Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 9

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: 99.9% 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%

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

   2. associate-*r/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-*l/ 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-*r/ 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-*l/ 100.0%

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

   9. *-commutative 100.0%

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

   10. associate-*l* 100.0%

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

   11. metadata-eval 100.0%

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

   12. associate-/l* 100.0%

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

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: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t}\\ t_2 := \frac{4 \cdot \left(t \cdot t\right)}{1 + t}\\ \mathbf{if}\;t \leq -1.15:\\ \;\;\;\;t_1 + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{t \cdot t}}\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;\frac{1 + t_2}{2 + t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 0.037037037037037035 (* t t)))
        (t_2 (/ (* 4.0 (* t t)) (+ 1.0 t))))
   (if (<= t -1.15)
     (+ t_1 (/ 1.0 (+ (+ 1.2 (/ 0.32 t)) (/ 0.08533333333333333 (* t t)))))
     (if (<= t 0.6)
       (/ (+ 1.0 t_2) (+ 2.0 t_2))
       (+ t_1 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))))

C

double code(double t) {
	double t_1 = 0.037037037037037035 / (t * t);
	double t_2 = (4.0 * (t * t)) / (1.0 + t);
	double tmp;
	if (t <= -1.15) {
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))));
	} else if (t <= 0.6) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.037037037037037035d0 / (t * t)
    t_2 = (4.0d0 * (t * t)) / (1.0d0 + t)
    if (t <= (-1.15d0)) then
        tmp = t_1 + (1.0d0 / ((1.2d0 + (0.32d0 / t)) + (0.08533333333333333d0 / (t * t))))
    else if (t <= 0.6d0) then
        tmp = (1.0d0 + t_2) / (2.0d0 + t_2)
    else
        tmp = t_1 + (0.8333333333333334d0 - (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.037037037037037035 / (t * t);
	double t_2 = (4.0 * (t * t)) / (1.0 + t);
	double tmp;
	if (t <= -1.15) {
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))));
	} else if (t <= 0.6) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.037037037037037035 / (t * t)
	t_2 = (4.0 * (t * t)) / (1.0 + t)
	tmp = 0
	if t <= -1.15:
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))))
	elif t <= 0.6:
		tmp = (1.0 + t_2) / (2.0 + t_2)
	else:
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(0.037037037037037035 / Float64(t * t))
	t_2 = Float64(Float64(4.0 * Float64(t * t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -1.15)
		tmp = Float64(t_1 + Float64(1.0 / Float64(Float64(1.2 + Float64(0.32 / t)) + Float64(0.08533333333333333 / Float64(t * t)))));
	elseif (t <= 0.6)
		tmp = Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2));
	else
		tmp = Float64(t_1 + Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.037037037037037035 / (t * t);
	t_2 = (4.0 * (t * t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -1.15)
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))));
	elseif (t <= 0.6)
		tmp = (1.0 + t_2) / (2.0 + t_2);
	else
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.15], N[(t$95$1 + N[(1.0 / N[(N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision] + N[(0.08533333333333333 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.6], N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1499999999999999

   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. Taylor expanded in t around inf 98.2%

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

      1. associate--l+ 98.2%

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

      2. associate-*r/ 98.2%

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

      3. metadata-eval 98.2%

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

      4. unpow2 98.2%

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

      5. associate-*r/ 98.2%

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

      6. metadata-eval 98.2%

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

   4. Simplified 98.2%

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

      1. flip-- 98.2%

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

      2. clear-num 98.2%

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

      3. metadata-eval 96.8%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{\color{blue}{0.6944444444444444} - \frac{0.2222222222222222}{t} \cdot \frac{0.2222222222222222}{t}}} \]

      4. frac-times 96.8%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{0.6944444444444444 - \color{blue}{\frac{0.2222222222222222 \cdot 0.2222222222222222}{t \cdot t}}}} \]

      5. metadata-eval 96.8%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{0.6944444444444444 - \frac{\color{blue}{0.04938271604938271}}{t \cdot t}}} \]

   6. Applied egg-rr 96.8%

      \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \color{blue}{\frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{0.6944444444444444 - \frac{0.04938271604938271}{t \cdot t}}}} \]

   7. Taylor expanded in t around inf 98.3%

      \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\color{blue}{0.08533333333333333 \cdot \frac{1}{{t}^{2}} + \left(1.2 + 0.32 \cdot \frac{1}{t}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\color{blue}{\left(1.2 + 0.32 \cdot \frac{1}{t}\right) + 0.08533333333333333 \cdot \frac{1}{{t}^{2}}}} \]

      2. associate-*r/ 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}\right) + 0.08533333333333333 \cdot \frac{1}{{t}^{2}}} \]

      3. metadata-eval 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{\color{blue}{0.32}}{t}\right) + 0.08533333333333333 \cdot \frac{1}{{t}^{2}}} \]

      4. associate-*r/ 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \color{blue}{\frac{0.08533333333333333 \cdot 1}{{t}^{2}}}} \]

      5. metadata-eval 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{\color{blue}{0.08533333333333333}}{{t}^{2}}} \]

      6. unpow2 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{\color{blue}{t \cdot t}}} \]

   9. Simplified 98.3%

      \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\color{blue}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{t \cdot t}}} \]

   if -1.1499999999999999 < t < 0.599999999999999978

   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%

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

      2. associate-*r/ 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in t around 0 98.9%

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

      1. unpow2 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in t around 0 98.9%

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

       1. unpow2 98.9%

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

   11. Simplified 98.9%

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

   if 0.599999999999999978 < 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. Taylor expanded in t around inf 99.0%

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

      1. associate--l+ 99.0%

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

      2. associate-*r/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. unpow2 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

   4. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{t \cdot t}}\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;\frac{1 + \frac{4 \cdot \left(t \cdot t\right)}{1 + t}}{2 + \frac{4 \cdot \left(t \cdot t\right)}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t}\\ t_2 := t \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;t_1 + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{t \cdot t}}\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;\frac{1 + t_2}{2 + t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 0.037037037037037035 (* t t))) (t_2 (* t (* t 4.0))))
   (if (<= t -0.6)
     (+ t_1 (/ 1.0 (+ (+ 1.2 (/ 0.32 t)) (/ 0.08533333333333333 (* t t)))))
     (if (<= t 0.44)
       (/ (+ 1.0 t_2) (+ 2.0 t_2))
       (+ t_1 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))))

C

double code(double t) {
	double t_1 = 0.037037037037037035 / (t * t);
	double t_2 = t * (t * 4.0);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))));
	} else if (t <= 0.44) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.037037037037037035d0 / (t * t)
    t_2 = t * (t * 4.0d0)
    if (t <= (-0.6d0)) then
        tmp = t_1 + (1.0d0 / ((1.2d0 + (0.32d0 / t)) + (0.08533333333333333d0 / (t * t))))
    else if (t <= 0.44d0) then
        tmp = (1.0d0 + t_2) / (2.0d0 + t_2)
    else
        tmp = t_1 + (0.8333333333333334d0 - (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.037037037037037035 / (t * t);
	double t_2 = t * (t * 4.0);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))));
	} else if (t <= 0.44) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.037037037037037035 / (t * t)
	t_2 = t * (t * 4.0)
	tmp = 0
	if t <= -0.6:
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))))
	elif t <= 0.44:
		tmp = (1.0 + t_2) / (2.0 + t_2)
	else:
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(0.037037037037037035 / Float64(t * t))
	t_2 = Float64(t * Float64(t * 4.0))
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(t_1 + Float64(1.0 / Float64(Float64(1.2 + Float64(0.32 / t)) + Float64(0.08533333333333333 / Float64(t * t)))));
	elseif (t <= 0.44)
		tmp = Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2));
	else
		tmp = Float64(t_1 + Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.037037037037037035 / (t * t);
	t_2 = t * (t * 4.0);
	tmp = 0.0;
	if (t <= -0.6)
		tmp = t_1 + (1.0 / ((1.2 + (0.32 / t)) + (0.08533333333333333 / (t * t))));
	elseif (t <= 0.44)
		tmp = (1.0 + t_2) / (2.0 + t_2);
	else
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.6], N[(t$95$1 + N[(1.0 / N[(N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision] + N[(0.08533333333333333 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.44], N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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. Taylor expanded in t around inf 98.2%

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

      1. associate--l+ 98.2%

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

      2. associate-*r/ 98.2%

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

      3. metadata-eval 98.2%

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

      4. unpow2 98.2%

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

      5. associate-*r/ 98.2%

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

      6. metadata-eval 98.2%

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

   4. Simplified 98.2%

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

      1. flip-- 98.2%

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

      2. clear-num 98.2%

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

      3. metadata-eval 96.8%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{\color{blue}{0.6944444444444444} - \frac{0.2222222222222222}{t} \cdot \frac{0.2222222222222222}{t}}} \]

      4. frac-times 96.8%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{0.6944444444444444 - \color{blue}{\frac{0.2222222222222222 \cdot 0.2222222222222222}{t \cdot t}}}} \]

      5. metadata-eval 96.8%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{0.6944444444444444 - \frac{\color{blue}{0.04938271604938271}}{t \cdot t}}} \]

   6. Applied egg-rr 96.8%

      \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \color{blue}{\frac{1}{\frac{0.8333333333333334 + \frac{0.2222222222222222}{t}}{0.6944444444444444 - \frac{0.04938271604938271}{t \cdot t}}}} \]

   7. Taylor expanded in t around inf 98.3%

      \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\color{blue}{0.08533333333333333 \cdot \frac{1}{{t}^{2}} + \left(1.2 + 0.32 \cdot \frac{1}{t}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\color{blue}{\left(1.2 + 0.32 \cdot \frac{1}{t}\right) + 0.08533333333333333 \cdot \frac{1}{{t}^{2}}}} \]

      2. associate-*r/ 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}\right) + 0.08533333333333333 \cdot \frac{1}{{t}^{2}}} \]

      3. metadata-eval 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{\color{blue}{0.32}}{t}\right) + 0.08533333333333333 \cdot \frac{1}{{t}^{2}}} \]

      4. associate-*r/ 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \color{blue}{\frac{0.08533333333333333 \cdot 1}{{t}^{2}}}} \]

      5. metadata-eval 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{\color{blue}{0.08533333333333333}}{{t}^{2}}} \]

      6. unpow2 98.3%

         \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{\color{blue}{t \cdot t}}} \]

   9. Simplified 98.3%

      \[\leadsto \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\color{blue}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{t \cdot t}}} \]

   if -0.599999999999999978 < t < 0.440000000000000002

   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%

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

      2. associate-*r/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*l/ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l/ 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l* 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

   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 98.8%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 0.440000000000000002 < 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. Taylor expanded in t around inf 99.0%

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

      1. associate--l+ 99.0%

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

      2. associate-*r/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. unpow2 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

   4. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{t \cdot t}}\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;t \leq -0.62 \lor \neg \left(t \leq 0.44\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (* t 4.0))))
   (if (or (<= t -0.62) (not (<= t 0.44)))
     (+
      (/ 0.037037037037037035 (* t t))
      (- 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 <= -0.62) || !(t <= 0.44)) {
		tmp = (0.037037037037037035 / (t * t)) + (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 <= (-0.62d0)) .or. (.not. (t <= 0.44d0))) then
        tmp = (0.037037037037037035d0 / (t * t)) + (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 <= -0.62) || !(t <= 0.44)) {
		tmp = (0.037037037037037035 / (t * t)) + (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 <= -0.62) or not (t <= 0.44):
		tmp = (0.037037037037037035 / (t * t)) + (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 <= -0.62) || !(t <= 0.44))
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + 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 <= -0.62) || ~((t <= 0.44)))
		tmp = (0.037037037037037035 / (t * t)) + (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, -0.62], N[Not[LessEqual[t, 0.44]], $MachinePrecision]], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $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 < -0.619999999999999996 or 0.440000000000000002 < 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. Taylor expanded in t around inf 98.7%

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

      1. associate--l+ 98.7%

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

      2. associate-*r/ 98.7%

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

      3. metadata-eval 98.7%

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

      4. unpow2 98.7%

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

      5. associate-*r/ 98.7%

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

      6. metadata-eval 98.7%

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

   4. Simplified 98.7%

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

   if -0.619999999999999996 < t < 0.440000000000000002

   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%

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

      2. associate-*r/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*l/ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l/ 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l* 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

   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 98.8%

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

   5. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.62 \lor \neg \left(t \leq 0.44\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \]

Alternative 5: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.32\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.8) (not (<= t 0.32)))
   (+
    (/ 0.037037037037037035 (* t t))
    (- 0.8333333333333334 (/ 0.2222222222222222 t)))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.32)) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t));
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.8d0)) .or. (.not. (t <= 0.32d0))) then
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 - (0.2222222222222222d0 / t))
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.32)) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t));
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.8) or not (t <= 0.32):
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t))
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.8) || !(t <= 0.32))
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.8) || ~((t <= 0.32)))
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t));
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.8], N[Not[LessEqual[t, 0.32]], $MachinePrecision]], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.320000000000000007 < 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. Taylor expanded in t around inf 98.7%

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

      1. associate--l+ 98.7%

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

      2. associate-*r/ 98.7%

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

      3. metadata-eval 98.7%

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

      4. unpow2 98.7%

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

      5. associate-*r/ 98.7%

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

      6. metadata-eval 98.7%

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

   4. Simplified 98.7%

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

   if -0.80000000000000004 < t < 0.320000000000000007

   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. Taylor expanded in t around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. unpow2 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.32\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 6: 99.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.58)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.58)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.78d0)) .or. (.not. (t <= 0.58d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.58)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.78) or not (t <= 0.58):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.78) || !(t <= 0.58))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.78) || ~((t <= 0.58)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.78], N[Not[LessEqual[t, 0.58]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.78000000000000003 or 0.57999999999999996 < 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. Taylor expanded in t around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. metadata-eval 98.5%

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

   4. Simplified 98.5%

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

   if -0.78000000000000003 < t < 0.57999999999999996

   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. Taylor expanded in t around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. unpow2 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 7: 98.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.41:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.41)
   0.8333333333333334
   (if (<= t 0.58) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.41) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.41d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.41) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.41:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.41)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.41)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.41], 0.8333333333333334, If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.409999999999999976 or 0.57999999999999996 < 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. Taylor expanded in t around inf 97.9%

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

   if -0.409999999999999976 < t < 0.57999999999999996

   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. Taylor expanded in t around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. unpow2 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.41:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8: 98.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		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 <= (-0.34d0)) 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 <= -0.34) {
		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 <= -0.34:
		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 <= -0.34)
		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 <= -0.34)
		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, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 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. Taylor expanded in t around inf 97.9%

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

   if -0.340000000000000024 < 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. Taylor expanded in t around 0 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 9: 59.6% 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. Taylor expanded in t around 0 60.2%

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

3. Final simplification 60.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023216 
(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))))))