Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 14.1s

Alternatives: 10

Speedup: 1.0×

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: 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]]]

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. Final simplification 100.0%

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

Alternative 2: 75.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t (* t 4.0)) (* (+ 1.0 t) (+ 1.0 t)))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2.0)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (+
      (/ 0.037037037037037035 (* t t))
      (+ 0.8333333333333334 (/ -0.2222222222222222 t))))))

C

double code(double t) {
	double t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 2.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	}
	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)) / ((1.0d0 + t) * (1.0d0 + t))
    if (((2.0d0 * t) / (1.0d0 + t)) <= 2.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

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

Python

def code(t):
	t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t))
	tmp = 0
	if ((2.0 * t) / (1.0 + t)) <= 2.0:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(t * Float64(t * 4.0)) / Float64(Float64(1.0 + t) * Float64(1.0 + t)))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 2.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	tmp = 0.0;
	if (((2.0 * t) / (1.0 + t)) <= 2.0)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 2

   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. times-frac 73.9%

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

      2. sqr-neg 73.9%

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

      3. distribute-rgt-neg-out 73.9%

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

      4. distribute-rgt-neg-out 73.9%

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

      5. swap-sqr 73.9%

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

      6. *-commutative 73.9%

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

      7. sqr-neg 73.9%

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

      8. associate-*r* 73.9%

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

      9. metadata-eval 73.9%

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

      10. times-frac 73.9%

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

   3. Simplified 73.9%

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

   if 2 < (/.f64 (*.f64 2 t) (+.f64 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 62.4%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. associate--l+ 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{12 \cdot \frac{1}{{t}^{2}} + \left(6 - 8 \cdot \frac{1}{t}\right)}} \]

      3. unpow2 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{12 \cdot \frac{1}{\color{blue}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-*r/ 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      5. metadata-eval 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{\color{blue}{12}}{t \cdot t} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      6. sub-neg 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)}} \]

      7. associate-*r/ 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\color{blue}{\frac{8 \cdot 1}{t}}\right)\right)} \]

      8. metadata-eval 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\frac{\color{blue}{8}}{t}\right)\right)} \]

      9. distribute-neg-frac 62.4%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \color{blue}{\frac{-8}{t}}\right)} \]

      10. metadata-eval 62.4%

          \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \frac{\color{blue}{-8}}{t}\right)} \]

   4. Simplified 62.4%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12}{t \cdot t} + \left(6 + \frac{-8}{t}\right)}} \]

   5. Taylor expanded in t around inf 68.5%

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

      1. +-commutative 68.5%

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

      2. associate--l+ 68.5%

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

      3. associate-*r/ 68.5%

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

      4. metadata-eval 68.5%

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

      5. unpow2 68.5%

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

      6. sub-neg 68.5%

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

      7. associate-*r/ 68.5%

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

      8. metadata-eval 68.5%

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

      9. distribute-neg-frac 68.5%

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

      10. metadata-eval 68.5%

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

   7. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot \left(t \cdot 4\right)}{1 + 2 \cdot t}\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t (* t 4.0)) (+ 1.0 (* 2.0 t)))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.005)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (+
      (/ 0.037037037037037035 (* t t))
      (+ 0.8333333333333334 (/ -0.2222222222222222 t))))))

C

double code(double t) {
	double t_1 = (t * (t * 4.0)) / (1.0 + (2.0 * t));
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	}
	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)) / (1.0d0 + (2.0d0 * t))
    if (((2.0d0 * t) / (1.0d0 + t)) <= 0.005d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (t * (t * 4.0)) / (1.0 + (2.0 * t));
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = (t * (t * 4.0)) / (1.0 + (2.0 * t))
	tmp = 0
	if ((2.0 * t) / (1.0 + t)) <= 0.005:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + Float64(2.0 * t)))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.005)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (t * (t * 4.0)) / (1.0 + (2.0 * t));
	tmp = 0.0;
	if (((2.0 * t) / (1.0 + t)) <= 0.005)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 0.0050000000000000001

   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. times-frac 100.0%

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

      2. sqr-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. swap-sqr 100.0%

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

      6. *-commutative 100.0%

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

      7. sqr-neg 100.0%

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

      8. associate-*r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.8%

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

      1. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 99.8%

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

      1. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   if 0.0050000000000000001 < (/.f64 (*.f64 2 t) (+.f64 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 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. associate--l+ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{12 \cdot \frac{1}{{t}^{2}} + \left(6 - 8 \cdot \frac{1}{t}\right)}} \]

      3. unpow2 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{12 \cdot \frac{1}{\color{blue}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      5. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{\color{blue}{12}}{t \cdot t} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)}} \]

      7. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\color{blue}{\frac{8 \cdot 1}{t}}\right)\right)} \]

      8. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\frac{\color{blue}{8}}{t}\right)\right)} \]

      9. distribute-neg-frac 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \color{blue}{\frac{-8}{t}}\right)} \]

      10. metadata-eval 98.8%

          \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \frac{\color{blue}{-8}}{t}\right)} \]

   4. Simplified 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12}{t \cdot t} + \left(6 + \frac{-8}{t}\right)}} \]

   5. Taylor expanded in t around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. associate--l+ 99.0%

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

      3. associate-*r/ 99.0%

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

      4. metadata-eval 99.0%

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

      5. unpow2 99.0%

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

      6. sub-neg 99.0%

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

      7. associate-*r/ 99.0%

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

      8. metadata-eval 99.0%

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

      9. distribute-neg-frac 99.0%

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

      10. metadata-eval 99.0%

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

   7. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + \frac{t \cdot \left(t \cdot 4\right)}{1 + 2 \cdot t}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{1 + 2 \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + \frac{\left(t \cdot t\right) \cdot -4}{-1 - t}}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.005)
   (/ (+ 1.0 (/ (* (* t t) -4.0) (- -1.0 t))) (+ 2.0 (* 4.0 (* t t))))
   (+
    (/ 0.037037037037037035 (* t t))
    (+ 0.8333333333333334 (/ -0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + (((t * t) * -4.0) / (-1.0 - t))) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 * t) / (1.0d0 + t)) <= 0.005d0) then
        tmp = (1.0d0 + (((t * t) * (-4.0d0)) / ((-1.0d0) - t))) / (2.0d0 + (4.0d0 * (t * t)))
    else
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + (((t * t) * -4.0) / (-1.0 - t))) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 * t) / (1.0 + t)) <= 0.005:
		tmp = (1.0 + (((t * t) * -4.0) / (-1.0 - t))) / (2.0 + (4.0 * (t * t)))
	else:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.005)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(t * t) * -4.0) / Float64(-1.0 - t))) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	else
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 * t) / (1.0 + t)) <= 0.005)
		tmp = (1.0 + (((t * t) * -4.0) / (-1.0 - t))) / (2.0 + (4.0 * (t * t)));
	else
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(1.0 + N[(N[(N[(t * t), $MachinePrecision] * -4.0), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 0.0050000000000000001

   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 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in t around 0 99.8%

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

      1. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-commutative 99.8%

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

      3. frac-2neg 99.8%

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

      4. swap-sqr 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-*r* 99.8%

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

       2. unpow2 99.8%

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

       3. distribute-rgt-neg-in 99.8%

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

       4. unpow2 99.8%

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

       5. metadata-eval 99.8%

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

       6. +-commutative 99.8%

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

       7. unsub-neg 99.8%

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

   11. Simplified 99.8%

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

   if 0.0050000000000000001 < (/.f64 (*.f64 2 t) (+.f64 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 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. associate--l+ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{12 \cdot \frac{1}{{t}^{2}} + \left(6 - 8 \cdot \frac{1}{t}\right)}} \]

      3. unpow2 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{12 \cdot \frac{1}{\color{blue}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      5. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{\color{blue}{12}}{t \cdot t} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)}} \]

      7. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\color{blue}{\frac{8 \cdot 1}{t}}\right)\right)} \]

      8. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\frac{\color{blue}{8}}{t}\right)\right)} \]

      9. distribute-neg-frac 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \color{blue}{\frac{-8}{t}}\right)} \]

      10. metadata-eval 98.8%

          \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \frac{\color{blue}{-8}}{t}\right)} \]

   4. Simplified 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12}{t \cdot t} + \left(6 + \frac{-8}{t}\right)}} \]

   5. Taylor expanded in t around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. associate--l+ 99.0%

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

      3. associate-*r/ 99.0%

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

      4. metadata-eval 99.0%

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

      5. unpow2 99.0%

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

      6. sub-neg 99.0%

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

      7. associate-*r/ 99.0%

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

      8. metadata-eval 99.0%

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

      9. distribute-neg-frac 99.0%

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

      10. metadata-eval 99.0%

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

   7. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + \frac{\left(t \cdot t\right) \cdot -4}{-1 - t}}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 5: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* 4.0 (* t t))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.005)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (+
      (/ 0.037037037037037035 (* t t))
      (+ 0.8333333333333334 (/ -0.2222222222222222 t))))))

C

double code(double t) {
	double t_1 = 4.0 * (t * t);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 4.0d0 * (t * t)
    if (((2.0d0 * t) / (1.0d0 + t)) <= 0.005d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 4.0 * (t * t);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 4.0 * (t * t)
	tmp = 0
	if ((2.0 * t) / (1.0 + t)) <= 0.005:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(4.0 * Float64(t * t))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.005)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 4.0 * (t * t);
	tmp = 0.0;
	if (((2.0 * t) / (1.0 + t)) <= 0.005)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 0.0050000000000000001

   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 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in t around 0 99.7%

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

      1. unpow2 99.7%

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

   7. Simplified 99.7%

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

   if 0.0050000000000000001 < (/.f64 (*.f64 2 t) (+.f64 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 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. associate--l+ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{12 \cdot \frac{1}{{t}^{2}} + \left(6 - 8 \cdot \frac{1}{t}\right)}} \]

      3. unpow2 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{12 \cdot \frac{1}{\color{blue}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      5. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{\color{blue}{12}}{t \cdot t} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)}} \]

      7. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\color{blue}{\frac{8 \cdot 1}{t}}\right)\right)} \]

      8. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\frac{\color{blue}{8}}{t}\right)\right)} \]

      9. distribute-neg-frac 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \color{blue}{\frac{-8}{t}}\right)} \]

      10. metadata-eval 98.8%

          \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \frac{\color{blue}{-8}}{t}\right)} \]

   4. Simplified 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12}{t \cdot t} + \left(6 + \frac{-8}{t}\right)}} \]

   5. Taylor expanded in t around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. associate--l+ 99.0%

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

      3. associate-*r/ 99.0%

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

      4. metadata-eval 99.0%

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

      5. unpow2 99.0%

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

      6. sub-neg 99.0%

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

      7. associate-*r/ 99.0%

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

      8. metadata-eval 99.0%

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

      9. distribute-neg-frac 99.0%

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

      10. metadata-eval 99.0%

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

   7. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot t\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.8) (not (<= t 0.235)))
   (+
    (/ 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 <= 0.235)) {
		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 <= 0.235d0))) 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 <= 0.235)) {
		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 <= 0.235):
		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 <= 0.235))
		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 <= 0.235)))
		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, 0.235]], $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 0.23499999999999999 < 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 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. associate--l+ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{12 \cdot \frac{1}{{t}^{2}} + \left(6 - 8 \cdot \frac{1}{t}\right)}} \]

      3. unpow2 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{12 \cdot \frac{1}{\color{blue}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      5. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{\color{blue}{12}}{t \cdot t} + \left(6 - 8 \cdot \frac{1}{t}\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)}} \]

      7. associate-*r/ 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\color{blue}{\frac{8 \cdot 1}{t}}\right)\right)} \]

      8. metadata-eval 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \left(-\frac{\color{blue}{8}}{t}\right)\right)} \]

      9. distribute-neg-frac 98.8%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \color{blue}{\frac{-8}{t}}\right)} \]

      10. metadata-eval 98.8%

          \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\frac{12}{t \cdot t} + \left(6 + \frac{\color{blue}{-8}}{t}\right)} \]

   4. Simplified 98.8%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{12}{t \cdot t} + \left(6 + \frac{-8}{t}\right)}} \]

   5. Taylor expanded in t around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. associate--l+ 99.0%

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

      3. associate-*r/ 99.0%

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

      4. metadata-eval 99.0%

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

      5. unpow2 99.0%

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

      6. sub-neg 99.0%

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

      7. associate-*r/ 99.0%

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

      8. metadata-eval 99.0%

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

      9. distribute-neg-frac 99.0%

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

      10. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

   if -0.80000000000000004 < t < 0.23499999999999999

   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 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in t around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 7: 99.1% 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. Step-by-step derivation

      1. times-frac 47.6%

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

      2. sqr-neg 47.6%

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

      3. distribute-rgt-neg-out 47.6%

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

      4. distribute-rgt-neg-out 47.6%

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

      5. swap-sqr 47.6%

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

      6. *-commutative 47.6%

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

      7. sqr-neg 47.6%

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

      8. associate-*r* 47.6%

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

      9. metadata-eval 47.6%

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

      10. times-frac 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in t around inf 46.2%

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

      1. unpow2 46.2%

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

      2. distribute-rgt-out 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in t around inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

   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 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in t around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.2%

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

Alternative 8: 98.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.58) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		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 <= (-0.9d0)) 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 <= -0.9) {
		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 <= -0.9:
		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 <= -0.9)
		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 <= -0.9)
		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, -0.9], 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 < -0.900000000000000022 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. Step-by-step derivation

      1. times-frac 47.6%

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

      2. sqr-neg 47.6%

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

      3. distribute-rgt-neg-out 47.6%

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

      4. distribute-rgt-neg-out 47.6%

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

      5. swap-sqr 47.6%

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

      6. *-commutative 47.6%

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

      7. sqr-neg 47.6%

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

      8. associate-*r* 47.6%

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

      9. metadata-eval 47.6%

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

      10. times-frac 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in t around inf 46.2%

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

      1. unpow2 46.2%

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

      2. distribute-rgt-out 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in t around inf 98.0%

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

   if -0.900000000000000022 < 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 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in t around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 9: 98.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

      1. times-frac 47.6%

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

      2. sqr-neg 47.6%

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

      3. distribute-rgt-neg-out 47.6%

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

      4. distribute-rgt-neg-out 47.6%

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

      5. swap-sqr 47.6%

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

      6. *-commutative 47.6%

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

      7. sqr-neg 47.6%

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

      8. associate-*r* 47.6%

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

      9. metadata-eval 47.6%

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

      10. times-frac 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in t around inf 46.2%

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

      1. unpow2 46.2%

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

      2. distribute-rgt-out 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in t around inf 98.0%

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

   if -0.330000000000000016 < 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. times-frac 100.0%

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

      2. sqr-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. swap-sqr 100.0%

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

      6. *-commutative 100.0%

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

      7. sqr-neg 100.0%

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

      8. associate-*r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.5%

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

      1. unpow2 98.5%

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

      2. distribute-rgt-out 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in t around 0 99.4%

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

4. Final simplification 98.7%

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

Alternative 10: 59.1% 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. times-frac 74.2%

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

   2. sqr-neg 74.2%

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

   3. distribute-rgt-neg-out 74.2%

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

   4. distribute-rgt-neg-out 74.2%

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

   5. swap-sqr 74.2%

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

   6. *-commutative 74.2%

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

   7. sqr-neg 74.2%

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

   8. associate-*r* 74.2%

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

   9. metadata-eval 74.2%

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

   10. times-frac 74.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in t around inf 72.8%

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

   1. unpow2 72.8%

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

   2. distribute-rgt-out 72.8%

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

6. Simplified 72.8%

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

7. Taylor expanded in t around 0 60.1%

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

8. Final simplification 60.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023293 
(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))))))