Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 12.8s

Alternatives: 7

Speedup: 1.4×

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{2}{-1 - t}\\ 1 + \frac{-1}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

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

Python

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

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(2.0 / 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 / (-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[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 100.0%

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

   2. associate-/l* 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

   1. div-inv 100.0%

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

   2. associate-/l* 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. rgt-mult-inverse 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.64 \lor \neg \left(t \leq 0.24\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 + \frac{-1}{2 + t \cdot \left(t \cdot 4\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.64) (not (<= t 0.24)))
   (+
    1.0
    (-
     (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)
     0.16666666666666666))
   (+ -1.0 (+ 2.0 (/ -1.0 (+ 2.0 (* t (* t 4.0))))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.64) || !(t <= 0.24)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = -1.0 + (2.0 + (-1.0 / (2.0 + (t * (t * 4.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.64d0)) .or. (.not. (t <= 0.24d0))) then
        tmp = 1.0d0 + ((((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t) - 0.16666666666666666d0)
    else
        tmp = (-1.0d0) + (2.0d0 + ((-1.0d0) / (2.0d0 + (t * (t * 4.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.64) || !(t <= 0.24)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = -1.0 + (2.0 + (-1.0 / (2.0 + (t * (t * 4.0)))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.64) or not (t <= 0.24):
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666)
	else:
		tmp = -1.0 + (2.0 + (-1.0 / (2.0 + (t * (t * 4.0)))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.64) || !(t <= 0.24))
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666));
	else
		tmp = Float64(-1.0 + Float64(2.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(t * 4.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.64) || ~((t <= 0.24)))
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	else
		tmp = -1.0 + (2.0 + (-1.0 / (2.0 + (t * (t * 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.64], N[Not[LessEqual[t, 0.24]], $MachinePrecision]], N[(1.0 + N[(N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 + N[(-1.0 / N[(2.0 + N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.640000000000000013 or 0.23999999999999999 < 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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. sub-neg 99.5%

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

      4. associate-*r/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if -0.640000000000000013 < t < 0.23999999999999999

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 100.0%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{t + 1}, 2 \cdot t, 2\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 100.0%

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

       2. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.64 \lor \neg \left(t \leq 0.24\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 + \frac{-1}{2 + t \cdot \left(t \cdot 4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.23)))
   (+
    1.0
    (-
     (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)
     0.16666666666666666))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 1.0d0 + ((((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t) - 0.16666666666666666d0)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.23):
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.23))
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.23)))
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.23]], $MachinePrecision]], N[(1.0 + N[(N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < 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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. sub-neg 99.5%

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

      4. associate-*r/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if -0.52000000000000002 < t < 0.23000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 100.0%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{t + 1}, 2 \cdot t, 2\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 100.0%

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

       2. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around 0 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.48)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.66)
     0.5
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.66) {
		tmp = 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 <= (-0.48d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.66d0) then
        tmp = 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 <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.48:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.48)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.66)
		tmp = 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 <= -0.48)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.48], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.66], 0.5, N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.47999999999999998

   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. Add Preprocessing

   3. Taylor expanded in t around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in t around 0 98.3%

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

      1. div-sub 98.3%

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

      2. associate-/l* 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. associate-*r/ 98.6%

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

      5. rgt-mult-inverse 98.7%

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

      6. metadata-eval 98.7%

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

   8. Simplified 98.7%

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

   if -0.47999999999999998 < t < 0.660000000000000031

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 100.0%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{t + 1}, 2 \cdot t, 2\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 100.0%

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

       2. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around 0 99.7%

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

   if 0.660000000000000031 < 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. Add Preprocessing

   3. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.48) (not (<= t 0.66)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.48d0)) .or. (.not. (t <= 0.66d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.48) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.48) || !(t <= 0.66))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.48) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.48], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.47999999999999998 or 0.660000000000000031 < 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. Add Preprocessing

   3. Taylor expanded in t around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in t around 0 99.0%

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

      1. div-sub 99.0%

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

      2. associate-/l* 99.4%

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

      3. *-rgt-identity 99.4%

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

      4. associate-*r/ 99.2%

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

      5. rgt-mult-inverse 99.4%

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

      6. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

   if -0.47999999999999998 < t < 0.660000000000000031

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 100.0%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{t + 1}, 2 \cdot t, 2\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 100.0%

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

       2. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around 0 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 2.6× 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%

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

   3. Taylor expanded in t around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in t around inf 97.0%

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

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 100.0%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{t + 1}, 2 \cdot t, 2\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 100.0%

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

       2. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around 0 99.7%

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

4. Final simplification 98.3%

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

Alternative 7: 59.9% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. div-inv 100.0%

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

   2. associate-/l* 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in t around 0 59.1%

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

   1. expm1-log1p-u 59.1%

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

   2. sub-neg 59.1%

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

   3. distribute-neg-frac 59.1%

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

   4. metadata-eval 59.1%

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

   5. +-commutative 59.1%

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

   6. *-commutative 59.1%

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

   7. fma-define 59.1%

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

   8. sub-neg 59.1%

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

   9. distribute-neg-frac 59.1%

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

   10. metadata-eval 59.1%

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

9. Applied egg-rr 59.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{t + 1}, 2 \cdot t, 2\right)}\right)\right)} \]
10. Step-by-step derivation

    1. expm1-undefine 59.1%

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

    2. sub-neg 59.1%

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

11. Simplified 59.1%

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

12. Taylor expanded in t around 0 58.1%

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

13. Final simplification 58.1%

    \[\leadsto 0.5 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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)))))))))