Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 8.1s

Alternatives: 8

Speedup: 1.6×

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return (5.0 + (t_1 * (-4.0 - (-2.0 / (1.0 + t))))) / (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 = (5.0d0 + (t_1 * ((-4.0d0) - ((-2.0d0) / (1.0d0 + t))))) / (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 (5.0 + (t_1 * (-4.0 - (-2.0 / (1.0 + t))))) / (6.0 + (t_1 * (t_1 - 4.0)));
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	return (5.0 + (t_1 * (-4.0 - (-2.0 / (1.0 + t))))) / (6.0 + (t_1 * (t_1 - 4.0)))

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	return Float64(Float64(5.0 + Float64(t_1 * Float64(-4.0 - Float64(-2.0 / Float64(1.0 + t))))) / 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 = (5.0 + (t_1 * (-4.0 - (-2.0 / (1.0 + t))))) / (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[(N[(5.0 + N[(t$95$1 * N[(-4.0 - N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(t$95$1 * N[(t$95$1 - 4.0), $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%

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

4. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{4}{t} + -8\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{t_1}{-1 - t}}{6 + \frac{t_1}{1 + t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ (/ 4.0 t) -8.0)))
   (if (<= t -0.82)
     (+
      (/ 0.037037037037037035 (* t t))
      (+ 0.8333333333333334 (/ -0.2222222222222222 t)))
     (if (<= t 0.56)
       (+ (* t t) 0.5)
       (/ (- 5.0 (/ t_1 (- -1.0 t))) (+ 6.0 (/ t_1 (+ 1.0 t))))))))

C

double code(double t) {
	double t_1 = (4.0 / t) + -8.0;
	double tmp;
	if (t <= -0.82) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (5.0 - (t_1 / (-1.0 - t))) / (6.0 + (t_1 / (1.0 + t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (4.0d0 / t) + (-8.0d0)
    if (t <= (-0.82d0)) then
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    else if (t <= 0.56d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = (5.0d0 - (t_1 / ((-1.0d0) - t))) / (6.0d0 + (t_1 / (1.0d0 + t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (4.0 / t) + -8.0;
	double tmp;
	if (t <= -0.82) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (5.0 - (t_1 / (-1.0 - t))) / (6.0 + (t_1 / (1.0 + t)));
	}
	return tmp;
}

Python

def code(t):
	t_1 = (4.0 / t) + -8.0
	tmp = 0
	if t <= -0.82:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	elif t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = (5.0 - (t_1 / (-1.0 - t))) / (6.0 + (t_1 / (1.0 + t)))
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(4.0 / t) + -8.0)
	tmp = 0.0
	if (t <= -0.82)
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.56)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(Float64(5.0 - Float64(t_1 / Float64(-1.0 - t))) / Float64(6.0 + Float64(t_1 / Float64(1.0 + t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (4.0 / t) + -8.0;
	tmp = 0.0;
	if (t <= -0.82)
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	elseif (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = (5.0 - (t_1 / (-1.0 - t))) / (6.0 + (t_1 / (1.0 + t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(4.0 / t), $MachinePrecision] + -8.0), $MachinePrecision]}, If[LessEqual[t, -0.82], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.56], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(5.0 - N[(t$95$1 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(t$95$1 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.819999999999999951

   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%

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

   4. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} + 0.8333333333333334\right)} - 0.2222222222222222 \cdot \frac{1}{t} \]

      2. associate--l+ 100.0%

         \[\leadsto \color{blue}{0.037037037037037035 \cdot \frac{1}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      3. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0.037037037037037035}}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      5. unpow2 100.0%

         \[\leadsto \frac{0.037037037037037035}{\color{blue}{t \cdot t}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      6. sub-neg 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      7. associate-*r/ 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)} \]

   if -0.819999999999999951 < t < 0.56000000000000005

   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%

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

   4. Taylor expanded in t around 0 99.1%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.1%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   6. Simplified 99.1%

      \[\leadsto \color{blue}{t \cdot t + 0.5} \]

   if 0.56000000000000005 < 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%

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

   4. Taylor expanded in t around inf 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{1 + t} \cdot \left(\frac{2}{t} - 4\right)\right)\right)}} \]

      2. expm1-udef 100.0%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{1 + t} \cdot \left(\frac{2}{t} - 4\right)\right)} - 1\right)}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{t + 1}} \cdot \left(\frac{2}{t} - 4\right)\right)} - 1\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(e^{\mathsf{log1p}\left(\frac{2}{t + 1} \cdot \color{blue}{\left(\frac{2}{t} + \left(-4\right)\right)}\right)} - 1\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(e^{\mathsf{log1p}\left(\frac{2}{t + 1} \cdot \left(\frac{2}{t} + \color{blue}{-4}\right)\right)} - 1\right)} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{t + 1} \cdot \left(\frac{2}{t} + -4\right)\right)} - 1\right)}} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{t + 1} \cdot \left(\frac{2}{t} + -4\right)\right)\right)}} \]

      2. expm1-log1p 100.0%

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

      3. associate-*l/ 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

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

       1. associate-*l/ 100.0%

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

       2. frac-2neg 100.0%

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

       3. sub-neg 100.0%

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

       4. metadata-eval 100.0%

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

       5. distribute-lft-in 100.0%

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

       6. metadata-eval 100.0%

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

       7. metadata-eval 100.0%

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

       8. metadata-eval 100.0%

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

       9. fma-def 100.0%

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

       10. metadata-eval 100.0%

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

       11. metadata-eval 100.0%

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

       12. +-commutative 100.0%

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

       13. distribute-neg-in 100.0%

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

       14. neg-mul-1 100.0%

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

       15. metadata-eval 100.0%

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

       16. metadata-eval 100.0%

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

       17. metadata-eval 100.0%

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

       18. fma-def 100.0%

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

       19. metadata-eval 100.0%

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

       20. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. fma-udef 100.0%

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

       2. associate-*r/ 100.0%

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

       3. metadata-eval 100.0%

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

       4. metadata-eval 100.0%

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

       5. distribute-neg-frac 100.0%

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

       6. metadata-eval 100.0%

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

       7. associate-*r/ 100.0%

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

       8. metadata-eval 100.0%

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

       9. distribute-neg-in 100.0%

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

       10. associate-*r/ 100.0%

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

       11. metadata-eval 100.0%

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

       12. remove-double-neg 100.0%

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

       13. fma-udef 100.0%

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

       14. neg-mul-1 100.0%

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

       15. +-commutative 100.0%

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

       16. unsub-neg 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \frac{5 - \color{blue}{\frac{\frac{4}{t} + -8}{-1 - t}}}{6 + \frac{\frac{4}{t} + -8}{t + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{\frac{4}{t} + -8}{-1 - t}}{6 + \frac{\frac{4}{t} + -8}{1 + t}}\\ \end{array} \]

Alternative 3: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 1.65:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.82)
   (+
    (/ 0.037037037037037035 (* t t))
    (+ 0.8333333333333334 (/ -0.2222222222222222 t)))
   (if (<= t 1.65) (+ (* t t) 0.5) (/ (- 5.0 (/ 8.0 t)) (+ 6.0 (/ -8.0 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.82) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 1.65) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.82d0)) then
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    else if (t <= 1.65d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = (5.0d0 - (8.0d0 / t)) / (6.0d0 + ((-8.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.82) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 1.65) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.82:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	elif t <= 1.65:
		tmp = (t * t) + 0.5
	else:
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.82)
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 1.65)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(Float64(5.0 - Float64(8.0 / t)) / Float64(6.0 + Float64(-8.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.82)
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	elseif (t <= 1.65)
		tmp = (t * t) + 0.5;
	else
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.82], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(5.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.819999999999999951

   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%

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

   4. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} + 0.8333333333333334\right)} - 0.2222222222222222 \cdot \frac{1}{t} \]

      2. associate--l+ 100.0%

         \[\leadsto \color{blue}{0.037037037037037035 \cdot \frac{1}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      3. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0.037037037037037035}}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      5. unpow2 100.0%

         \[\leadsto \frac{0.037037037037037035}{\color{blue}{t \cdot t}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      6. sub-neg 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      7. associate-*r/ 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)} \]

   if -0.819999999999999951 < t < 1.6499999999999999

   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%

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

   4. Taylor expanded in t around 0 99.1%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.1%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   6. Simplified 99.1%

      \[\leadsto \color{blue}{t \cdot t + 0.5} \]

   if 1.6499999999999999 < 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%

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

   4. Taylor expanded in t around inf 100.0%

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

   5. Taylor expanded in t around inf 99.9%

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

   6. Taylor expanded in t around inf 99.9%

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

   7. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 - \color{blue}{\frac{8}{t}}}{6 + \frac{-8}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 1.65:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \]

Alternative 4: 99.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.82)
   (+
    (/ 0.037037037037037035 (* t t))
    (+ 0.8333333333333334 (/ -0.2222222222222222 t)))
   (if (<= t 0.56)
     (+ (* t t) 0.5)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.82) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 0.56) {
		tmp = (t * t) + 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 <= (-0.82d0)) then
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    else if (t <= 0.56d0) then
        tmp = (t * t) + 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 <= -0.82) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.82:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	elif t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.82)
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.56)
		tmp = Float64(Float64(t * t) + 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 <= -0.82)
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	elseif (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.82], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.56], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.819999999999999951

   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%

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

   4. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} + 0.8333333333333334\right)} - 0.2222222222222222 \cdot \frac{1}{t} \]

      2. associate--l+ 100.0%

         \[\leadsto \color{blue}{0.037037037037037035 \cdot \frac{1}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      3. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0.037037037037037035}}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      5. unpow2 100.0%

         \[\leadsto \frac{0.037037037037037035}{\color{blue}{t \cdot t}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      6. sub-neg 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      7. associate-*r/ 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)} \]

   if -0.819999999999999951 < t < 0.56000000000000005

   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%

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

   4. Taylor expanded in t around 0 99.1%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.1%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   6. Simplified 99.1%

      \[\leadsto \color{blue}{t \cdot t + 0.5} \]

   if 0.56000000000000005 < 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%

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

   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 99.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.8) (not (<= t 0.56)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.8d0)) .or. (.not. (t <= 0.56d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.8) or not (t <= 0.56):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.8) || !(t <= 0.56))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.8) || ~((t <= 0.56)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.8], N[Not[LessEqual[t, 0.56]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.56000000000000005 < 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%

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

   4. Taylor expanded in t around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   if -0.80000000000000004 < t < 0.56000000000000005

   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%

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

   4. Taylor expanded in t around 0 99.1%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.1%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   6. Simplified 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 6: 98.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.56) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.9d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.56d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.9:
		tmp = 0.8333333333333334
	elif t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.9], 0.8333333333333334, If[LessEqual[t, 0.56], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.56000000000000005 < 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%

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

   4. Taylor expanded in t around inf 99.1%

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

   if -0.900000000000000022 < t < 0.56000000000000005

   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%

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

   4. Taylor expanded in t around 0 99.1%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   5. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.1%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   6. Simplified 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 7: 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%

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

   4. Taylor expanded in t around inf 99.1%

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

   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%

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

   4. Taylor expanded in t around 0 98.8%

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

4. Final simplification 99.0%

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

Alternative 8: 59.8% 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%

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

4. Taylor expanded in t around 0 56.2%

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

5. Final simplification 56.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023271 
(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))))))))