Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 12.7s

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: 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, 1.0× 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: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 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[(0.8333333333333334 + N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + 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. 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 99.7%

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

      1. unpow2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.7%

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

       1. unpow2 99.7%

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

   11. Simplified 99.7%

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

   if 0.0050000000000000001 < (/.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 98.5%

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

      1. associate--l+ 98.5%

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

      2. associate-*r/ 98.5%

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

      3. metadata-eval 98.5%

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

      4. unpow2 98.5%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in t around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate--l+ 98.9%

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

      5. sub-neg 98.9%

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

      6. associate-*r/ 98.9%

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

      7. metadata-eval 98.9%

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

      8. unpow2 98.9%

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

      9. distribute-neg-frac 98.9%

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

      10. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.3%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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) / (2.0d0 + t_1)
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) / (2.0 + t_1);
}

Python

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

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(2.0 + t_1))
end

MATLAB

function tmp = code(t)
	t_1 = t * (((t * 4.0) / (1.0 + t)) / (1.0 + t));
	tmp = (1.0 + t_1) / (2.0 + t_1);
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[(2.0 + t$95$1), $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%

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

Alternative 4: 99.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(t)
	t_1 = Float64(t * Float64(t * 4.0))
	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(0.8333333333333334 + Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(t * 4.0), $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[(0.8333333333333334 + N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + 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. 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 99.7%

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

   5. Taylor expanded in t around 0 99.7%

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

   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 98.5%

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

      1. associate--l+ 98.5%

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

      2. associate-*r/ 98.5%

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

      3. metadata-eval 98.5%

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

      4. unpow2 98.5%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in t around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate--l+ 98.9%

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

      5. sub-neg 98.9%

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

      6. associate-*r/ 98.9%

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

      7. metadata-eval 98.9%

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

      8. unpow2 98.9%

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

      9. distribute-neg-frac 98.9%

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

      10. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.3%

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

Alternative 5: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.82) (not (<= t 0.235)))
   (+
    0.8333333333333334
    (+ (/ 0.037037037037037035 (* t t)) (/ -0.2222222222222222 t)))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.82) || !(t <= 0.235)) {
		tmp = 0.8333333333333334 + ((0.037037037037037035 / (t * t)) + (-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 <= (-0.82d0)) .or. (.not. (t <= 0.235d0))) then
        tmp = 0.8333333333333334d0 + ((0.037037037037037035d0 / (t * t)) + ((-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 <= -0.82) || !(t <= 0.235)) {
		tmp = 0.8333333333333334 + ((0.037037037037037035 / (t * t)) + (-0.2222222222222222 / t));
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.82) or not (t <= 0.235):
		tmp = 0.8333333333333334 + ((0.037037037037037035 / (t * t)) + (-0.2222222222222222 / t))
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.82) || !(t <= 0.235))
		tmp = Float64(0.8333333333333334 + Float64(Float64(0.037037037037037035 / Float64(t * t)) + 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 <= -0.82) || ~((t <= 0.235)))
		tmp = 0.8333333333333334 + ((0.037037037037037035 / (t * t)) + (-0.2222222222222222 / t));
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.82], N[Not[LessEqual[t, 0.235]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + 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 < -0.819999999999999951 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 98.5%

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

      1. associate--l+ 98.5%

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

      2. associate-*r/ 98.5%

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

      3. metadata-eval 98.5%

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

      4. unpow2 98.5%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in t around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate--l+ 98.9%

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

      5. sub-neg 98.9%

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

      6. associate-*r/ 98.9%

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

      7. metadata-eval 98.9%

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

      8. unpow2 98.9%

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

      9. distribute-neg-frac 98.9%

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

      10. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

   if -0.819999999999999951 < 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. 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 99.7%

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

   5. Taylor expanded in t around 0 99.7%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 6: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t_1 + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(t_1 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 0.037037037037037035 (* t t))))
   (if (<= t -0.82)
     (+ t_1 (- 0.8333333333333334 (/ 0.2222222222222222 t)))
     (if (<= t 0.235)
       (+ (* t t) 0.5)
       (+ 0.8333333333333334 (+ t_1 (/ -0.2222222222222222 t)))))))

C

double code(double t) {
	double t_1 = 0.037037037037037035 / (t * t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 + (t_1 + (-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 = 0.037037037037037035d0 / (t * t)
    if (t <= (-0.82d0)) then
        tmp = t_1 + (0.8333333333333334d0 - (0.2222222222222222d0 / t))
    else if (t <= 0.235d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0 + (t_1 + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.037037037037037035 / (t * t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 + (t_1 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.037037037037037035 / (t * t)
	tmp = 0
	if t <= -0.82:
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t))
	elif t <= 0.235:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334 + (t_1 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(0.037037037037037035 / Float64(t * t))
	tmp = 0.0
	if (t <= -0.82)
		tmp = Float64(t_1 + Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)));
	elseif (t <= 0.235)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(t_1 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.037037037037037035 / (t * t);
	tmp = 0.0;
	if (t <= -0.82)
		tmp = t_1 + (0.8333333333333334 - (0.2222222222222222 / t));
	elseif (t <= 0.235)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334 + (t_1 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.82], N[(t$95$1 + N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.235], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(t$95$1 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.819999999999999951

   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 98.9%

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

      1. associate--l+ 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. unpow2 98.9%

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

      5. associate-*r/ 98.9%

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

      6. metadata-eval 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in t around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

      3. associate-+r- 99.3%

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

      4. associate-*r/ 99.3%

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

      5. metadata-eval 99.3%

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

      6. unpow2 99.3%

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

   7. Simplified 99.3%

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

   if -0.819999999999999951 < 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. 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 99.7%

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

   5. Taylor expanded in t around 0 99.7%

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

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

   if 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 98.0%

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

      1. associate--l+ 98.1%

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

      2. associate-*r/ 98.1%

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

      3. metadata-eval 98.1%

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

      4. unpow2 98.1%

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

      5. associate-*r/ 98.1%

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

      6. metadata-eval 98.1%

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

   4. Simplified 98.1%

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

   5. Taylor expanded in t around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-*r/ 98.5%

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

      3. metadata-eval 98.5%

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

      4. associate--l+ 98.5%

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

      5. sub-neg 98.5%

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

      6. associate-*r/ 98.5%

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

      7. metadata-eval 98.5%

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

      8. unpow2 98.5%

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

      9. distribute-neg-frac 98.5%

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

      10. metadata-eval 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 7: 99.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.56)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.56)) {
		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 <= 0.56d0))) 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 <= 0.56)) {
		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 <= 0.56):
		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 <= 0.56))
		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 <= 0.56)))
		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, 0.56]], $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 0.56000000000000005 < 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 97.6%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 97.6%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 97.6%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \frac{\color{blue}{8}}{t}} \]

   4. Simplified 97.6%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - \frac{8}{t}}} \]

   5. Taylor expanded in t around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

   if -0.78000000000000003 < t < 0.56000000000000005

   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 99.7%

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

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   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 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 8: 98.7% 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. Taylor expanded in t around inf 95.9%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6}} \]

   3. Taylor expanded in t around inf 96.2%

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

   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. 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 99.7%

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

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   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 98.1%

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

Alternative 9: 98.5% 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. Taylor expanded in t around inf 95.9%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6}} \]

   3. Taylor expanded in t around inf 96.2%

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

   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. 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 99.7%

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

   5. Taylor expanded in t around 0 99.6%

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

4. Final simplification 98.0%

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

Alternative 10: 59.7% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   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 55.9%

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

5. Taylor expanded in t around 0 62.8%

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

6. Final simplification 62.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023224 
(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))))))