Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 17.4s

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ \frac{1 + \left(\left(\left(2 + {t_1}^{2}\right) + -1\right) + -1\right)}{2 + t_1 \cdot t_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(N[(N[(2.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $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. frac-times 76.9%

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

   2. *-commutative 76.9%

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

   3. *-commutative 76.9%

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

   4. swap-sqr 76.9%

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

   5. metadata-eval 76.9%

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

   6. associate-*r* 76.9%

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

   7. associate-/l/ 77.6%

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

   8. expm1-log1p-u 77.6%

      \[\leadsto \frac{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\right)\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   9. expm1-udef 77.0%

      \[\leadsto \frac{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\right)} - 1\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

3. Applied egg-rr 99.2%

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

   1. associate-*r/ 99.2%

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

   2. +-commutative 99.2%

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

   3. hypot-1-def 99.2%

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

   4. sqrt-pow2 100.0%

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

   5. metadata-eval 100.0%

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

   6. expm1-log1p-u 100.0%

      \[\leadsto \frac{1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}^{1}\right)\right)} - 1\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   7. pow1 100.0%

      \[\leadsto \frac{1 + \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\right)\right) - 1\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   8. expm1-udef 100.0%

      \[\leadsto \frac{1 + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)} - 1\right)} - 1\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ \frac{1 + \left(1 + \left({t_1}^{2} + -1\right)\right)}{2 + t_1 \cdot t_1} \end{array} \end{array} \]

FPCore

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

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	return (1.0 + (1.0 + (pow(t_1, 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
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    code = (1.0d0 + (1.0d0 + ((t_1 ** 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) / (1.0 + t);
	return (1.0 + (1.0 + (Math.pow(t_1, 2.0) + -1.0))) / (2.0 + (t_1 * t_1));
}

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	tmp = (1.0 + (1.0 + ((t_1 ^ 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[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(1.0 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $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. frac-times 76.9%

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

   2. *-commutative 76.9%

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

   3. *-commutative 76.9%

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

   4. swap-sqr 76.9%

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

   5. metadata-eval 76.9%

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

   6. associate-*r* 76.9%

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

   7. associate-/l/ 77.6%

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

   8. expm1-log1p-u 77.6%

      \[\leadsto \frac{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\right)\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   9. expm1-udef 77.0%

      \[\leadsto \frac{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\right)} - 1\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

3. Applied egg-rr 99.2%

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

   1. associate-*r/ 99.2%

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

   2. +-commutative 99.2%

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

   3. hypot-1-def 99.2%

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

   4. sqrt-pow2 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow1 100.0%

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

   7. associate--l+ 100.0%

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

   8. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+33}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 20000000:\\ \;\;\;\;\frac{1 + \frac{\left(t \cdot 4\right) \cdot \frac{t}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -2e+33)
   0.8333333333333334
   (if (<= t 20000000.0)
     (/
      (+ 1.0 (/ (* (* t 4.0) (/ t (+ 1.0 t))) (+ 1.0 t)))
      (+ 2.0 (/ (/ (* t (* t 4.0)) (+ 1.0 t)) (+ 1.0 t))))
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -2e+33) {
		tmp = 0.8333333333333334;
	} else if (t <= 20000000.0) {
		tmp = (1.0 + (((t * 4.0) * (t / (1.0 + t))) / (1.0 + t))) / (2.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)));
	} 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 <= (-2d+33)) then
        tmp = 0.8333333333333334d0
    else if (t <= 20000000.0d0) then
        tmp = (1.0d0 + (((t * 4.0d0) * (t / (1.0d0 + t))) / (1.0d0 + t))) / (2.0d0 + (((t * (t * 4.0d0)) / (1.0d0 + t)) / (1.0d0 + t)))
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -2e+33) {
		tmp = 0.8333333333333334;
	} else if (t <= 20000000.0) {
		tmp = (1.0 + (((t * 4.0) * (t / (1.0 + t))) / (1.0 + t))) / (2.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)));
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -2e+33:
		tmp = 0.8333333333333334
	elif t <= 20000000.0:
		tmp = (1.0 + (((t * 4.0) * (t / (1.0 + t))) / (1.0 + t))) / (2.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)))
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -2e+33)
		tmp = 0.8333333333333334;
	elseif (t <= 20000000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(t * 4.0) * Float64(t / Float64(1.0 + t))) / Float64(1.0 + t))) / Float64(2.0 + Float64(Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + t)) / Float64(1.0 + t))));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -2e+33)
		tmp = 0.8333333333333334;
	elseif (t <= 20000000.0)
		tmp = (1.0 + (((t * 4.0) * (t / (1.0 + t))) / (1.0 + t))) / (2.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)));
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -2e+33], 0.8333333333333334, If[LessEqual[t, 20000000.0], N[(N[(1.0 + N[(N[(N[(t * 4.0), $MachinePrecision] * N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9999999999999999e33

   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 100.0%

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

   if -1.9999999999999999e33 < t < 2e7

   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/ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. associate-*r* 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*l/ 99.9%

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

   3. Simplified 100.0%

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

      1. div-inv 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. div-inv 99.9%

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

      5. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 2e7 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+33}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 20000000:\\ \;\;\;\;\frac{1 + \frac{\left(t \cdot 4\right) \cdot \frac{t}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 5000000:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (/ (* t (* t 4.0)) (+ 1.0 t)) (+ 1.0 t))))
   (if (<= t -2e+154)
     0.8333333333333334
     (if (<= t 5000000.0)
       (/ (+ 1.0 t_1) (+ 2.0 t_1))
       (- 0.8333333333333334 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -2e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 5000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t * (t * 4.0d0)) / (1.0d0 + t)) / (1.0d0 + t)
    if (t <= (-2d+154)) then
        tmp = 0.8333333333333334d0
    else if (t <= 5000000.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -2e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 5000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)
	tmp = 0
	if t <= -2e+154:
		tmp = 0.8333333333333334
	elif t <= 5000000.0:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -2e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 5000000.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -2e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 5000000.0)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+154], 0.8333333333333334, If[LessEqual[t, 5000000.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.00000000000000007e154

   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 100.0%

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

   if -2.00000000000000007e154 < t < 5e6

   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/ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. associate-*r* 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*l/ 99.9%

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

   3. Simplified 100.0%

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

   if 5e6 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 5000000:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 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 6: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.56 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{t \cdot 4}{\frac{1 + t}{t}}}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.56) (not (<= t 0.75)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (/ (+ 1.0 (/ (* t 4.0) (/ (+ 1.0 t) t))) (+ 2.0 (* (* 2.0 t) (* 2.0 t))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.56) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + ((t * 4.0) / ((1.0 + t) / t))) / (2.0 + ((2.0 * t) * (2.0 * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.56d0)) .or. (.not. (t <= 0.75d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (1.0d0 + ((t * 4.0d0) / ((1.0d0 + t) / t))) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.56) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + ((t * 4.0) / ((1.0 + t) / t))) / (2.0 + ((2.0 * t) * (2.0 * t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.56) or not (t <= 0.75):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (1.0 + ((t * 4.0) / ((1.0 + t) / t))) / (2.0 + ((2.0 * t) * (2.0 * t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.56) || !(t <= 0.75))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t * 4.0) / Float64(Float64(1.0 + t) / t))) / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.56) || ~((t <= 0.75)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (1.0 + ((t * 4.0) / ((1.0 + t) / t))) / (2.0 + ((2.0 * t) * (2.0 * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.56], N[Not[LessEqual[t, 0.75]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t * 4.0), $MachinePrecision] / N[(N[(1.0 + t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.56000000000000005 or 0.75 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

   if -0.56000000000000005 < t < 0.75

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 98.5%

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

      1. *-commutative 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in t around 0 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in t around 0 98.9%

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

      1. *-commutative 98.5%

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

   10. Simplified 98.9%

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

       1. associate-*l/ 98.9%

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

       2. *-commutative 98.9%

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

       3. swap-sqr 98.9%

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

       4. metadata-eval 98.9%

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

       5. associate-*l* 98.9%

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

       6. *-commutative 98.9%

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

       7. associate-*r/ 98.9%

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

       8. clear-num 98.9%

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

       9. un-div-inv 98.9%

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

   12. Applied egg-rr 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.56 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{t \cdot 4}{\frac{1 + t}{t}}}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 7: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)\\ \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* (* 2.0 t) (* 2.0 t))))
   (if (or (<= t -0.58) (not (<= t 0.68)))
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (/ (+ 1.0 t_1) (+ 2.0 t_1)))))

C

double code(double t) {
	double t_1 = (2.0 * t) * (2.0 * t);
	double tmp;
	if ((t <= -0.58) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * t) * (2.0d0 * t)
    if ((t <= (-0.58d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) * (2.0 * t);
	double tmp;
	if ((t <= -0.58) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = (2.0 * t) * (2.0 * t)
	tmp = 0
	if (t <= -0.58) or not (t <= 0.68):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) * Float64(2.0 * t))
	tmp = 0.0
	if ((t <= -0.58) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (2.0 * t) * (2.0 * t);
	tmp = 0.0;
	if ((t <= -0.58) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -0.58], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

   if -0.57999999999999996 < t < 0.680000000000000049

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 98.5%

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

      1. *-commutative 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in t around 0 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in t around 0 98.9%

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

      1. *-commutative 98.5%

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

   10. Simplified 98.9%

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

   11. Taylor expanded in t around 0 98.5%

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

       1. *-commutative 98.5%

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

   13. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 8: 99.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.68):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 98.6% 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 97.8%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 97.7%

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

4. Final simplification 97.7%

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

Alternative 10: 59.5% 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 57.2%

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

3. Final simplification 57.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023314 
(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))))))