Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 23.8s

Alternatives: 9

Speedup: 1.0×

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, 1.0× 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]]]

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. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 1.99999999:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (/ (* t (* t 4.0)) (+ 1.0 t)) (+ 1.0 t))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1.99999999)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (+
      0.8333333333333334
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)))))

C

double code(double t) {
	double t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1.99999999) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 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 (((2.0d0 * t) / (1.0d0 + t)) <= 1.99999999d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) - 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 (((2.0 * t) / (1.0 + t)) <= 1.99999999) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)
	tmp = 0
	if ((2.0 * t) / (1.0 + t)) <= 1.99999999:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + t)) / Float64(1.0 + t))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1.99999999)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 / t) - 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 (((2.0 * t) / (1.0 + t)) <= 1.99999999)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1.99999999], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 1.9999999900000001

   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

   3. Simplified 100.0%

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

   if 1.9999999900000001 < (/.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. Taylor expanded in t around inf 99.9%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. unpow2 99.9%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(12 \cdot \frac{1}{\color{blue}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      3. associate-*r/ 99.9%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{\color{blue}{12}}{t \cdot t} + 6\right) - 8 \cdot \frac{1}{t}} \]

      5. associate-*r/ 99.9%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{\color{blue}{8}}{t}} \]

   4. Simplified 99.9%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{8}{t}}} \]

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{0.8333333333333334 + \left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      2. unpow2 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} - \frac{0.2222222222222222}{t}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 100.0%

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

      2. sub-div 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 1.99999999:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ \mathbf{if}\;t_1 \leq 0.01:\\ \;\;\;\;\frac{1 + t_1 \cdot \left(2 \cdot t\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))))
   (if (<= t_1 0.01)
     (/ (+ 1.0 (* t_1 (* 2.0 t))) (+ 2.0 (* 4.0 (* t t))))
     (+
      0.8333333333333334
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double tmp;
	if (t_1 <= 0.01) {
		tmp = (1.0 + (t_1 * (2.0 * t))) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 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 = (2.0d0 * t) / (1.0d0 + t)
    if (t_1 <= 0.01d0) then
        tmp = (1.0d0 + (t_1 * (2.0d0 * t))) / (2.0d0 + (4.0d0 * (t * t)))
    else
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double tmp;
	if (t_1 <= 0.01) {
		tmp = (1.0 + (t_1 * (2.0 * t))) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	tmp = 0
	if t_1 <= 0.01:
		tmp = (1.0 + (t_1 * (2.0 * t))) / (2.0 + (4.0 * (t * t)))
	else:
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	tmp = 0.0
	if (t_1 <= 0.01)
		tmp = Float64(Float64(1.0 + Float64(t_1 * Float64(2.0 * t))) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	tmp = 0.0;
	if (t_1 <= 0.01)
		tmp = (1.0 + (t_1 * (2.0 * t))) / (2.0 + (4.0 * (t * t)));
	else
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.01], N[(N[(1.0 + N[(t$95$1 * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 0.0100000000000000002

   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. Taylor expanded in t around 0 99.3%

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

      1. unpow2 99.1%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in t around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 0.0100000000000000002 < (/.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. Taylor expanded in t around inf 99.7%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. unpow2 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(12 \cdot \frac{1}{\color{blue}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      3. associate-*r/ 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{\color{blue}{12}}{t \cdot t} + 6\right) - 8 \cdot \frac{1}{t}} \]

      5. associate-*r/ 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{\color{blue}{8}}{t}} \]

   4. Simplified 99.7%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{8}{t}}} \]

   5. Taylor expanded in t around inf 99.8%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \color{blue}{0.8333333333333334 + \left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \color{blue}{0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} - \frac{0.2222222222222222}{t}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 99.8%

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

      2. sub-div 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.01:\\ \;\;\;\;\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.01:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* 4.0 (* t t))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.01)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (+
      0.8333333333333334
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)))))

C

double code(double t) {
	double t_1 = 4.0 * (t * t);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.01) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 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 = 4.0d0 * (t * t)
    if (((2.0d0 * t) / (1.0d0 + t)) <= 0.01d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 4.0 * (t * t);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.01) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = 4.0 * (t * t)
	tmp = 0
	if ((2.0 * t) / (1.0 + t)) <= 0.01:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(4.0 * Float64(t * t))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.01)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 4.0 * (t * t);
	tmp = 0.0;
	if (((2.0 * t) / (1.0 + t)) <= 0.01)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 0.0100000000000000002

   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. Taylor expanded in t around 0 99.3%

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

      1. unpow2 99.1%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in t around 0 99.3%

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

      1. unpow2 99.1%

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

   7. Simplified 99.3%

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

   if 0.0100000000000000002 < (/.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. Taylor expanded in t around inf 99.7%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. unpow2 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(12 \cdot \frac{1}{\color{blue}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      3. associate-*r/ 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{\color{blue}{12}}{t \cdot t} + 6\right) - 8 \cdot \frac{1}{t}} \]

      5. associate-*r/ 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{\color{blue}{8}}{t}} \]

   4. Simplified 99.7%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{8}{t}}} \]

   5. Taylor expanded in t around inf 99.8%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \color{blue}{0.8333333333333334 + \left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \color{blue}{0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} - \frac{0.2222222222222222}{t}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 99.8%

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

      2. sub-div 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.01:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot t\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.8) (not (<= t 0.33)))
   (+
    0.8333333333333334
    (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.33)) {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.8d0)) .or. (.not. (t <= 0.33d0))) then
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) - 0.2222222222222222d0) / t)
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.33)) {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.8) or not (t <= 0.33):
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t)
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.8) || !(t <= 0.33))
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.8) || ~((t <= 0.33)))
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.8], N[Not[LessEqual[t, 0.33]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.330000000000000016 < 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. Taylor expanded in t around inf 99.7%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(6 + 12 \cdot \frac{1}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} + 6\right)} - 8 \cdot \frac{1}{t}} \]

      2. unpow2 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(12 \cdot \frac{1}{\color{blue}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      3. associate-*r/ 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\color{blue}{\frac{12 \cdot 1}{t \cdot t}} + 6\right) - 8 \cdot \frac{1}{t}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{\color{blue}{12}}{t \cdot t} + 6\right) - 8 \cdot \frac{1}{t}} \]

      5. associate-*r/ 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{\color{blue}{8}}{t}} \]

   4. Simplified 99.7%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\left(\frac{12}{t \cdot t} + 6\right) - \frac{8}{t}}} \]

   5. Taylor expanded in t around inf 99.8%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \color{blue}{0.8333333333333334 + \left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \color{blue}{0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} - \frac{0.2222222222222222}{t}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 99.8%

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

      2. sub-div 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -0.80000000000000004 < t < 0.330000000000000016

   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. Taylor expanded in t around 0 99.3%

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

      1. unpow2 99.1%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in t around 0 99.3%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.3%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   7. Simplified 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 6: 99.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.56)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.78d0)) .or. (.not. (t <= 0.56d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.78) or not (t <= 0.56):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.78) || !(t <= 0.56))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.78) || ~((t <= 0.56)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.78], N[Not[LessEqual[t, 0.56]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.78000000000000003 or 0.56000000000000005 < 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. Taylor expanded in t around inf 99.2%

      \[\leadsto \frac{\color{blue}{5 - 8 \cdot \frac{1}{t}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.2%

         \[\leadsto \frac{5 - \color{blue}{\frac{8 \cdot 1}{t}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

      2. metadata-eval 99.2%

         \[\leadsto \frac{5 - \frac{\color{blue}{8}}{t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   4. Simplified 99.2%

      \[\leadsto \frac{\color{blue}{5 - \frac{8}{t}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   5. Taylor expanded in t around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   if -0.78000000000000003 < t < 0.56000000000000005

   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. Taylor expanded in t around 0 99.3%

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

      1. unpow2 99.1%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in t around 0 99.3%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.3%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   7. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 7: 98.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.58) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.9d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.9:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.9], 0.8333333333333334, If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.57999999999999996 < 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. Taylor expanded in t around inf 97.0%

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

   3. Taylor expanded in t around inf 97.3%

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

   if -0.900000000000000022 < t < 0.57999999999999996

   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. Taylor expanded in t around 0 99.3%

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

      1. unpow2 99.1%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in t around 0 99.3%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 99.3%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   7. Simplified 99.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8: 98.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   2. Taylor expanded in t around inf 97.0%

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

   3. Taylor expanded in t around inf 97.3%

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

   if -0.340000000000000024 < t < 1

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   2. Taylor expanded in t around 0 99.1%

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

   3. Taylor expanded in t around 0 99.1%

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

      1. unpow2 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

4. Final simplification 98.2%

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

Alternative 9: 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. Taylor expanded in t around 0 56.3%

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

3. Taylor expanded in t around 0 49.8%

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

   1. unpow2 49.8%

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

5. Simplified 49.8%

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

6. Taylor expanded in t around 0 57.8%

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

7. Final simplification 57.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023297 
(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))))))