Kahan p13 Example 1

Percentage Accurate: 100.0% → 99.9%

Time: 17.3s

Alternatives: 6

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{\frac{t \cdot 4}{1 + t}}{1 + t}\\ \frac{1 + t_1}{t_1 + 2} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-*r* 100.0%

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

   6. *-commutative 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/l* 100.0%

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

   9. associate-*l/ 100.0%

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

   10. associate-/r/ 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \frac{1 + t \cdot \frac{\frac{t \cdot 4}{1 + t}}{1 + t}}{t \cdot \frac{\frac{t \cdot 4}{1 + t}}{1 + t} + 2} \]

Alternative 2: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.55 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \frac{t \cdot 4}{1 + t}}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.55) (not (<= t 0.75)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (/ (+ 1.0 (* t (/ (* t 4.0) (+ 1.0 t)))) (+ 2.0 (* t (* t 4.0))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.55) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + (t * ((t * 4.0) / (1.0 + t)))) / (2.0 + (t * (t * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.55d0)) .or. (.not. (t <= 0.75d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (1.0d0 + (t * ((t * 4.0d0) / (1.0d0 + t)))) / (2.0d0 + (t * (t * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.55) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + (t * ((t * 4.0) / (1.0 + t)))) / (2.0 + (t * (t * 4.0)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.55) or not (t <= 0.75):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (1.0 + (t * ((t * 4.0) / (1.0 + t)))) / (2.0 + (t * (t * 4.0)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.55) || !(t <= 0.75))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(1.0 + Float64(t * Float64(Float64(t * 4.0) / Float64(1.0 + t)))) / Float64(2.0 + Float64(t * Float64(t * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.55) || ~((t <= 0.75)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (1.0 + (t * ((t * 4.0) / (1.0 + t)))) / (2.0 + (t * (t * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.55], N[Not[LessEqual[t, 0.75]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t * N[(N[(t * 4.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.55000000000000004 or 0.75 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 97.9%

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

      1. associate-*r/ 97.9%

         \[\leadsto \frac{5 - \color{blue}{\frac{8 \cdot 1}{t}}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

      2. metadata-eval 97.9%

         \[\leadsto \frac{5 - \frac{\color{blue}{8}}{t}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

   6. Simplified 97.9%

      \[\leadsto \frac{\color{blue}{5 - \frac{8}{t}}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

   7. Taylor expanded in t around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   9. Simplified 98.2%

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

   if -0.55000000000000004 < t < 0.75

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.3%

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

   5. Taylor expanded in t around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. associate-*r* 99.3%

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

      4. *-commutative 99.3%

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

      5. distribute-lft-in 99.3%

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

      6. +-commutative 99.3%

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

      7. metadata-eval 99.3%

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

      8. metadata-eval 99.3%

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

      9. associate-*l* 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-*l* 99.3%

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

      12. *-commutative 99.3%

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

      13. neg-mul-1 99.3%

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

      14. distribute-lft-in 99.3%

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

      15. sub-neg 99.3%

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

      16. associate-*l* 99.3%

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

      17. *-commutative 99.3%

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

      18. associate-*l* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in t around 0 99.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.55 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \frac{t \cdot 4}{1 + t}}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (* t 4.0))))
   (if (or (<= t -0.6) (not (<= t 0.66)))
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (/ (+ 1.0 t_1) (+ 2.0 t_1)))))

C

double code(double t) {
	double t_1 = t * (t * 4.0);
	double tmp;
	if ((t <= -0.6) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (t * 4.0d0)
    if ((t <= (-0.6d0)) .or. (.not. (t <= 0.66d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (t * 4.0);
	double tmp;
	if ((t <= -0.6) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (t * 4.0)
	tmp = 0
	if (t <= -0.6) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(t * 4.0))
	tmp = 0.0
	if ((t <= -0.6) || !(t <= 0.66))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (t * 4.0);
	tmp = 0.0;
	if ((t <= -0.6) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -0.6], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.599999999999999978 or 0.660000000000000031 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 97.9%

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

      1. associate-*r/ 97.9%

         \[\leadsto \frac{5 - \color{blue}{\frac{8 \cdot 1}{t}}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

      2. metadata-eval 97.9%

         \[\leadsto \frac{5 - \frac{\color{blue}{8}}{t}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

   6. Simplified 97.9%

      \[\leadsto \frac{\color{blue}{5 - \frac{8}{t}}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

   7. Taylor expanded in t around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   9. Simplified 98.2%

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

   if -0.599999999999999978 < t < 0.660000000000000031

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.3%

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

   5. Taylor expanded in t around 0 99.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \]

Alternative 4: 99.1% accurate, 3.8× 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%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 97.9%

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

      1. associate-*r/ 97.9%

         \[\leadsto \frac{5 - \color{blue}{\frac{8 \cdot 1}{t}}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

      2. metadata-eval 97.9%

         \[\leadsto \frac{5 - \frac{\color{blue}{8}}{t}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

   6. Simplified 97.9%

      \[\leadsto \frac{\color{blue}{5 - \frac{8}{t}}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

   7. Taylor expanded in t around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   9. Simplified 98.2%

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

   if -0.47999999999999998 < t < 0.660000000000000031

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.5%

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

   5. Taylor expanded in t around 0 98.6%

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

4. Final simplification 98.4%

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

Alternative 5: 98.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.32:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.32) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.32) {
		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 <= (-0.32d0)) 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 <= -0.32) {
		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 <= -0.32:
		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 <= -0.32)
		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 <= -0.32)
		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, -0.32], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.320000000000000007 or 1 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 97.5%

      \[\leadsto \frac{\color{blue}{5}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]

   5. Taylor expanded in t around inf 97.7%

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

   if -0.320000000000000007 < t < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.5%

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

   5. Taylor expanded in t around 0 98.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.32:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 6: 59.3% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-*r* 100.0%

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

   6. *-commutative 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/l* 100.0%

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

   9. associate-*l/ 100.0%

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

   10. associate-/r/ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in t around 0 58.9%

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

5. Taylor expanded in t around 0 60.3%

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

6. Final simplification 60.3%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))