Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 14.0s

Alternatives: 5

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{4}{t + 1} + -8}{t + 1}\\ \frac{5 + t\_1}{t\_1 + 6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ 4.0 (+ t 1.0)) -8.0) (+ t 1.0))))
   (/ (+ 5.0 t_1) (+ t_1 6.0))))

C

double code(double t) {
	double t_1 = ((4.0 / (t + 1.0)) + -8.0) / (t + 1.0);
	return (5.0 + t_1) / (t_1 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((4.0d0 / (t + 1.0d0)) + (-8.0d0)) / (t + 1.0d0)
    code = (5.0d0 + t_1) / (t_1 + 6.0d0)
end function

Java

public static double code(double t) {
	double t_1 = ((4.0 / (t + 1.0)) + -8.0) / (t + 1.0);
	return (5.0 + t_1) / (t_1 + 6.0);
}

Python

def code(t):
	t_1 = ((4.0 / (t + 1.0)) + -8.0) / (t + 1.0)
	return (5.0 + t_1) / (t_1 + 6.0)

Julia

function code(t)
	t_1 = Float64(Float64(Float64(4.0 / Float64(t + 1.0)) + -8.0) / Float64(t + 1.0))
	return Float64(Float64(5.0 + t_1) / Float64(t_1 + 6.0))
end

MATLAB

function tmp = code(t)
	t_1 = ((4.0 / (t + 1.0)) + -8.0) / (t + 1.0);
	tmp = (5.0 + t_1) / (t_1 + 6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(t$95$1 + 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 100.0%

      \[\leadsto \frac{5 + \color{blue}{\log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{5 + \log \color{blue}{\left(1 \cdot e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   3. log-prod 100.0%

      \[\leadsto \frac{5 + \color{blue}{\left(\log 1 + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{5 + \left(\color{blue}{0} + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. associate-*l/ 100.0%

      \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2 \cdot \left(\frac{2}{1 + t} - 4\right)}{1 + t}}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   7. sub-neg 100.0%

      \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \color{blue}{\left(\frac{2}{1 + t} + \left(-4\right)\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   8. distribute-lft-in 100.0%

      \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + 2 \cdot \left(-4\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   9. metadata-eval 100.0%

      \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + 2 \cdot \color{blue}{-4}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   10. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{-8}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{{-2}^{3}}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   12. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + {\color{blue}{\left(-2\right)}}^{3}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   13. fma-define 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\left(-2\right)}^{3}\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   14. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\color{blue}{-2}}^{3}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   15. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \color{blue}{-8}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{5 + \color{blue}{\left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{5 + \color{blue}{\frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   2. fma-undefine 100.0%

      \[\leadsto \frac{5 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + -8}}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{5 + \frac{\color{blue}{\frac{2 \cdot 2}{1 + t}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{5 + \frac{\frac{\color{blue}{4}}{1 + t} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   5. +-commutative 100.0%

      \[\leadsto \frac{5 + \frac{\frac{4}{\color{blue}{t + 1}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. +-commutative 100.0%

      \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{\color{blue}{t + 1}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

8. Simplified 100.0%

   \[\leadsto \frac{5 + \color{blue}{\frac{\frac{4}{t + 1} + -8}{t + 1}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
9. Step-by-step derivation

   1. add-log-exp 100.0%

      \[\leadsto \frac{5 + \color{blue}{\log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{5 + \log \color{blue}{\left(1 \cdot e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   3. log-prod 100.0%

      \[\leadsto \frac{5 + \color{blue}{\left(\log 1 + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{5 + \left(\color{blue}{0} + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. associate-*l/ 100.0%

      \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2 \cdot \left(\frac{2}{1 + t} - 4\right)}{1 + t}}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   7. sub-neg 100.0%

      \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \color{blue}{\left(\frac{2}{1 + t} + \left(-4\right)\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   8. distribute-lft-in 100.0%

      \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + 2 \cdot \left(-4\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   9. metadata-eval 100.0%

      \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + 2 \cdot \color{blue}{-4}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   10. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{-8}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{{-2}^{3}}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   12. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + {\color{blue}{\left(-2\right)}}^{3}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   13. fma-define 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\left(-2\right)}^{3}\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   14. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\color{blue}{-2}}^{3}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   15. metadata-eval 100.0%

       \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \color{blue}{-8}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{t + 1}}{6 + \color{blue}{\left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}\right)}} \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

       \[\leadsto \frac{5 + \color{blue}{\frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

    2. fma-undefine 100.0%

       \[\leadsto \frac{5 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + -8}}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

    3. associate-*r/ 100.0%

       \[\leadsto \frac{5 + \frac{\color{blue}{\frac{2 \cdot 2}{1 + t}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{5 + \frac{\frac{\color{blue}{4}}{1 + t} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

    5. +-commutative 100.0%

       \[\leadsto \frac{5 + \frac{\frac{4}{\color{blue}{t + 1}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

    6. +-commutative 100.0%

       \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{\color{blue}{t + 1}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

12. Simplified 100.0%

    \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{t + 1}}{6 + \color{blue}{\frac{\frac{4}{t + 1} + -8}{t + 1}}} \]

13. Final simplification 100.0%

    \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{t + 1}}{\frac{\frac{4}{t + 1} + -8}{t + 1} + 6} \]
14. Add Preprocessing

Alternative 2: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-8 + \frac{4}{t}}{t + 1}\\ \mathbf{if}\;t \leq -0.34 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;\frac{5 + t\_1}{6 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot \left(-2 + t \cdot 2\right)\right) \cdot \left(\frac{2}{t + 1} - 4\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ -8.0 (/ 4.0 t)) (+ t 1.0))))
   (if (or (<= t -0.34) (not (<= t 0.56)))
     (/ (+ 5.0 t_1) (+ 6.0 t_1))
     (/
      (+ 5.0 (* (+ 2.0 (* t (+ -2.0 (* t 2.0)))) (- (/ 2.0 (+ t 1.0)) 4.0)))
      2.0))))

C

double code(double t) {
	double t_1 = (-8.0 + (4.0 / t)) / (t + 1.0);
	double tmp;
	if ((t <= -0.34) || !(t <= 0.56)) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else {
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-8.0d0) + (4.0d0 / t)) / (t + 1.0d0)
    if ((t <= (-0.34d0)) .or. (.not. (t <= 0.56d0))) then
        tmp = (5.0d0 + t_1) / (6.0d0 + t_1)
    else
        tmp = (5.0d0 + ((2.0d0 + (t * ((-2.0d0) + (t * 2.0d0)))) * ((2.0d0 / (t + 1.0d0)) - 4.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (-8.0 + (4.0 / t)) / (t + 1.0);
	double tmp;
	if ((t <= -0.34) || !(t <= 0.56)) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else {
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (-8.0 + (4.0 / t)) / (t + 1.0)
	tmp = 0
	if (t <= -0.34) or not (t <= 0.56):
		tmp = (5.0 + t_1) / (6.0 + t_1)
	else:
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(-8.0 + Float64(4.0 / t)) / Float64(t + 1.0))
	tmp = 0.0
	if ((t <= -0.34) || !(t <= 0.56))
		tmp = Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1));
	else
		tmp = Float64(Float64(5.0 + Float64(Float64(2.0 + Float64(t * Float64(-2.0 + Float64(t * 2.0)))) * Float64(Float64(2.0 / Float64(t + 1.0)) - 4.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (-8.0 + (4.0 / t)) / (t + 1.0);
	tmp = 0.0;
	if ((t <= -0.34) || ~((t <= 0.56)))
		tmp = (5.0 + t_1) / (6.0 + t_1);
	else
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-8.0 + N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -0.34], N[Not[LessEqual[t, 0.56]], $MachinePrecision]], N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 + N[(N[(2.0 + N[(t * N[(-2.0 + N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 0.56000000000000005 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \frac{5 + \color{blue}{\log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \frac{5 + \log \color{blue}{\left(1 \cdot e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      3. log-prod 100.0%

         \[\leadsto \frac{5 + \color{blue}{\left(\log 1 + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{5 + \left(\color{blue}{0} + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      5. add-log-exp 100.0%

         \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      6. associate-*l/ 100.0%

         \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2 \cdot \left(\frac{2}{1 + t} - 4\right)}{1 + t}}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      7. sub-neg 100.0%

         \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \color{blue}{\left(\frac{2}{1 + t} + \left(-4\right)\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      8. distribute-lft-in 100.0%

         \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + 2 \cdot \left(-4\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + 2 \cdot \color{blue}{-4}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{-8}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{{-2}^{3}}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + {\color{blue}{\left(-2\right)}}^{3}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      13. fma-define 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\left(-2\right)}^{3}\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\color{blue}{-2}}^{3}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \color{blue}{-8}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{5 + \color{blue}{\left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{5 + \color{blue}{\frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. fma-undefine 100.0%

         \[\leadsto \frac{5 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + -8}}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{5 + \frac{\color{blue}{\frac{2 \cdot 2}{1 + t}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{\frac{\color{blue}{4}}{1 + t} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{5 + \frac{\frac{4}{\color{blue}{t + 1}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      6. +-commutative 100.0%

         \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{\color{blue}{t + 1}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   8. Simplified 100.0%

      \[\leadsto \frac{5 + \color{blue}{\frac{\frac{4}{t + 1} + -8}{t + 1}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
   9. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \frac{5 + \color{blue}{\log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \frac{5 + \log \color{blue}{\left(1 \cdot e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      3. log-prod 100.0%

         \[\leadsto \frac{5 + \color{blue}{\left(\log 1 + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{5 + \left(\color{blue}{0} + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      5. add-log-exp 100.0%

         \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      6. associate-*l/ 100.0%

         \[\leadsto \frac{5 + \left(0 + \color{blue}{\frac{2 \cdot \left(\frac{2}{1 + t} - 4\right)}{1 + t}}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      7. sub-neg 100.0%

         \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \color{blue}{\left(\frac{2}{1 + t} + \left(-4\right)\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      8. distribute-lft-in 100.0%

         \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + 2 \cdot \left(-4\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + 2 \cdot \color{blue}{-4}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{-8}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{{-2}^{3}}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + {\color{blue}{\left(-2\right)}}^{3}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      13. fma-define 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{\color{blue}{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\left(-2\right)}^{3}\right)}}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, {\color{blue}{-2}}^{3}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{5 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \color{blue}{-8}\right)}{1 + t}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{t + 1}}{6 + \color{blue}{\left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}\right)}} \]
   11. Step-by-step derivation

       1. +-lft-identity 100.0%

          \[\leadsto \frac{5 + \color{blue}{\frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

       2. fma-undefine 100.0%

          \[\leadsto \frac{5 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + -8}}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

       3. associate-*r/ 100.0%

          \[\leadsto \frac{5 + \frac{\color{blue}{\frac{2 \cdot 2}{1 + t}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{\frac{\color{blue}{4}}{1 + t} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

       5. +-commutative 100.0%

          \[\leadsto \frac{5 + \frac{\frac{4}{\color{blue}{t + 1}} + -8}{1 + t}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

       6. +-commutative 100.0%

          \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{\color{blue}{t + 1}}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   12. Simplified 100.0%

       \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{t + 1}}{6 + \color{blue}{\frac{\frac{4}{t + 1} + -8}{t + 1}}} \]

   13. Taylor expanded in t around inf 100.0%

       \[\leadsto \frac{5 + \frac{\frac{4}{t + 1} + -8}{t + 1}}{6 + \frac{\color{blue}{\frac{4}{t}} + -8}{t + 1}} \]

   14. Taylor expanded in t around inf 100.0%

       \[\leadsto \frac{5 + \frac{\color{blue}{\frac{4}{t}} + -8}{t + 1}}{6 + \frac{\frac{4}{t} + -8}{t + 1}} \]

   if -0.340000000000000024 < t < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.6%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{-4}} \]

   6. Taylor expanded in t around 0 99.6%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + \left(-2 \cdot t + 2 \cdot {t}^{2}\right)\right)} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(\color{blue}{t \cdot -2} + 2 \cdot {t}^{2}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{{t}^{2} \cdot 2}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      3. unpow2 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{\left(t \cdot t\right)} \cdot 2\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      4. associate-*l* 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{t \cdot \left(t \cdot 2\right)}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      5. distribute-lft-out 99.6%

         \[\leadsto \frac{5 + \left(2 + \color{blue}{t \cdot \left(-2 + t \cdot 2\right)}\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

   8. Simplified 99.6%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + t \cdot \left(-2 + t \cdot 2\right)\right)} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;\frac{5 + \frac{-8 + \frac{4}{t}}{t + 1}}{6 + \frac{-8 + \frac{4}{t}}{t + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot \left(-2 + t \cdot 2\right)\right) \cdot \left(\frac{2}{t + 1} - 4\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.39 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot \left(-2 + t \cdot 2\right)\right) \cdot \left(\frac{2}{t + 1} - 4\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.39) (not (<= t 0.56)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (/
    (+ 5.0 (* (+ 2.0 (* t (+ -2.0 (* t 2.0)))) (- (/ 2.0 (+ t 1.0)) 4.0)))
    2.0)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.39) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.39d0)) .or. (.not. (t <= 0.56d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (5.0d0 + ((2.0d0 + (t * ((-2.0d0) + (t * 2.0d0)))) * ((2.0d0 / (t + 1.0d0)) - 4.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.39) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.39) or not (t <= 0.56):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.39) || !(t <= 0.56))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(5.0 + Float64(Float64(2.0 + Float64(t * Float64(-2.0 + Float64(t * 2.0)))) * Float64(Float64(2.0 / Float64(t + 1.0)) - 4.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.39) || ~((t <= 0.56)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (5.0 + ((2.0 + (t * (-2.0 + (t * 2.0)))) * ((2.0 / (t + 1.0)) - 4.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.39], N[Not[LessEqual[t, 0.56]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 + N[(N[(2.0 + N[(t * N[(-2.0 + N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.39000000000000001 or 0.56000000000000005 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 99.4%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

   6. Taylor expanded in t around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   if -0.39000000000000001 < t < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.6%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{-4}} \]

   6. Taylor expanded in t around 0 99.6%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + \left(-2 \cdot t + 2 \cdot {t}^{2}\right)\right)} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(\color{blue}{t \cdot -2} + 2 \cdot {t}^{2}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{{t}^{2} \cdot 2}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      3. unpow2 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{\left(t \cdot t\right)} \cdot 2\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      4. associate-*l* 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{t \cdot \left(t \cdot 2\right)}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      5. distribute-lft-out 99.6%

         \[\leadsto \frac{5 + \left(2 + \color{blue}{t \cdot \left(-2 + t \cdot 2\right)}\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

   8. Simplified 99.6%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + t \cdot \left(-2 + t \cdot 2\right)\right)} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.39 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot \left(-2 + t \cdot 2\right)\right) \cdot \left(\frac{2}{t + 1} - 4\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.48) (not (<= t 0.66)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.66)) {
		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 <= (-0.48d0)) .or. (.not. (t <= 0.66d0))) 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 <= -0.48) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.48) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.48) || !(t <= 0.66))
		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 <= -0.48) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.48], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.47999999999999998 or 0.660000000000000031 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 99.4%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

   6. Taylor expanded in t around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   if -0.47999999999999998 < t < 0.660000000000000031

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.6%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{-4}} \]

   6. Taylor expanded in t around 0 99.6%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + \left(-2 \cdot t + 2 \cdot {t}^{2}\right)\right)} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(\color{blue}{t \cdot -2} + 2 \cdot {t}^{2}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{{t}^{2} \cdot 2}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      3. unpow2 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{\left(t \cdot t\right)} \cdot 2\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      4. associate-*l* 99.6%

         \[\leadsto \frac{5 + \left(2 + \left(t \cdot -2 + \color{blue}{t \cdot \left(t \cdot 2\right)}\right)\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

      5. distribute-lft-out 99.6%

         \[\leadsto \frac{5 + \left(2 + \color{blue}{t \cdot \left(-2 + t \cdot 2\right)}\right) \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

   8. Simplified 99.6%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + t \cdot \left(-2 + t \cdot 2\right)\right)} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + -4} \]

   9. Taylor expanded in t around 0 99.6%

      \[\leadsto \frac{\color{blue}{1}}{6 + -4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.7% accurate, 51.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0
end function

Java

public static double code(double t) {
	return 0.8333333333333334;
}

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334;
end

Wolfram

code[t_] := 0.8333333333333334

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 52.2%

   \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

6. Taylor expanded in t around inf 59.7%

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

7. Final simplification 59.7%

   \[\leadsto 0.8333333333333334 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024035 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))