Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 14.8s

Alternatives: 5

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-/l* 100.0%

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

   3. frac-times 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-*l/ 100.0%

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

   6. associate-/l* 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/r* 100.0%

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

   9. +-commutative 100.0%

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

   10. +-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-/l* 100.0%

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

   3. frac-times 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-*l/ 100.0%

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

   6. associate-/l* 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/r* 100.0%

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

   9. +-commutative 100.0%

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

   10. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot 4}{\frac{1}{t}}}{1 + t}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ 1.0 t)) 0.005)
   (/
    (+ 1.0 (/ (/ (* t 4.0) (/ 1.0 t)) (+ 1.0 t)))
    (+ 2.0 (/ (* t (* t 4.0)) (+ 1.0 t))))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + (((t * 4.0) / (1.0 / t)) / (1.0 + t))) / (2.0 + ((t * (t * 4.0)) / (1.0 + t)));
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t * 2.0d0) / (1.0d0 + t)) <= 0.005d0) then
        tmp = (1.0d0 + (((t * 4.0d0) / (1.0d0 / t)) / (1.0d0 + t))) / (2.0d0 + ((t * (t * 4.0d0)) / (1.0d0 + t)))
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + (((t * 4.0) / (1.0 / t)) / (1.0 + t))) / (2.0 + ((t * (t * 4.0)) / (1.0 + t)));
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (1.0 + t)) <= 0.005:
		tmp = (1.0 + (((t * 4.0) / (1.0 / t)) / (1.0 + t))) / (2.0 + ((t * (t * 4.0)) / (1.0 + t)))
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(1.0 + t)) <= 0.005)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(t * 4.0) / Float64(1.0 / t)) / Float64(1.0 + t))) / Float64(2.0 + Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + t))));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((t * 2.0) / (1.0 + t)) <= 0.005)
		tmp = (1.0 + (((t * 4.0) / (1.0 / t)) / (1.0 + t))) / (2.0 + ((t * (t * 4.0)) / (1.0 + t)));
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(1.0 + N[(N[(N[(t * 4.0), $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 0.0050000000000000001

   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. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*l/ 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.3%

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

   6. Taylor expanded in t around 0 99.3%

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

      1. div-inv 99.3%

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

      2. inv-pow 99.3%

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

      3. pow-flip 99.3%

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

      4. metadata-eval 99.3%

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

      5. pow1 99.3%

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

   8. Applied egg-rr 99.3%

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

   if 0.0050000000000000001 < (/.f64 (*.f64 2 t) (+.f64 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. Add Preprocessing

   3. Taylor expanded in t around inf 98.2%

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

      1. associate-*r/ 98.2%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 98.2%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \frac{\color{blue}{8}}{t}} \]

   5. Simplified 98.2%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - \frac{8}{t}}} \]

   6. Taylor expanded in t around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot 4}{\frac{1}{t}}}{1 + t}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 0.005:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ 1.0 t)) 0.005)
   0.5
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 0.005) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t * 2.0d0) / (1.0d0 + t)) <= 0.005d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 0.005) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (1.0 + t)) <= 0.005:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(1.0 + t)) <= 0.005)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((t * 2.0) / (1.0 + t)) <= 0.005)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.005], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 0.0050000000000000001

   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. Add Preprocessing

   3. Taylor expanded in t around 0 98.6%

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

   4. Taylor expanded in t around 0 98.7%

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

   if 0.0050000000000000001 < (/.f64 (*.f64 2 t) (+.f64 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. Add Preprocessing

   3. Taylor expanded in t around inf 98.2%

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

      1. associate-*r/ 98.2%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 98.2%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \frac{\color{blue}{8}}{t}} \]

   5. Simplified 98.2%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - \frac{8}{t}}} \]

   6. Taylor expanded in t around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 0.005:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.2%

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

      1. associate-*r/ 98.2%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 98.2%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{6 - \frac{\color{blue}{8}}{t}} \]

   5. Simplified 98.2%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - \frac{8}{t}}} \]

   6. Taylor expanded in t around inf 97.1%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.6%

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

   4. Taylor expanded in t around 0 98.7%

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

4. Final simplification 97.9%

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

Alternative 5: 59.2% 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. Add Preprocessing

3. Taylor expanded in t around 0 56.4%

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

4. Taylor expanded in t around 0 57.9%

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

5. Final simplification 57.9%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))))