Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 7.0s

Alternatives: 10

Speedup: 1.1×

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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{t + 1}\\ t_2 := {t\_1}^{2}\\ \frac{\mathsf{fma}\left(4, t\_2, 1\right)}{4 - {t\_1}^{4} \cdot 16} \cdot \left(2 + t\_2 \cdot -4\right) \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (+ t 1.0))) (t_2 (pow t_1 2.0)))
   (*
    (/ (fma 4.0 t_2 1.0) (- 4.0 (* (pow t_1 4.0) 16.0)))
    (+ 2.0 (* t_2 -4.0)))))

C

double code(double t) {
	double t_1 = t / (t + 1.0);
	double t_2 = pow(t_1, 2.0);
	return (fma(4.0, t_2, 1.0) / (4.0 - (pow(t_1, 4.0) * 16.0))) * (2.0 + (t_2 * -4.0));
}

Julia

function code(t)
	t_1 = Float64(t / Float64(t + 1.0))
	t_2 = t_1 ^ 2.0
	return Float64(Float64(fma(4.0, t_2, 1.0) / Float64(4.0 - Float64((t_1 ^ 4.0) * 16.0))) * Float64(2.0 + Float64(t_2 * -4.0)))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[(N[(4.0 * t$95$2 + 1.0), $MachinePrecision] / N[(4.0 - N[(N[Power[t$95$1, 4.0], $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(t$95$2 * -4.0), $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%

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

   2. associate-/l* 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l* 100.0%

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

   6. associate-/l* 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. flip-+ 100.0%

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

   2. associate-/r/ 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, {\left(\frac{t}{t + 1}\right)}^{2}, 1\right)}{4 - {\left(\frac{t}{t + 1}\right)}^{4} \cdot 16} \cdot \left(2 + -4 \cdot {\left(\frac{t}{t + 1}\right)}^{2}\right)} \]

7. Final simplification 100.0%

   \[\leadsto \frac{\mathsf{fma}\left(4, {\left(\frac{t}{t + 1}\right)}^{2}, 1\right)}{4 - {\left(\frac{t}{t + 1}\right)}^{4} \cdot 16} \cdot \left(2 + {\left(\frac{t}{t + 1}\right)}^{2} \cdot -4\right) \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{t + 1}\\ \frac{1 + 4 \cdot \left(t\_1 \cdot t\_1\right)}{2 + \left(\mathsf{fma}\left(4, {t\_1}^{2}, 1\right) + -1\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-/l* 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l* 100.0%

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

   6. associate-/l* 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. log1p-define 100.0%

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

   3. expm1-undefine 99.2%

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

   4. add-exp-log 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-define 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. pow2 100.0%

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

   9. sqrt-prod 72.6%

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

   10. add-sqr-sqrt 100.0%

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

   11. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{t + 1}\right)}{2 + \left(\mathsf{fma}\left(4, {\left(\frac{t}{t + 1}\right)}^{2}, 1\right) + -1\right)} \]
8. Add Preprocessing

Alternative 3: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(\frac{t}{t + 1} \cdot \left(t \cdot \left(1 - t\right)\right)\right)}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ t 1.0)) 2e-9)
   (/
    (+ 1.0 (* 4.0 (* (/ t (+ t 1.0)) (* t (- 1.0 t)))))
    (- 2.0 (* 4.0 (* t (/ t (- -1.0 t))))))
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = (1.0 + (4.0 * ((t / (t + 1.0)) * (t * (1.0 - t))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t * 2.0d0) / (t + 1.0d0)) <= 2d-9) then
        tmp = (1.0d0 + (4.0d0 * ((t / (t + 1.0d0)) * (t * (1.0d0 - t))))) / (2.0d0 - (4.0d0 * (t * (t / ((-1.0d0) - t)))))
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = (1.0 + (4.0 * ((t / (t + 1.0)) * (t * (1.0 - t))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (t + 1.0)) <= 2e-9:
		tmp = (1.0 + (4.0 * ((t / (t + 1.0)) * (t * (1.0 - t))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))))
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t + 1.0)) <= 2e-9)
		tmp = Float64(Float64(1.0 + Float64(4.0 * Float64(Float64(t / Float64(t + 1.0)) * Float64(t * Float64(1.0 - t))))) / Float64(2.0 - Float64(4.0 * Float64(t * Float64(t / Float64(-1.0 - t))))));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9)
		tmp = (1.0 + (4.0 * ((t / (t + 1.0)) * (t * (1.0 - t))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 2e-9], N[(N[(1.0 + N[(4.0 * N[(N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(4.0 * N[(t * N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2.00000000000000012e-9

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.2%

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

   6. Taylor expanded in t around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. sub-neg 99.2%

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

   8. Simplified 99.2%

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

   if 2.00000000000000012e-9 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) 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%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

      3. mul-1-neg 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}\right)}}{t} \]

      4. unsub-neg 100.0%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}}{t} \]

      5. associate-*r/ 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t} \]

      6. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(\frac{t}{t + 1} \cdot \left(t \cdot \left(1 - t\right)\right)\right)}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \frac{t}{t + 1}\right)}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ t 1.0)) 2e-9)
   (/
    (+ 1.0 (* 4.0 (* t (/ t (+ t 1.0)))))
    (- 2.0 (* 4.0 (* t (/ t (- -1.0 t))))))
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = (1.0 + (4.0 * (t * (t / (t + 1.0))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t * 2.0d0) / (t + 1.0d0)) <= 2d-9) then
        tmp = (1.0d0 + (4.0d0 * (t * (t / (t + 1.0d0))))) / (2.0d0 - (4.0d0 * (t * (t / ((-1.0d0) - t)))))
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = (1.0 + (4.0 * (t * (t / (t + 1.0))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (t + 1.0)) <= 2e-9:
		tmp = (1.0 + (4.0 * (t * (t / (t + 1.0))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))))
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t + 1.0)) <= 2e-9)
		tmp = Float64(Float64(1.0 + Float64(4.0 * Float64(t * Float64(t / Float64(t + 1.0))))) / Float64(2.0 - Float64(4.0 * Float64(t * Float64(t / Float64(-1.0 - t))))));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9)
		tmp = (1.0 + (4.0 * (t * (t / (t + 1.0))))) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 2e-9], N[(N[(1.0 + N[(4.0 * N[(t * N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(4.0 * N[(t * N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2.00000000000000012e-9

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.2%

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

   6. Taylor expanded in t around 0 99.2%

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

   if 2.00000000000000012e-9 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) 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%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

      3. mul-1-neg 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}\right)}}{t} \]

      4. unsub-neg 100.0%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}}{t} \]

      5. associate-*r/ 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t} \]

      6. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \frac{t}{t + 1}\right)}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{t + 1}\\ t_2 := 4 \cdot \left(t\_1 \cdot t\_1\right)\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

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

C

double code(double t) {
	double t_1 = t / (t + 1.0);
	double t_2 = 4.0 * (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 = t / (t + 1.0d0)
    t_2 = 4.0d0 * (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 = t / (t + 1.0);
	double t_2 = 4.0 * (t_1 * t_1);
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = t / (t + 1.0)
	t_2 = 4.0 * (t_1 * t_1)
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(t / Float64(t + 1.0))
	t_2 = Float64(4.0 * 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 = t / (t + 1.0);
	t_2 = 4.0 * (t_1 * t_1);
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-/l* 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l* 100.0%

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

   6. associate-/l* 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 6: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ t 1.0)) 2e-9)
   0.5
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t * 2.0d0) / (t + 1.0d0)) <= 2d-9) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (t + 1.0)) <= 2e-9:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t + 1.0)) <= 2e-9)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 2e-9], 0.5, N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2.00000000000000012e-9

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 2.00000000000000012e-9 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) 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%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

      3. mul-1-neg 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}\right)}}{t} \]

      4. unsub-neg 100.0%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}}{t} \]

      5. associate-*r/ 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t} \]

      6. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ t 1.0)) 2e-9)
   0.5
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t * 2.0d0) / (t + 1.0d0)) <= 2d-9) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (t + 1.0)) <= 2e-9:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t + 1.0)) <= 2e-9)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 2e-9], 0.5, N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2.00000000000000012e-9

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 2.00000000000000012e-9 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) 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%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 99.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]

      3. sub-neg 99.8%

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

      4. associate-*r/ 99.8%

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

      5. metadata-eval 99.8%

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

      6. distribute-neg-frac 99.8%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} \]

      7. metadata-eval 99.8%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} \]

   7. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ t 1.0)) 2e-9)
   0.5
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 2e-9) {
		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 * 2.0d0) / (t + 1.0d0)) <= 2d-9) 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 * 2.0) / (t + 1.0)) <= 2e-9) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (t + 1.0)) <= 2e-9:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t + 1.0)) <= 2e-9)
		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 * 2.0) / (t + 1.0)) <= 2e-9)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 2e-9], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2.00000000000000012e-9

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 2.00000000000000012e-9 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) 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%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.6%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. 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%

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

   7. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ t 1.0)) 1.0) 0.5 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t + 1.0)) <= 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 * 2.0d0) / (t + 1.0d0)) <= 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 * 2.0) / (t + 1.0)) <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (t + 1.0)) <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t + 1.0)) <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((t * 2.0) / (t + 1.0)) <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1.0], 0.5, 0.8333333333333334]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 1 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) 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%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t + 1} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-/l* 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l* 100.0%

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

   6. associate-/l* 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around 0 58.3%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))))