Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 13.6s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 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. frac-times 73.4%

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

   2. swap-sqr 73.4%

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

   3. metadata-eval 73.4%

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

   4. *-commutative 73.4%

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

   5. associate-*r* 73.4%

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

   6. times-frac 100.0%

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

   7. associate-*r/ 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. frac-times 73.4%

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

   2. swap-sqr 73.4%

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

   3. metadata-eval 73.4%

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

   4. *-commutative 73.4%

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

   5. associate-*r* 73.4%

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

   6. times-frac 100.0%

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

   7. associate-*r/ 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 74.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t (* t 4.0)) (* (+ 1.0 t) (+ 1.0 t)))))
   (if (<= (/ (* t 2.0) (+ 1.0 t)) 2.0)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 2.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * (t * 4.0d0)) / ((1.0d0 + t) * (1.0d0 + t))
    if (((t * 2.0d0) / (1.0d0 + t)) <= 2.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 2.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t))
	tmp = 0
	if ((t * 2.0) / (1.0 + t)) <= 2.0:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(t * Float64(t * 4.0)) / Float64(Float64(1.0 + t) * Float64(1.0 + t)))
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(1.0 + t)) <= 2.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	tmp = 0.0;
	if (((t * 2.0) / (1.0 + t)) <= 2.0)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t), $MachinePrecision] * N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   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. times-frac 72.8%

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

      2. sqr-neg 72.8%

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

      3. distribute-rgt-neg-out 72.8%

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

      4. distribute-rgt-neg-out 72.8%

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

      5. swap-sqr 72.8%

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

      6. *-commutative 72.8%

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

      7. sqr-neg 72.8%

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

      8. associate-*r* 72.8%

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

      9. metadata-eval 72.8%

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

      10. times-frac 72.8%

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

   3. Simplified 72.8%

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

   if 2 < (/.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 75.6%

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

      1. associate-*r/ 75.6%

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

      2. metadata-eval 75.6%

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

   5. Simplified 75.6%

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

   6. Taylor expanded in t around inf 80.6%

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

      1. associate-*r/ 80.6%

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

      2. metadata-eval 80.6%

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

   8. Simplified 80.6%

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

4. Final simplification 73.0%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot \left(t \cdot 4\right)}{1 + t \cdot 2}\\ \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 0.005:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t (* t 4.0)) (+ 1.0 (* t 2.0)))))
   (if (<= (/ (* t 2.0) (+ 1.0 t)) 0.005)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double t_1 = (t * (t * 4.0)) / (1.0 + (t * 2.0));
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * (t * 4.0d0)) / (1.0d0 + (t * 2.0d0))
    if (((t * 2.0d0) / (1.0d0 + t)) <= 0.005d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (t * (t * 4.0)) / (1.0 + (t * 2.0));
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 0.005) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = (t * (t * 4.0)) / (1.0 + (t * 2.0))
	tmp = 0
	if ((t * 2.0) / (1.0 + t)) <= 0.005:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + Float64(t * 2.0)))
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(1.0 + t)) <= 0.005)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (t * (t * 4.0)) / (1.0 + (t * 2.0));
	tmp = 0.0;
	if (((t * 2.0) / (1.0 + t)) <= 0.005)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 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. times-frac 100.0%

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

      2. sqr-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. swap-sqr 100.0%

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

      6. *-commutative 100.0%

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

      7. sqr-neg 100.0%

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

      8. associate-*r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. times-frac 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 99.7%

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

      1. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   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 99.1%

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

Alternative 4: 99.0% accurate, 2.7× 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 5: 98.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. 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 6: 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))))))