Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 13.5s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 99.2%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-2}{1 + t}\\ t_2 := t_1 \cdot \left(t_1 - -4\right)\\ \frac{5 + t_2}{t_2 + 6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ -2.0 (+ 1.0 t))) (t_2 (* t_1 (- t_1 -4.0))))
   (/ (+ 5.0 t_2) (+ t_2 6.0))))

C

double code(double t) {
	double t_1 = -2.0 / (1.0 + t);
	double t_2 = t_1 * (t_1 - -4.0);
	return (5.0 + t_2) / (t_2 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (-2.0d0) / (1.0d0 + t)
    t_2 = t_1 * (t_1 - (-4.0d0))
    code = (5.0d0 + t_2) / (t_2 + 6.0d0)
end function

Java

public static double code(double t) {
	double t_1 = -2.0 / (1.0 + t);
	double t_2 = t_1 * (t_1 - -4.0);
	return (5.0 + t_2) / (t_2 + 6.0);
}

Python

def code(t):
	t_1 = -2.0 / (1.0 + t)
	t_2 = t_1 * (t_1 - -4.0)
	return (5.0 + t_2) / (t_2 + 6.0)

Julia

function code(t)
	t_1 = Float64(-2.0 / Float64(1.0 + t))
	t_2 = Float64(t_1 * Float64(t_1 - -4.0))
	return Float64(Float64(5.0 + t_2) / Float64(t_2 + 6.0))
end

MATLAB

function tmp = code(t)
	t_1 = -2.0 / (1.0 + t);
	t_2 = t_1 * (t_1 - -4.0);
	tmp = (5.0 + t_2) / (t_2 + 6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 - -4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$2), $MachinePrecision] / N[(t$95$2 + 6.0), $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%

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

Alternative 3: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-8 + \frac{4}{t}}{1 + t}\\ \mathbf{if}\;t \leq -0.4:\\ \;\;\;\;\frac{5 + t_1}{6 + t_1}\\ \mathbf{elif}\;t \leq 0.62:\\ \;\;\;\;\frac{5 + \frac{-8 + \frac{4}{1 + t}}{1 + t}}{6 + \left(4 + t \cdot 4\right) \cdot \left(t + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ -8.0 (/ 4.0 t)) (+ 1.0 t))))
   (if (<= t -0.4)
     (/ (+ 5.0 t_1) (+ 6.0 t_1))
     (if (<= t 0.62)
       (/
        (+ 5.0 (/ (+ -8.0 (/ 4.0 (+ 1.0 t))) (+ 1.0 t)))
        (+ 6.0 (* (+ 4.0 (* t 4.0)) (+ t -1.0))))
       0.8333333333333334))))

C

double code(double t) {
	double t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	double tmp;
	if (t <= -0.4) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else if (t <= 0.62) {
		tmp = (5.0 + ((-8.0 + (4.0 / (1.0 + t))) / (1.0 + t))) / (6.0 + ((4.0 + (t * 4.0)) * (t + -1.0)));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-8.0d0) + (4.0d0 / t)) / (1.0d0 + t)
    if (t <= (-0.4d0)) then
        tmp = (5.0d0 + t_1) / (6.0d0 + t_1)
    else if (t <= 0.62d0) then
        tmp = (5.0d0 + (((-8.0d0) + (4.0d0 / (1.0d0 + t))) / (1.0d0 + t))) / (6.0d0 + ((4.0d0 + (t * 4.0d0)) * (t + (-1.0d0))))
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	double tmp;
	if (t <= -0.4) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else if (t <= 0.62) {
		tmp = (5.0 + ((-8.0 + (4.0 / (1.0 + t))) / (1.0 + t))) / (6.0 + ((4.0 + (t * 4.0)) * (t + -1.0)));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (-8.0 + (4.0 / t)) / (1.0 + t)
	tmp = 0
	if t <= -0.4:
		tmp = (5.0 + t_1) / (6.0 + t_1)
	elif t <= 0.62:
		tmp = (5.0 + ((-8.0 + (4.0 / (1.0 + t))) / (1.0 + t))) / (6.0 + ((4.0 + (t * 4.0)) * (t + -1.0)))
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(-8.0 + Float64(4.0 / t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -0.4)
		tmp = Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1));
	elseif (t <= 0.62)
		tmp = Float64(Float64(5.0 + Float64(Float64(-8.0 + Float64(4.0 / Float64(1.0 + t))) / Float64(1.0 + t))) / Float64(6.0 + Float64(Float64(4.0 + Float64(t * 4.0)) * Float64(t + -1.0))));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -0.4)
		tmp = (5.0 + t_1) / (6.0 + t_1);
	elseif (t <= 0.62)
		tmp = (5.0 + ((-8.0 + (4.0 / (1.0 + t))) / (1.0 + t))) / (6.0 + ((4.0 + (t * 4.0)) * (t + -1.0)));
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-8.0 + N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.4], N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.62], N[(N[(5.0 + N[(N[(-8.0 + N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(N[(4.0 + N[(t * 4.0), $MachinePrecision]), $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.40000000000000002

   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. Step-by-step derivation

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 100.0%

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

      3. fma-udef 100.0%

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

      4. associate-*l/ 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-udef 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 99.6%

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

      1. sub-neg 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. metadata-eval 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in t around inf 99.6%

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

       1. sub-neg 99.6%

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

       2. associate-*r/ 99.6%

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

       3. metadata-eval 99.6%

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

       4. metadata-eval 99.6%

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

   13. Simplified 99.6%

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

   if -0.40000000000000002 < t < 0.619999999999999996

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. expm1-def 99.9%

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

      2. expm1-log1p 100.0%

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

      3. fma-udef 100.0%

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

      4. associate-*l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

      10. fma-udef 99.9%

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

   7. Simplified 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in t around 0 98.3%

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

       1. *-commutative 98.3%

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

   12. Simplified 98.3%

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

   if 0.619999999999999996 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.4:\\ \;\;\;\;\frac{5 + \frac{-8 + \frac{4}{t}}{1 + t}}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{elif}\;t \leq 0.62:\\ \;\;\;\;\frac{5 + \frac{-8 + \frac{4}{1 + t}}{1 + t}}{6 + \left(4 + t \cdot 4\right) \cdot \left(t + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

5. Applied egg-rr 99.2%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. fma-udef 100.0%

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

   4. associate-*l/ 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-*r/ 100.0%

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

   9. metadata-eval 100.0%

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

   10. fma-udef 100.0%

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

7. Simplified 100.0%

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

   1. flip-+ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. fma-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. sub-neg 100.0%

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

   7. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

    1. associate-/r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. fma-neg 100.0%

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

    4. metadata-eval 100.0%

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

    5. flip-- 100.0%

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

    6. sub-neg 100.0%

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

    7. metadata-eval 100.0%

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

    8. clear-num 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 5: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = (-8.0 + (4.0 / (1.0 + t))) / (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 / (1.0d0 + t))) / (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 / (1.0 + t))) / (1.0 + t);
	return (5.0 + t_1) / (6.0 + t_1);
}

Python

def code(t):
	t_1 = (-8.0 + (4.0 / (1.0 + t))) / (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(1.0 + t))) / 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 / (1.0 + t))) / (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[(1.0 + t), $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%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

5. Applied egg-rr 99.2%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. fma-udef 100.0%

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

   4. associate-*l/ 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-*r/ 100.0%

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

   9. metadata-eval 100.0%

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

   10. fma-udef 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 6: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-8 + \frac{4}{t}}{1 + t}\\ \mathbf{if}\;t \leq -0.43:\\ \;\;\;\;\frac{5 + t_1}{6 + t_1}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ -8.0 (/ 4.0 t)) (+ 1.0 t))))
   (if (<= t -0.43)
     (/ (+ 5.0 t_1) (+ 6.0 t_1))
     (if (<= t 1.0) 0.5 0.8333333333333334))))

C

double code(double t) {
	double t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	double tmp;
	if (t <= -0.43) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} 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) :: t_1
    real(8) :: tmp
    t_1 = ((-8.0d0) + (4.0d0 / t)) / (1.0d0 + t)
    if (t <= (-0.43d0)) then
        tmp = (5.0d0 + t_1) / (6.0d0 + t_1)
    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 t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	double tmp;
	if (t <= -0.43) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (-8.0 + (4.0 / t)) / (1.0 + t)
	tmp = 0
	if t <= -0.43:
		tmp = (5.0 + t_1) / (6.0 + t_1)
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(-8.0 + Float64(4.0 / t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -0.43)
		tmp = Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1));
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -0.43)
		tmp = (5.0 + t_1) / (6.0 + t_1);
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-8.0 + N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.43], N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.429999999999999993

   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. Step-by-step derivation

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 100.0%

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

      3. fma-udef 100.0%

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

      4. associate-*l/ 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-udef 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 99.6%

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

      1. sub-neg 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. metadata-eval 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in t around inf 99.6%

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

       1. sub-neg 99.6%

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

       2. associate-*r/ 99.6%

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

       3. metadata-eval 99.6%

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

       4. metadata-eval 99.6%

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

   13. Simplified 99.6%

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

   if -0.429999999999999993 < t < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.9%

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

   if 1 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.43:\\ \;\;\;\;\frac{5 + \frac{-8 + \frac{4}{t}}{1 + t}}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 7: 98.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

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

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} 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 <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    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 <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

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

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	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 <= -0.49)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

   if -0.48999999999999999 < t < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.9%

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

   if 1 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8: 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%

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

   2. Taylor expanded in t around inf 98.2%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 98.5%

      \[\leadsto \color{blue}{0.5} \]
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 9: 58.8% 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%

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

2. Taylor expanded in t around 0 59.1%

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

3. Final simplification 59.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023315 
(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))))))))