Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 8

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-/l* 100.0%

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

   3. associate-*r/ 100.0%

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

   4. associate-/r/ 100.0%

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

   5. fma-define 100.0%

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

   6. associate-/l* 100.0%

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

   7. associate-*l/ 100.0%

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

   8. metadata-eval 100.0%

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

   9. +-commutative 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. div-sub 100.0%

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

   13. *-inverses 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{\frac{4}{\frac{1 + t}{t}}}{1 + t}\\ \frac{1 + t\_1}{2 + t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*r/ 100.0%

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

   2. associate-*r* 100.0%

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

   3. associate-*l/ 100.0%

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

   4. associate-/l* 100.0%

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

   5. associate-*l/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r/ 100.0%

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

   8. associate-*r* 100.0%

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

   9. associate-*l/ 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.63 \lor \neg \left(t \leq 0.53\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \frac{4 \cdot t}{1 + t}}{2 + t \cdot \left(4 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.63) (not (<= t 0.53)))
   (/ 1.0 (+ 1.2 (/ 0.32 t)))
   (/ (+ 1.0 (* t (/ (* 4.0 t) (+ 1.0 t)))) (+ 2.0 (* t (* 4.0 t))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.63) || !(t <= 0.53)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = (1.0 + (t * ((4.0 * t) / (1.0 + t)))) / (2.0 + (t * (4.0 * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.63d0)) .or. (.not. (t <= 0.53d0))) then
        tmp = 1.0d0 / (1.2d0 + (0.32d0 / t))
    else
        tmp = (1.0d0 + (t * ((4.0d0 * t) / (1.0d0 + t)))) / (2.0d0 + (t * (4.0d0 * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.63) || !(t <= 0.53)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = (1.0 + (t * ((4.0 * t) / (1.0 + t)))) / (2.0 + (t * (4.0 * t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.63) or not (t <= 0.53):
		tmp = 1.0 / (1.2 + (0.32 / t))
	else:
		tmp = (1.0 + (t * ((4.0 * t) / (1.0 + t)))) / (2.0 + (t * (4.0 * t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.63) || !(t <= 0.53))
		tmp = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)));
	else
		tmp = Float64(Float64(1.0 + Float64(t * Float64(Float64(4.0 * t) / Float64(1.0 + t)))) / Float64(2.0 + Float64(t * Float64(4.0 * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.63) || ~((t <= 0.53)))
		tmp = 1.0 / (1.2 + (0.32 / t));
	else
		tmp = (1.0 + (t * ((4.0 * t) / (1.0 + t)))) / (2.0 + (t * (4.0 * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.63], N[Not[LessEqual[t, 0.53]], $MachinePrecision]], N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t * N[(N[(4.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t * N[(4.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.630000000000000004 or 0.53000000000000003 < 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. +-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/r/ 100.0%

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

      5. fma-define 100.0%

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

      6. associate-/l* 99.9%

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

      7. associate-*l/ 99.9%

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

      8. metadata-eval 99.9%

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

      9. +-commutative 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. div-sub 99.9%

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

      13. *-inverses 99.9%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{4}{e^{\mathsf{log1p}\left(t\right) + \mathsf{log1p}\left(\frac{1}{t}\right)}}, t, 2\right)}{\mathsf{fma}\left(\frac{4}{e^{\mathsf{log1p}\left(t\right) + \mathsf{log1p}\left(\frac{1}{t}\right)}}, t, 1\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 51.9%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{1.2 + 0.32 \cdot \frac{1}{t}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.2%

          \[\leadsto \frac{1}{1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}} \]

       2. metadata-eval 99.2%

          \[\leadsto \frac{1}{1.2 + \frac{\color{blue}{0.32}}{t}} \]

   11. Simplified 99.2%

       \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.630000000000000004 < t < 0.53000000000000003

   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-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*l/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

   7. Taylor expanded in t around 0 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.63 \lor \neg \left(t \leq 0.53\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \frac{4 \cdot t}{1 + t}}{2 + t \cdot \left(4 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(4 \cdot t\right)\\ \mathbf{if}\;t \leq -0.7 \lor \neg \left(t \leq 0.44\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_1}{2 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (* 4.0 t))))
   (if (or (<= t -0.7) (not (<= t 0.44)))
     (/ 1.0 (+ 1.2 (/ 0.32 t)))
     (/ (+ 1.0 t_1) (+ 2.0 t_1)))))

C

double code(double t) {
	double t_1 = t * (4.0 * t);
	double tmp;
	if ((t <= -0.7) || !(t <= 0.44)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (4.0d0 * t)
    if ((t <= (-0.7d0)) .or. (.not. (t <= 0.44d0))) then
        tmp = 1.0d0 / (1.2d0 + (0.32d0 / t))
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (4.0 * t);
	double tmp;
	if ((t <= -0.7) || !(t <= 0.44)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (4.0 * t)
	tmp = 0
	if (t <= -0.7) or not (t <= 0.44):
		tmp = 1.0 / (1.2 + (0.32 / t))
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(4.0 * t))
	tmp = 0.0
	if ((t <= -0.7) || !(t <= 0.44))
		tmp = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (4.0 * t);
	tmp = 0.0;
	if ((t <= -0.7) || ~((t <= 0.44)))
		tmp = 1.0 / (1.2 + (0.32 / t));
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(4.0 * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -0.7], N[Not[LessEqual[t, 0.44]], $MachinePrecision]], N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.69999999999999996 or 0.440000000000000002 < 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. +-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/r/ 100.0%

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

      5. fma-define 100.0%

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

      6. associate-/l* 99.9%

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

      7. associate-*l/ 99.9%

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

      8. metadata-eval 99.9%

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

      9. +-commutative 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. div-sub 99.9%

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

      13. *-inverses 99.9%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{4}{e^{\mathsf{log1p}\left(t\right) + \mathsf{log1p}\left(\frac{1}{t}\right)}}, t, 2\right)}{\mathsf{fma}\left(\frac{4}{e^{\mathsf{log1p}\left(t\right) + \mathsf{log1p}\left(\frac{1}{t}\right)}}, t, 1\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 51.9%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{1.2 + 0.32 \cdot \frac{1}{t}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.2%

          \[\leadsto \frac{1}{1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}} \]

       2. metadata-eval 99.2%

          \[\leadsto \frac{1}{1.2 + \frac{\color{blue}{0.32}}{t}} \]

   11. Simplified 99.2%

       \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.69999999999999996 < t < 0.440000000000000002

   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-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*l/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

   7. Taylor expanded in t around 0 99.3%

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

   8. Taylor expanded in t around 0 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.7 \lor \neg \left(t \leq 0.44\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(4 \cdot t\right)}{2 + t \cdot \left(4 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.4\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.58) (not (<= t 0.4))) (/ 1.0 (+ 1.2 (/ 0.32 t))) 0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.4)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.58d0)) .or. (.not. (t <= 0.4d0))) then
        tmp = 1.0d0 / (1.2d0 + (0.32d0 / t))
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.4)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.58) or not (t <= 0.4):
		tmp = 1.0 / (1.2 + (0.32 / t))
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.58) || !(t <= 0.4))
		tmp = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.58) || ~((t <= 0.4)))
		tmp = 1.0 / (1.2 + (0.32 / t));
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.58], N[Not[LessEqual[t, 0.4]], $MachinePrecision]], N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 0.40000000000000002 < 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. +-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/r/ 100.0%

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

      5. fma-define 100.0%

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

      6. associate-/l* 99.9%

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

      7. associate-*l/ 99.9%

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

      8. metadata-eval 99.9%

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

      9. +-commutative 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. div-sub 99.9%

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

      13. *-inverses 99.9%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{4}{e^{\mathsf{log1p}\left(t\right) + \mathsf{log1p}\left(\frac{1}{t}\right)}}, t, 2\right)}{\mathsf{fma}\left(\frac{4}{e^{\mathsf{log1p}\left(t\right) + \mathsf{log1p}\left(\frac{1}{t}\right)}}, t, 1\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 51.9%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{1.2 + 0.32 \cdot \frac{1}{t}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.2%

          \[\leadsto \frac{1}{1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}} \]

       2. metadata-eval 99.2%

          \[\leadsto \frac{1}{1.2 + \frac{\color{blue}{0.32}}{t}} \]

   11. Simplified 99.2%

       \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.57999999999999996 < t < 0.40000000000000002

   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-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*l/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.4\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 99.9%

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

      5. associate-*l/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

   if -0.48999999999999999 < 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-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*l/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.6%

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

4. Final simplification 98.9%

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

Alternative 7: 98.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		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 <= (-0.34d0)) 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 <= -0.34) {
		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 <= -0.34:
		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 <= -0.34)
		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 <= -0.34)
		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, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 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-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 99.9%

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

      5. associate-*l/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.2%

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

   if -0.340000000000000024 < 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-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*l/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% 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-*r/ 100.0%

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

   2. associate-*r* 100.0%

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

   3. associate-*l/ 100.0%

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

   4. associate-/l* 100.0%

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

   5. associate-*l/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r/ 100.0%

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

   8. associate-*r* 100.0%

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

   9. associate-*l/ 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around 0 58.5%

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

6. Final simplification 58.5%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))))