Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 8.9s

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: 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-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%

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

   3. associate-*r* 100.0%

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

   4. *-commutative 100.0%

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

   5. associate-*l/ 100.0%

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

   6. associate-/l* 100.0%

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

   7. fma-define 100.0%

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

   8. associate-/l* 100.0%

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

   9. *-commutative 100.0%

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

   10. associate-*l* 100.0%

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

   11. metadata-eval 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (+ t 1.0))) (t_2 (* 4.0 (* t_1 t_1))))
   (/ (+ t_2 1.0) (+ 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 (t_2 + 1.0) / (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 = (t_2 + 1.0d0) / (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 (t_2 + 1.0) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = t / (t + 1.0)
	t_2 = 4.0 * (t_1 * t_1)
	return (t_2 + 1.0) / (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(t_2 + 1.0) / 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 = (t_2 + 1.0) / (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[(t$95$2 + 1.0), $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%

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

Alternative 3: 59.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;t \cdot 0.125 + 1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.78)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.45)
     (+ (* t 0.125) 1.0)
     (+
      0.8333333333333334
      (/
       (-
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        0.2222222222222222)
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.45) {
		tmp = (t * 0.125) + 1.0;
	} 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 <= (-0.78d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.45d0) then
        tmp = (t * 0.125d0) + 1.0d0
    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 <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.45) {
		tmp = (t * 0.125) + 1.0;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.78:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.45:
		tmp = (t * 0.125) + 1.0
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.45)
		tmp = Float64(Float64(t * 0.125) + 1.0);
	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 <= -0.78)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.45)
		tmp = (t * 0.125) + 1.0;
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.78], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.45], N[(N[(t * 0.125), $MachinePrecision] + 1.0), $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 3 regimes

2. if t < -0.78000000000000003

   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%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. fma-define 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l* 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.78000000000000003 < t < 0.450000000000000011

   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 3.8%

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

      1. associate-*r/ 3.8%

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

      2. metadata-eval 3.8%

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

   7. Simplified 3.8%

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

   8. Taylor expanded in t around inf 18.8%

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

      1. associate-*r/ 18.8%

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

      2. metadata-eval 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in t around 0 18.8%

       \[\leadsto \color{blue}{1 + 0.125 \cdot t} \]
   12. Step-by-step derivation

       1. *-commutative 18.8%

          \[\leadsto 1 + \color{blue}{t \cdot 0.125} \]

   13. Simplified 18.8%

       \[\leadsto \color{blue}{1 + t \cdot 0.125} \]

   if 0.450000000000000011 < t

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. fma-define 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. unsub-neg 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;t \cdot 0.125 + 1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.15:\\ \;\;\;\;t \cdot 0.125 + 1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.78)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.15)
     (+ (* t 0.125) 1.0)
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.15) {
		tmp = (t * 0.125) + 1.0;
	} 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 <= (-0.78d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.15d0) then
        tmp = (t * 0.125d0) + 1.0d0
    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 <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.15) {
		tmp = (t * 0.125) + 1.0;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.78:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.15:
		tmp = (t * 0.125) + 1.0
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.15)
		tmp = Float64(Float64(t * 0.125) + 1.0);
	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 <= -0.78)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.15)
		tmp = (t * 0.125) + 1.0;
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.78], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.15], N[(N[(t * 0.125), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.78000000000000003

   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%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. fma-define 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l* 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.78000000000000003 < t < 0.149999999999999994

   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 3.8%

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

      1. associate-*r/ 3.8%

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

      2. metadata-eval 3.8%

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

   7. Simplified 3.8%

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

   8. Taylor expanded in t around inf 18.8%

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

      1. associate-*r/ 18.8%

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

      2. metadata-eval 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in t around 0 18.8%

       \[\leadsto \color{blue}{1 + 0.125 \cdot t} \]
   12. Step-by-step derivation

       1. *-commutative 18.8%

          \[\leadsto 1 + \color{blue}{t \cdot 0.125} \]

   13. Simplified 18.8%

       \[\leadsto \color{blue}{1 + t \cdot 0.125} \]

   if 0.149999999999999994 < t

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. fma-define 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. sub-neg 99.0%

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

      4. associate-*r/ 99.0%

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

      5. metadata-eval 99.0%

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

      6. distribute-neg-frac 99.0%

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

      7. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.15:\\ \;\;\;\;t \cdot 0.125 + 1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.42\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot 0.125 + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.42)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t 0.125) 1.0)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.42)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * 0.125) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.78d0)) .or. (.not. (t <= 0.42d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (t * 0.125d0) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.42)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * 0.125) + 1.0;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.78) or not (t <= 0.42):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (t * 0.125) + 1.0
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.78) || !(t <= 0.42))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(t * 0.125) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.78) || ~((t <= 0.42)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (t * 0.125) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.78], N[Not[LessEqual[t, 0.42]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * 0.125), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.78000000000000003 or 0.419999999999999984 < 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%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. fma-define 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l* 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, \frac{\frac{t}{1 + t} \cdot 4}{1 + t}, 1\right)}{\mathsf{fma}\left(t, \frac{\frac{t}{1 + t} \cdot 4}{1 + t}, 2\right)}} \]
   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 -0.78000000000000003 < t < 0.419999999999999984

   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 3.8%

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

      1. associate-*r/ 3.8%

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

      2. metadata-eval 3.8%

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

   7. Simplified 3.8%

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

   8. Taylor expanded in t around inf 18.8%

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

      1. associate-*r/ 18.8%

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

      2. metadata-eval 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in t around 0 18.8%

       \[\leadsto \color{blue}{1 + 0.125 \cdot t} \]
   12. Step-by-step derivation

       1. *-commutative 18.8%

          \[\leadsto 1 + \color{blue}{t \cdot 0.125} \]

   13. Simplified 18.8%

       \[\leadsto \color{blue}{1 + t \cdot 0.125} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.42\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot 0.125 + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(1 - \frac{2}{t}\right) + 1}{6 - \frac{8}{t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/ (+ (* 4.0 (- 1.0 (/ 2.0 t))) 1.0) (- 6.0 (/ 8.0 t))))

C

double code(double t) {
	return ((4.0 * (1.0 - (2.0 / t))) + 1.0) / (6.0 - (8.0 / t));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = ((4.0d0 * (1.0d0 - (2.0d0 / t))) + 1.0d0) / (6.0d0 - (8.0d0 / t))
end function

Java

public static double code(double t) {
	return ((4.0 * (1.0 - (2.0 / t))) + 1.0) / (6.0 - (8.0 / t));
}

Python

def code(t):
	return ((4.0 * (1.0 - (2.0 / t))) + 1.0) / (6.0 - (8.0 / t))

Julia

function code(t)
	return Float64(Float64(Float64(4.0 * Float64(1.0 - Float64(2.0 / t))) + 1.0) / Float64(6.0 - Float64(8.0 / t)))
end

MATLAB

function tmp = code(t)
	tmp = ((4.0 * (1.0 - (2.0 / t))) + 1.0) / (6.0 - (8.0 / t));
end

Wolfram

code[t_] := N[(N[(N[(4.0 * N[(1.0 - N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(6.0 - N[(8.0 / t), $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. Taylor expanded in t around inf 47.4%

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

   1. associate-*r/ 47.4%

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

   2. metadata-eval 47.4%

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

7. Simplified 47.4%

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

8. Taylor expanded in t around inf 55.5%

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

   1. associate-*r/ 55.5%

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

   2. metadata-eval 55.5%

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

10. Simplified 55.5%

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

11. Final simplification 55.5%

    \[\leadsto \frac{4 \cdot \left(1 - \frac{2}{t}\right) + 1}{6 - \frac{8}{t}} \]
12. Add Preprocessing

Alternative 7: 59.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{t \cdot 5 - 8}{t}}{6 - \frac{8}{t}} \end{array} \]

FPCore

(FPCore (t) :precision binary64 (/ (/ (- (* t 5.0) 8.0) t) (- 6.0 (/ 8.0 t))))

C

double code(double t) {
	return (((t * 5.0) - 8.0) / t) / (6.0 - (8.0 / t));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (((t * 5.0d0) - 8.0d0) / t) / (6.0d0 - (8.0d0 / t))
end function

Java

public static double code(double t) {
	return (((t * 5.0) - 8.0) / t) / (6.0 - (8.0 / t));
}

Python

def code(t):
	return (((t * 5.0) - 8.0) / t) / (6.0 - (8.0 / t))

Julia

function code(t)
	return Float64(Float64(Float64(Float64(t * 5.0) - 8.0) / t) / Float64(6.0 - Float64(8.0 / t)))
end

MATLAB

function tmp = code(t)
	tmp = (((t * 5.0) - 8.0) / t) / (6.0 - (8.0 / t));
end

Wolfram

code[t_] := N[(N[(N[(N[(t * 5.0), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision] / N[(6.0 - N[(8.0 / t), $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. Taylor expanded in t around inf 47.4%

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

   1. associate-*r/ 47.4%

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

   2. metadata-eval 47.4%

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

7. Simplified 47.4%

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

8. Taylor expanded in t around inf 55.5%

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

   1. associate-*r/ 55.5%

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

   2. metadata-eval 55.5%

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

10. Simplified 55.5%

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

11. Taylor expanded in t around 0 55.5%

    \[\leadsto \frac{\color{blue}{\frac{5 \cdot t - 8}{t}}}{6 - \frac{8}{t}} \]

12. Final simplification 55.5%

    \[\leadsto \frac{\frac{t \cdot 5 - 8}{t}}{6 - \frac{8}{t}} \]
13. Add Preprocessing

Alternative 8: 11.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ t \cdot 0.125 + 1 \end{array} \]

FPCore

(FPCore (t) :precision binary64 (+ (* t 0.125) 1.0))

C

double code(double t) {
	return (t * 0.125) + 1.0;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (t * 0.125d0) + 1.0d0
end function

Java

public static double code(double t) {
	return (t * 0.125) + 1.0;
}

Python

def code(t):
	return (t * 0.125) + 1.0

Julia

function code(t)
	return Float64(Float64(t * 0.125) + 1.0)
end

MATLAB

function tmp = code(t)
	tmp = (t * 0.125) + 1.0;
end

Wolfram

code[t_] := N[(N[(t * 0.125), $MachinePrecision] + 1.0), $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. Taylor expanded in t around inf 47.4%

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

   1. associate-*r/ 47.4%

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

   2. metadata-eval 47.4%

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

7. Simplified 47.4%

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

8. Taylor expanded in t around inf 55.5%

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

   1. associate-*r/ 55.5%

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

   2. metadata-eval 55.5%

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

10. Simplified 55.5%

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

11. Taylor expanded in t around 0 11.8%

    \[\leadsto \color{blue}{1 + 0.125 \cdot t} \]
12. Step-by-step derivation

    1. *-commutative 11.8%

       \[\leadsto 1 + \color{blue}{t \cdot 0.125} \]

13. Simplified 11.8%

    \[\leadsto \color{blue}{1 + t \cdot 0.125} \]

14. Final simplification 11.8%

    \[\leadsto t \cdot 0.125 + 1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(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))))))