Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 12

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. associate-/l/ 100.0%

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

   4. *-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. distribute-rgt-in 100.0%

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

   4. lft-mult-inverse 100.0%

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

   5. *-lft-identity 100.0%

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

5. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. lft-mult-inverse 100.0%

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

   4. *-lft-identity 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{6 + \frac{1}{1 + t} \cdot \left(\frac{4}{1 + t} + -8\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

   3. +-commutative 100.0%

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

   4. +-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;1 + \frac{-1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.52)
   (+ 1.0 (/ -1.0 (+ 6.0 (/ (+ -8.0 (/ 4.0 t)) (+ 1.0 t)))))
   (if (<= t 0.44)
     (+ 1.0 (/ -1.0 (+ 2.0 (* 4.0 (* t t)))))
     (-
      1.0
      (+
       0.16666666666666666
       (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	} else if (t <= 0.44) {
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.52d0)) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + (((-8.0d0) + (4.0d0 / t)) / (1.0d0 + t))))
    else if (t <= 0.44d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (4.0d0 * (t * t))))
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	} else if (t <= 0.44) {
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.52:
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))))
	elif t <= 0.44:
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))))
	else:
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(-8.0 + Float64(4.0 / t)) / Float64(1.0 + t)))));
	elseif (t <= 0.44)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(4.0 * Float64(t * t)))));
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.52)
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	elseif (t <= 0.44)
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))));
	else
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.52], N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(-8.0 + N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.44], N[(1.0 + N[(-1.0 / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.9%

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

      1. sub-neg 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

   if -0.52000000000000002 < t < 0.440000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 98.9%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{4 \cdot {t}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 98.9%

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

      2. unpow2 98.9%

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

   4. Simplified 98.9%

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

   if 0.440000000000000002 < t

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. lft-mult-inverse 100.0%

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

      5. *-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right)} \]

      2. associate-*r/ 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right) \]

      3. metadata-eval 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{\color{blue}{0.2222222222222222}}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right) \]

      4. associate-*r/ 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{\color{blue}{0.037037037037037035}}{{t}^{2}}\right)\right) \]

      6. unpow2 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right)\right) \]

      7. associate-/r* 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{\frac{0.037037037037037035}{t}}{t}\right)\right)} \]

   9. Taylor expanded in t around 0 100.0%

      \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. sub-neg 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + \left(-0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right)}\right) \]

       2. distribute-lft-neg-in 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\left(-0.037037037037037035\right) \cdot \frac{1}{{t}^{2}}}\right)\right) \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{-0.037037037037037035} \cdot \frac{1}{{t}^{2}}\right)\right) \]

       4. unpow2 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + -0.037037037037037035 \cdot \frac{1}{\color{blue}{t \cdot t}}\right)\right) \]

       5. associate-/r* 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + -0.037037037037037035 \cdot \color{blue}{\frac{\frac{1}{t}}{t}}\right)\right) \]

       6. associate-*r/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\frac{-0.037037037037037035 \cdot \frac{1}{t}}{t}}\right)\right) \]

       7. associate-*l/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\frac{-0.037037037037037035}{t} \cdot \frac{1}{t}}\right)\right) \]

       8. distribute-rgt-in 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{1}{t} \cdot \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right)}\right) \]

       9. associate-*l/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{1 \cdot \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right)}{t}}\right) \]

       10. *-lft-identity 100.0%

           \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.2222222222222222 + \frac{-0.037037037037037035}{t}}}{t}\right) \]

   11. Simplified 100.0%

       \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;1 + \frac{-1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 5: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.34:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.82)
   (+
    (/ (/ 0.037037037037037035 t) t)
    (+ 0.8333333333333334 (/ -0.2222222222222222 t)))
   (if (<= t 0.34)
     (+ (* t t) 0.5)
     (-
      1.0
      (+
       0.16666666666666666
       (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) 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 <= 0.34) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / 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 <= 0.34d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / 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 <= 0.34) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / 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 <= 0.34:
		tmp = (t * t) + 0.5
	else:
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.82)
		tmp = Float64(Float64(Float64(0.037037037037037035 / t) / t) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.34)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / 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 <= 0.34)
		tmp = (t * t) + 0.5;
	else
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.82], N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] / t), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.34], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.819999999999999951

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.8%

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

      1. associate--l+ 98.8%

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

      2. associate-*r/ 98.8%

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

      3. metadata-eval 98.8%

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

      4. unpow2 98.8%

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

      5. associate-/r* 98.8%

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

      6. sub-neg 98.8%

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

      7. associate-*r/ 98.8%

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

      8. metadata-eval 98.8%

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

      9. distribute-neg-frac 98.8%

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

      10. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   if -0.819999999999999951 < t < 0.340000000000000024

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. unpow2 98.9%

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

   6. Simplified 98.9%

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

   if 0.340000000000000024 < t

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. lft-mult-inverse 100.0%

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

      5. *-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right)} \]

      2. associate-*r/ 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right) \]

      3. metadata-eval 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{\color{blue}{0.2222222222222222}}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right) \]

      4. associate-*r/ 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{\color{blue}{0.037037037037037035}}{{t}^{2}}\right)\right) \]

      6. unpow2 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right)\right) \]

      7. associate-/r* 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{\frac{0.037037037037037035}{t}}{t}\right)\right)} \]

   9. Taylor expanded in t around 0 100.0%

      \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. sub-neg 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + \left(-0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right)}\right) \]

       2. distribute-lft-neg-in 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\left(-0.037037037037037035\right) \cdot \frac{1}{{t}^{2}}}\right)\right) \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{-0.037037037037037035} \cdot \frac{1}{{t}^{2}}\right)\right) \]

       4. unpow2 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + -0.037037037037037035 \cdot \frac{1}{\color{blue}{t \cdot t}}\right)\right) \]

       5. associate-/r* 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + -0.037037037037037035 \cdot \color{blue}{\frac{\frac{1}{t}}{t}}\right)\right) \]

       6. associate-*r/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\frac{-0.037037037037037035 \cdot \frac{1}{t}}{t}}\right)\right) \]

       7. associate-*l/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\frac{-0.037037037037037035}{t} \cdot \frac{1}{t}}\right)\right) \]

       8. distribute-rgt-in 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{1}{t} \cdot \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right)}\right) \]

       9. associate-*l/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{1 \cdot \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right)}{t}}\right) \]

       10. *-lft-identity 100.0%

           \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.2222222222222222 + \frac{-0.037037037037037035}{t}}}{t}\right) \]

   11. Simplified 100.0%

       \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.34:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \]

Alternative 6: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.65:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;1 + \frac{-1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.65)
   (+
    (/ (/ 0.037037037037037035 t) t)
    (+ 0.8333333333333334 (/ -0.2222222222222222 t)))
   (if (<= t 0.44)
     (+ 1.0 (/ -1.0 (+ 2.0 (* 4.0 (* t t)))))
     (-
      1.0
      (+
       0.16666666666666666
       (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.65) {
		tmp = ((0.037037037037037035 / t) / t) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 0.44) {
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.65d0)) then
        tmp = ((0.037037037037037035d0 / t) / t) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    else if (t <= 0.44d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (4.0d0 * (t * t))))
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.65) {
		tmp = ((0.037037037037037035 / t) / t) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 0.44) {
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.65:
		tmp = ((0.037037037037037035 / t) / t) + (0.8333333333333334 + (-0.2222222222222222 / t))
	elif t <= 0.44:
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))))
	else:
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.65)
		tmp = Float64(Float64(Float64(0.037037037037037035 / t) / t) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.44)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(4.0 * Float64(t * t)))));
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.65)
		tmp = ((0.037037037037037035 / t) / t) + (0.8333333333333334 + (-0.2222222222222222 / t));
	elseif (t <= 0.44)
		tmp = 1.0 + (-1.0 / (2.0 + (4.0 * (t * t))));
	else
		tmp = 1.0 - (0.16666666666666666 + ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.65], N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] / t), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.44], N[(1.0 + N[(-1.0 / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.650000000000000022

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.8%

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

      1. associate--l+ 98.8%

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

      2. associate-*r/ 98.8%

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

      3. metadata-eval 98.8%

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

      4. unpow2 98.8%

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

      5. associate-/r* 98.8%

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

      6. sub-neg 98.8%

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

      7. associate-*r/ 98.8%

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

      8. metadata-eval 98.8%

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

      9. distribute-neg-frac 98.8%

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

      10. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   if -0.650000000000000022 < t < 0.440000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 98.9%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{4 \cdot {t}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 98.9%

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

      2. unpow2 98.9%

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

   4. Simplified 98.9%

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

   if 0.440000000000000002 < t

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. lft-mult-inverse 100.0%

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

      5. *-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right)} \]

      2. associate-*r/ 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right) \]

      3. metadata-eval 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{\color{blue}{0.2222222222222222}}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right) \]

      4. associate-*r/ 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{\color{blue}{0.037037037037037035}}{{t}^{2}}\right)\right) \]

      6. unpow2 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right)\right) \]

      7. associate-/r* 100.0%

         \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{\frac{0.037037037037037035}{t}}{t}\right)\right)} \]

   9. Taylor expanded in t around 0 100.0%

      \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. sub-neg 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + \left(-0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)\right)}\right) \]

       2. distribute-lft-neg-in 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\left(-0.037037037037037035\right) \cdot \frac{1}{{t}^{2}}}\right)\right) \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{-0.037037037037037035} \cdot \frac{1}{{t}^{2}}\right)\right) \]

       4. unpow2 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + -0.037037037037037035 \cdot \frac{1}{\color{blue}{t \cdot t}}\right)\right) \]

       5. associate-/r* 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + -0.037037037037037035 \cdot \color{blue}{\frac{\frac{1}{t}}{t}}\right)\right) \]

       6. associate-*r/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\frac{-0.037037037037037035 \cdot \frac{1}{t}}{t}}\right)\right) \]

       7. associate-*l/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} + \color{blue}{\frac{-0.037037037037037035}{t} \cdot \frac{1}{t}}\right)\right) \]

       8. distribute-rgt-in 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{1}{t} \cdot \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right)}\right) \]

       9. associate-*l/ 100.0%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{1 \cdot \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right)}{t}}\right) \]

       10. *-lft-identity 100.0%

           \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.2222222222222222 + \frac{-0.037037037037037035}{t}}}{t}\right) \]

   11. Simplified 100.0%

       \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.65:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;1 + \frac{-1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \]

Alternative 7: 99.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.82)
   (+
    (/ (/ 0.037037037037037035 t) t)
    (+ 0.8333333333333334 (/ -0.2222222222222222 t)))
   (if (<= t 0.55)
     (+ (* t t) 0.5)
     (- 1.0 (+ 0.16666666666666666 (/ 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 <= 0.55) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (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 <= 0.55d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (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 <= 0.55) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (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 <= 0.55:
		tmp = (t * t) + 0.5
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.82)
		tmp = Float64(Float64(Float64(0.037037037037037035 / t) / t) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.55)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + 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 <= 0.55)
		tmp = (t * t) + 0.5;
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.82], N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] / t), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.55], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.819999999999999951

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.8%

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

      1. associate--l+ 98.8%

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

      2. associate-*r/ 98.8%

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

      3. metadata-eval 98.8%

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

      4. unpow2 98.8%

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

      5. associate-/r* 98.8%

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

      6. sub-neg 98.8%

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

      7. associate-*r/ 98.8%

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

      8. metadata-eval 98.8%

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

      9. distribute-neg-frac 98.8%

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

      10. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   if -0.819999999999999951 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. unpow2 98.9%

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

   6. Simplified 98.9%

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

   if 0.55000000000000004 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 8: 99.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.78)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.55)
     (+ (* t t) 0.5)
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.55) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.78d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.55d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.55) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.78:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.55:
		tmp = (t * t) + 0.5
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.55)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.78)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.55)
		tmp = (t * t) + 0.5;
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.78], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.55], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.78000000000000003

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   if -0.78000000000000003 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. unpow2 98.9%

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

   6. Simplified 98.9%

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

   if 0.55000000000000004 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 9: 99.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.55)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.55)) {
		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 <= (-0.78d0)) .or. (.not. (t <= 0.55d0))) 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 <= -0.78) || !(t <= 0.55)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.78) or not (t <= 0.55):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.78) || !(t <= 0.55))
		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 <= -0.78) || ~((t <= 0.55)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.78], N[Not[LessEqual[t, 0.55]], $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 < -0.78000000000000003 or 0.55000000000000004 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if -0.78000000000000003 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. unpow2 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 10: 98.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.92)
   0.8333333333333334
   (if (<= t 0.58) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		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 <= (-0.92d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) 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 <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.92:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		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 <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.92], 0.8333333333333334, If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.92000000000000004 or 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 99.0%

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

   if -0.92000000000000004 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. unpow2 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 11: 98.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.5%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 98.4%

      \[\leadsto 1 - \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

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

Alternative 12: 58.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0
end function

Java

public static double code(double t) {
	return 0.8333333333333334;
}

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334;
end

Wolfram

code[t_] := 0.8333333333333334

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in t around inf 62.2%

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

5. Final simplification 62.2%

   \[\leadsto 0.8333333333333334 \]

Reproduce

herbie shell --seed 2023227 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))