Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 10.7s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 + t \cdot \frac{\frac{t \cdot 4}{1 + t}}{1 + t}}{2 + \left(1 + \left(-1 + t \cdot \frac{\frac{t \cdot -4}{-1 - t}}{1 + t}\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double t) {
	return (1.0 + (t * (((t * 4.0) / (1.0 + t)) / (1.0 + t)))) / (2.0 + (1.0 + (-1.0 + (t * (((t * -4.0) / (-1.0 - t)) / (1.0 + t))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(N[(1.0 + N[(t * N[(N[(N[(t * 4.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(-1.0 + N[(t * N[(N[(N[(t * -4.0), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. associate-/l* 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-*r* 100.0%

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

   6. *-commutative 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/l* 100.0%

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

   9. associate-*l/ 100.0%

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

   10. associate-/r/ 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. log1p-def 100.0%

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

   3. expm1-udef 99.3%

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

   4. add-exp-log 100.0%

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

   5. associate--l+ 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-/l/ 80.7%

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

   8. pow2 80.7%

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

   9. +-commutative 80.7%

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

5. Applied egg-rr 80.7%

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

   1. *-commutative 80.7%

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

   2. +-commutative 80.7%

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

   3. unpow2 80.7%

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

   4. associate-/l/ 100.0%

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

   5. *-commutative 100.0%

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

   6. frac-2neg 100.0%

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

   7. div-inv 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. *-commutative 100.0%

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

   11. distribute-rgt-neg-in 100.0%

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

   12. metadata-eval 100.0%

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

   13. +-commutative 100.0%

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

   14. distribute-neg-in 100.0%

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

   15. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. unsub-neg 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-*r* 100.0%

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

   6. *-commutative 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/l* 100.0%

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

   9. associate-*l/ 100.0%

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

   10. associate-/r/ 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;t \leq -0.39:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 3:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{t \cdot \frac{\frac{t \cdot 4}{1 + t}}{1 + t} + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (* t 4.0))))
   (if (<= t -0.39)
     0.8333333333333334
     (if (<= t 3.0)
       (/ (+ 1.0 t_1) (+ 2.0 t_1))
       (/
        (+ 1.0 (- 4.0 (/ 8.0 t)))
        (+ (* t (/ (/ (* t 4.0) (+ 1.0 t)) (+ 1.0 t))) 2.0))))))

C

double code(double t) {
	double t_1 = t * (t * 4.0);
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 3.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (1.0 + (4.0 - (8.0 / t))) / ((t * (((t * 4.0) / (1.0 + t)) / (1.0 + t))) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (t * 4.0d0)
    if (t <= (-0.39d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 3.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = (1.0d0 + (4.0d0 - (8.0d0 / t))) / ((t * (((t * 4.0d0) / (1.0d0 + t)) / (1.0d0 + t))) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (t * 4.0);
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 3.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (1.0 + (4.0 - (8.0 / t))) / ((t * (((t * 4.0) / (1.0 + t)) / (1.0 + t))) + 2.0);
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (t * 4.0)
	tmp = 0
	if t <= -0.39:
		tmp = 0.8333333333333334
	elif t <= 3.0:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = (1.0 + (4.0 - (8.0 / t))) / ((t * (((t * 4.0) / (1.0 + t)) / (1.0 + t))) + 2.0)
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(t * 4.0))
	tmp = 0.0
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 3.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(1.0 + Float64(4.0 - Float64(8.0 / t))) / Float64(Float64(t * Float64(Float64(Float64(t * 4.0) / Float64(1.0 + t)) / Float64(1.0 + t))) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (t * 4.0);
	tmp = 0.0;
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 3.0)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = (1.0 + (4.0 - (8.0 / t))) / ((t * (((t * 4.0) / (1.0 + t)) / (1.0 + t))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.39], 0.8333333333333334, If[LessEqual[t, 3.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(N[(N[(t * 4.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.39000000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 100.0%

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

   if -0.39000000000000001 < t < 3

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.6%

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

   5. Taylor expanded in t around 0 99.6%

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

   if 3 < t

   1. Initial program 100.0%

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

      1. associate-/l* 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 99.9%

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

      9. associate-*l/ 99.9%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8}{t}\right)}}{2 + \frac{\frac{t \cdot 4}{1 + t}}{1 + t} \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.39:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 3:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{t \cdot \frac{\frac{t \cdot 4}{1 + t}}{1 + t} + 2}\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;t \leq -0.39:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (* t 4.0))))
   (if (<= t -0.39)
     0.8333333333333334
     (if (<= t 0.68)
       (/ (+ 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);
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.68) {
		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)
    if (t <= (-0.39d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.68d0) 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);
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.68) {
		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)
	tmp = 0
	if t <= -0.39:
		tmp = 0.8333333333333334
	elif t <= 0.68:
		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(t * Float64(t * 4.0))
	tmp = 0.0
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 0.68)
		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);
	tmp = 0.0;
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 0.68)
		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[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.39], 0.8333333333333334, If[LessEqual[t, 0.68], 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 < -0.39000000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 100.0%

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

   if -0.39000000000000001 < t < 0.680000000000000049

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.6%

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

   5. Taylor expanded in t around 0 99.6%

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

   if 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. Step-by-step derivation

      1. associate-/l* 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 99.9%

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

      9. associate-*l/ 99.9%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.39:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 98.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33)
   0.8333333333333334
   (if (<= t 0.66) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 0.66], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.330000000000000016

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 100.0%

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

   if -0.330000000000000016 < t < 0.660000000000000031

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.0%

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

   if 0.660000000000000031 < t

   1. Initial program 100.0%

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

      1. associate-/l* 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 99.9%

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

      9. associate-*l/ 99.9%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 6: 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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 99.4%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. associate-*l/ 100.0%

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

      10. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.0%

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

4. Final simplification 99.2%

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

Alternative 7: 59.4% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-*r* 100.0%

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

   6. *-commutative 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/l* 100.0%

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

   9. associate-*l/ 100.0%

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

   10. associate-/r/ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in t around 0 63.1%

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

5. Final simplification 63.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023326 
(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))))))