t} \cdot -2\right)}}{-1 - t}} \]

   3. associate--r+ 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{\left(0 - -8\right) - \frac{-2}{1 + t} \cdot -2}}{-1 - t}} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{8} - \frac{-2}{1 + t} \cdot -2}{-1 - t}} \]

   5. associate-*l/ 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \color{blue}{\frac{-2 \cdot -2}{1 + t}}}{-1 - t}} \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{\color{blue}{4}}{1 + t}}{-1 - t}} \]

   7. +-commutative 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{4}{\color{blue}{t + 1}}}{-1 - t}} \]

7. Simplified 100.0%

   \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{8 - \frac{4}{t + 1}}{-1 - t}}} \]
8. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{-2 \cdot \left(4 - \frac{2}{1 + t}\right)}{1 + t}}} \]

   2. frac-2neg 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{\frac{--2 \cdot \left(4 - \frac{2}{1 + t}\right)}{-\left(1 + t\right)}}} \]

   3. sub-neg 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{--2 \cdot \color{blue}{\left(4 + \left(-\frac{2}{1 + t}\right)\right)}}{-\left(1 + t\right)}} \]

   4. distribute-rgt-in 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(4 \cdot -2 + \left(-\frac{2}{1 + t}\right) \cdot -2\right)}}{-\left(1 + t\right)}} \]

   5. distribute-neg-frac 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \color{blue}{\frac{-2}{1 + t}} \cdot -2\right)}{-\left(1 + t\right)}} \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(4 \cdot -2 + \frac{\color{blue}{-2}}{1 + t} \cdot -2\right)}{-\left(1 + t\right)}} \]

   7. +-commutative 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\color{blue}{\left(\frac{-2}{1 + t} \cdot -2 + 4 \cdot -2\right)}}{-\left(1 + t\right)}} \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + \color{blue}{-8}\right)}{-\left(1 + t\right)}} \]

   9. neg-sub0 100.0%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{0 - \left(1 + t\right)}}} \]

   10. associate--r+ 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{\left(0 - 1\right) - t}}} \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{\color{blue}{-1} - t}} \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{5 + \color{blue}{\frac{-\left(\frac{-2}{1 + t} \cdot -2 + -8\right)}{-1 - t}}}{6 + \frac{8 - \frac{4}{t + 1}}{-1 - t}} \]
10. Step-by-step derivation

    1. neg-sub0 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{0 - \left(\frac{-2}{1 + t} \cdot -2 + -8\right)}}{-1 - t}} \]

    2. +-commutative 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{0 - \color{blue}{\left(-8 + \frac{-2}{1 + t} \cdot -2\right)}}{-1 - t}} \]

    3. associate--r+ 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{\left(0 - -8\right) - \frac{-2}{1 + t} \cdot -2}}{-1 - t}} \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{\color{blue}{8} - \frac{-2}{1 + t} \cdot -2}{-1 - t}} \]

    5. associate-*l/ 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \color{blue}{\frac{-2 \cdot -2}{1 + t}}}{-1 - t}} \]

    6. metadata-eval 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{\color{blue}{4}}{1 + t}}{-1 - t}} \]

    7. +-commutative 100.0%

       \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{8 - \frac{4}{\color{blue}{t + 1}}}{-1 - t}} \]

11. Simplified 100.0%

    \[\leadsto \frac{5 + \color{blue}{\frac{8 - \frac{4}{t + 1}}{-1 - t}}}{6 + \frac{8 - \frac{4}{t + 1}}{-1 - t}} \]

12. Final simplification 100.0%

    \[\leadsto \frac{5 + \frac{8 - \frac{4}{t + 1}}{-1 - t}}{6 + \frac{8 - \frac{4}{t + 1}}{-1 - t}} \]

Alternative 4: 99.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \left(t + 1\right) \cdot \frac{4}{-1 - t}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.65)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (/ (+ 5.0 (* (+ t 1.0) (/ 4.0 (- -1.0 t)))) 2.0)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.65)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0;
	}
	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 = (5.0d0 + ((t + 1.0d0) * (4.0d0 / ((-1.0d0) - t)))) / 2.0d0
    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 = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0;
	}
	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 = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0
	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 = Float64(Float64(5.0 + Float64(Float64(t + 1.0) * Float64(4.0 / Float64(-1.0 - t)))) / 2.0);
	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 = (5.0 + ((t + 1.0) * (4.0 / (-1.0 - t)))) / 2.0;
	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], N[(N[(5.0 + N[(N[(t + 1.0), $MachinePrecision] * N[(4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.650000000000000022 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. Taylor expanded in t around inf 99.2%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   5. 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%

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

   6. Simplified 99.2%

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

   if -0.48999999999999999 < t < 0.650000000000000022

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. Taylor expanded in t around 0 98.1%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \color{blue}{-4}} \]

   5. Taylor expanded in t around 0 98.1%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \color{blue}{\left(2 + 2 \cdot t\right)}}{6 + -4} \]
   6. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \left(2 + \color{blue}{t \cdot 2}\right)}{6 + -4} \]

   7. Simplified 98.1%

      \[\leadsto \frac{5 + \frac{-2}{1 + t} \cdot \color{blue}{\left(2 + t \cdot 2\right)}}{6 + -4} \]
   8. Step-by-step derivation

      1. distribute-rgt-in 98.1%

         \[\leadsto \frac{5 + \color{blue}{\left(2 \cdot \frac{-2}{1 + t} + \left(t \cdot 2\right) \cdot \frac{-2}{1 + t}\right)}}{6 + -4} \]

      2. +-commutative 98.1%

         \[\leadsto \frac{5 + \color{blue}{\left(\left(t \cdot 2\right) \cdot \frac{-2}{1 + t} + 2 \cdot \frac{-2}{1 + t}\right)}}{6 + -4} \]

      3. associate-*l* 98.1%

         \[\leadsto \frac{5 + \left(\color{blue}{t \cdot \left(2 \cdot \frac{-2}{1 + t}\right)} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      4. frac-2neg 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \left(2 \cdot \color{blue}{\frac{--2}{-\left(1 + t\right)}}\right) + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      5. metadata-eval 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \left(2 \cdot \frac{\color{blue}{2}}{-\left(1 + t\right)}\right) + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      6. associate-*r/ 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \color{blue}{\frac{2 \cdot 2}{-\left(1 + t\right)}} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      7. metadata-eval 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \frac{\color{blue}{4}}{-\left(1 + t\right)} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      8. distribute-neg-in 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \frac{4}{\color{blue}{\left(-1\right) + \left(-t\right)}} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      9. metadata-eval 98.1%

         \[\leadsto \frac{5 + \left(t \cdot \frac{4}{\color{blue}{-1} + \left(-t\right)} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      10. sub-neg 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{\color{blue}{-1 - t}} + 2 \cdot \frac{-2}{1 + t}\right)}{6 + -4} \]

      11. frac-2neg 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + 2 \cdot \color{blue}{\frac{--2}{-\left(1 + t\right)}}\right)}{6 + -4} \]

      12. metadata-eval 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + 2 \cdot \frac{\color{blue}{2}}{-\left(1 + t\right)}\right)}{6 + -4} \]

      13. associate-*r/ 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \color{blue}{\frac{2 \cdot 2}{-\left(1 + t\right)}}\right)}{6 + -4} \]

      14. metadata-eval 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{\color{blue}{4}}{-\left(1 + t\right)}\right)}{6 + -4} \]

      15. distribute-neg-in 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{4}{\color{blue}{\left(-1\right) + \left(-t\right)}}\right)}{6 + -4} \]

      16. metadata-eval 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{4}{\color{blue}{-1} + \left(-t\right)}\right)}{6 + -4} \]

      17. sub-neg 98.1%

          \[\leadsto \frac{5 + \left(t \cdot \frac{4}{-1 - t} + \frac{4}{\color{blue}{-1 - t}}\right)}{6 + -4} \]

   9. Applied egg-rr 98.1%

      \[\leadsto \frac{5 + \color{blue}{\left(t \cdot \frac{4}{-1 - t} + \frac{4}{-1 - t}\right)}}{6 + -4} \]
   10. Step-by-step derivation

       1. distribute-lft1-in 98.1%

          \[\leadsto \frac{5 + \color{blue}{\left(t + 1\right) \cdot \frac{4}{-1 - t}}}{6 + -4} \]

       2. *-commutative 98.1%

          \[\leadsto \frac{5 + \color{blue}{\frac{4}{-1 - t} \cdot \left(t + 1\right)}}{6 + -4} \]

   11. Simplified 98.1%

       \[\leadsto \frac{5 + \color{blue}{\frac{4}{-1 - t} \cdot \left(t + 1\right)}}{6 + -4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \left(t + 1\right) \cdot \frac{4}{-1 - t}}{2}\\ \end{array} \]

Alternative 5: 99.0% accurate, 5.6× 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%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. Taylor expanded in t around inf 99.2%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   5. 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%

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

   6. Simplified 99.2%

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

   if -0.48999999999999999 < t < 0.650000000000000022

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. 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 6: 98.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. Taylor expanded in t around inf 98.6%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

   4. 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 7: 59.4% accurate, 51.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}{6 + \frac{-2}{1 + t} \cdot \left(4 - \frac{2}{1 + t}\right)}} \]

4. Taylor expanded in t around 0 61.4%

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

5. Final simplification 61.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))