Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 34.0s

Alternatives: 7

Speedup: 1.1×

• could not determine a ground truth

  1. t = -6.345523484422262e+49

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{t + 1}\\ t_2 := 2 \cdot \frac{t}{t + 1}\\ \frac{\frac{{t_2}^{4} + -1}{{t_2}^{2} + -1}}{2 + t_1 \cdot t_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ t 1.0))) (t_2 (* 2.0 (/ t (+ t 1.0)))))
   (/ (/ (+ (pow t_2 4.0) -1.0) (+ (pow t_2 2.0) -1.0)) (+ 2.0 (* t_1 t_1)))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (t + 1.0);
	double t_2 = 2.0 * (t / (t + 1.0));
	return ((pow(t_2, 4.0) + -1.0) / (pow(t_2, 2.0) + -1.0)) / (2.0 + (t_1 * t_1));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (2.0d0 * t) / (t + 1.0d0)
    t_2 = 2.0d0 * (t / (t + 1.0d0))
    code = (((t_2 ** 4.0d0) + (-1.0d0)) / ((t_2 ** 2.0d0) + (-1.0d0))) / (2.0d0 + (t_1 * t_1))
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (t + 1.0);
	double t_2 = 2.0 * (t / (t + 1.0));
	return ((Math.pow(t_2, 4.0) + -1.0) / (Math.pow(t_2, 2.0) + -1.0)) / (2.0 + (t_1 * t_1));
}

Python

def code(t):
	t_1 = (2.0 * t) / (t + 1.0)
	t_2 = 2.0 * (t / (t + 1.0))
	return ((math.pow(t_2, 4.0) + -1.0) / (math.pow(t_2, 2.0) + -1.0)) / (2.0 + (t_1 * t_1))

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(t + 1.0))
	t_2 = Float64(2.0 * Float64(t / Float64(t + 1.0)))
	return Float64(Float64(Float64((t_2 ^ 4.0) + -1.0) / Float64((t_2 ^ 2.0) + -1.0)) / Float64(2.0 + Float64(t_1 * t_1)))
end

MATLAB

function tmp = code(t)
	t_1 = (2.0 * t) / (t + 1.0);
	t_2 = 2.0 * (t / (t + 1.0));
	tmp = (((t_2 ^ 4.0) + -1.0) / ((t_2 ^ 2.0) + -1.0)) / (2.0 + (t_1 * t_1));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Power[t$95$2, 4.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Power[t$95$2, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $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. +-commutative 100.0%

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

   2. flip-+ 100.0%

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

   3. pow2 100.0%

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

   4. pow2 100.0%

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

   5. pow-prod-up 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. times-frac 100.0%

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

   8. metadata-eval 100.0%

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

   9. +-commutative 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ t 1.0))) (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) / (t + 1.0);
	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) / (t + 1.0d0)
    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) / (t + 1.0);
	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) / (t + 1.0)
	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(t + 1.0))
	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) / (t + 1.0);
	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[(t + 1.0), $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%

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

Alternative 3: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = t * (((t * 4.0) / (t + 1.0)) / (t + 1.0));
	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) / (t + 1.0d0)) / (t + 1.0d0))
    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) / (t + 1.0)) / (t + 1.0));
	return (1.0 + t_1) / (2.0 + t_1);
}

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = t * (((t * 4.0) / (t + 1.0)) / (t + 1.0));
	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[(t + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $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%

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

Alternative 4: 97.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;t \leq 0.24:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (* t 4.0))))
   (if (<= t 0.24)
     (/ (+ 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 (t <= 0.24) {
		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 (t <= 0.24d0) 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 (t <= 0.24) {
		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 t <= 0.24:
		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 (t <= 0.24)
		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 (t <= 0.24)
		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[t, 0.24], 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 t < 0.23999999999999999

   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 97.0%

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

   5. Taylor expanded in t around 0 97.3%

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

   if 0.23999999999999999 < 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 94.1%

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

      1. associate--l+ 94.1%

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

      2. associate-*r/ 94.1%

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

      3. metadata-eval 94.1%

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

      4. unpow2 94.1%

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

      5. associate-*r/ 94.1%

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

      6. metadata-eval 94.1%

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

   4. Simplified 94.1%

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

   5. Taylor expanded in t around 0 94.1%

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

      1. associate--l+ 94.1%

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

      2. sub-neg 94.1%

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

      3. associate-*r/ 94.1%

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

      4. metadata-eval 94.1%

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

      5. unpow2 94.1%

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

      6. associate-*r/ 94.1%

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

      7. metadata-eval 94.1%

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

      8. distribute-neg-frac 94.1%

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

      9. metadata-eval 94.1%

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

   7. Simplified 94.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.24:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.24:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t 0.24)
   (+ (* t t) 0.5)
   (+
    0.8333333333333334
    (+ (/ 0.037037037037037035 (* t t)) (/ -0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= 0.24) {
		tmp = (t * t) + 0.5;
	} 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) :: tmp
    if (t <= 0.24d0) then
        tmp = (t * t) + 0.5d0
    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 tmp;
	if (t <= 0.24) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 + ((0.037037037037037035 / (t * t)) + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= 0.24:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334 + ((0.037037037037037035 / (t * t)) + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= 0.24)
		tmp = Float64(Float64(t * t) + 0.5);
	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)
	tmp = 0.0;
	if (t <= 0.24)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334 + ((0.037037037037037035 / (t * t)) + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, 0.24], N[(N[(t * t), $MachinePrecision] + 0.5), $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 t < 0.23999999999999999

   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 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

   4. Simplified 97.0%

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

   if 0.23999999999999999 < 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 94.1%

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

      1. associate--l+ 94.1%

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

      2. associate-*r/ 94.1%

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

      3. metadata-eval 94.1%

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

      4. unpow2 94.1%

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

      5. associate-*r/ 94.1%

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

      6. metadata-eval 94.1%

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

   4. Simplified 94.1%

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

   5. Taylor expanded in t around 0 94.1%

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

      1. associate--l+ 94.1%

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

      2. sub-neg 94.1%

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

      3. associate-*r/ 94.1%

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

      4. metadata-eval 94.1%

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

      5. unpow2 94.1%

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

      6. associate-*r/ 94.1%

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

      7. metadata-eval 94.1%

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

      8. distribute-neg-frac 94.1%

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

      9. metadata-eval 94.1%

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

   7. Simplified 94.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.24:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 6: 96.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t 0.56)
   (+ (* t t) 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 0.56d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= 0.56)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, 0.56], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if 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. Taylor expanded in t around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

   4. Simplified 97.0%

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

   if 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 88.1%

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

      1. associate-*r/ 88.1%

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

      2. metadata-eval 88.1%

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

   4. Simplified 88.1%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 7: 95.1% 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. Taylor expanded in t around 0 95.2%

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

3. Final simplification 95.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023278 
(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))))))