Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.7%

Time: 13.8s

Alternatives: 8

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+158}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 100000000:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{-0.2222222222222222}{t} - \frac{1.962962962962963}{t \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t (* t 4.0)) (* (+ 1.0 t) (+ 1.0 t)))))
   (if (<= t -1e+158)
     0.8333333333333334
     (if (<= t 100000000.0)
       (/ (+ 1.0 t_1) (+ 2.0 t_1))
       (+
        0.8333333333333334
        (- (/ -0.2222222222222222 t) (/ 1.962962962962963 (* t t))))))))

C

double code(double t) {
	double t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	double tmp;
	if (t <= -1e+158) {
		tmp = 0.8333333333333334;
	} else if (t <= 100000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 / t) - (1.962962962962963 / (t * 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 <= (-1d+158)) then
        tmp = 0.8333333333333334d0
    else if (t <= 100000000.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = 0.8333333333333334d0 + (((-0.2222222222222222d0) / t) - (1.962962962962963d0 / (t * 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 <= -1e+158) {
		tmp = 0.8333333333333334;
	} else if (t <= 100000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 / t) - (1.962962962962963 / (t * t)));
	}
	return tmp;
}

Python

def code(t):
	t_1 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t))
	tmp = 0
	if t <= -1e+158:
		tmp = 0.8333333333333334
	elif t <= 100000000.0:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = 0.8333333333333334 + ((-0.2222222222222222 / t) - (1.962962962962963 / (t * 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 (t <= -1e+158)
		tmp = 0.8333333333333334;
	elseif (t <= 100000000.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 / t) - Float64(1.962962962962963 / Float64(t * 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 <= -1e+158)
		tmp = 0.8333333333333334;
	elseif (t <= 100000000.0)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = 0.8333333333333334 + ((-0.2222222222222222 / t) - (1.962962962962963 / (t * 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[t, -1e+158], 0.8333333333333334, If[LessEqual[t, 100000000.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(-0.2222222222222222 / t), $MachinePrecision] - N[(1.962962962962963 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.99999999999999953e157

   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 100.0%

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

   if -9.99999999999999953e157 < t < 1e8

   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%

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

   if 1e8 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \left(0.2222222222222222 \cdot \frac{1}{t} + 1.962962962962963 \cdot \frac{1}{{t}^{2}}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) - 1.962962962962963 \cdot \frac{1}{{t}^{2}}} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} - 1.962962962962963 \cdot \frac{1}{{t}^{2}} \]

      3. associate-*r/ 100.0%

         \[\leadsto \left(0.8333333333333334 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) - 1.962962962962963 \cdot \frac{1}{{t}^{2}} \]

      4. metadata-eval 100.0%

         \[\leadsto \left(0.8333333333333334 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) - 1.962962962962963 \cdot \frac{1}{{t}^{2}} \]

      5. associate--l+ 100.0%

         \[\leadsto \color{blue}{0.8333333333333334 + \left(\left(-\frac{0.2222222222222222}{t}\right) - 1.962962962962963 \cdot \frac{1}{{t}^{2}}\right)} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto 0.8333333333333334 + \left(\color{blue}{\frac{-0.2222222222222222}{t}} - 1.962962962962963 \cdot \frac{1}{{t}^{2}}\right) \]

      7. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 + \left(\frac{\color{blue}{-0.2222222222222222}}{t} - 1.962962962962963 \cdot \frac{1}{{t}^{2}}\right) \]

      8. associate-*r/ 100.0%

         \[\leadsto 0.8333333333333334 + \left(\frac{-0.2222222222222222}{t} - \color{blue}{\frac{1.962962962962963 \cdot 1}{{t}^{2}}}\right) \]

      9. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 + \left(\frac{-0.2222222222222222}{t} - \frac{\color{blue}{1.962962962962963}}{{t}^{2}}\right) \]

      10. unpow2 100.0%

          \[\leadsto 0.8333333333333334 + \left(\frac{-0.2222222222222222}{t} - \frac{1.962962962962963}{\color{blue}{t \cdot t}}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + \left(\frac{-0.2222222222222222}{t} - \frac{1.962962962962963}{t \cdot t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+158}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 100000000:\\ \;\;\;\;\frac{1 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{-0.2222222222222222}{t} - \frac{1.962962962962963}{t \cdot t}\right)\\ \end{array} \]

Alternative 2: 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 3: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \left(t + t\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.58)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.56)
     (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (+ t t))) (+ 2.0 (* 4.0 (* t t))))
     (+
      0.8333333333333334
      (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.58) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.56) {
		tmp = (1.0 + (((2.0 * t) / (1.0 + t)) * (t + t))) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.58d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.56d0) then
        tmp = (1.0d0 + (((2.0d0 * t) / (1.0d0 + t)) * (t + t))) / (2.0d0 + (4.0d0 * (t * t)))
    else
        tmp = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.58) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.56) {
		tmp = (1.0 + (((2.0 * t) / (1.0 + t)) * (t + t))) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.58:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.56:
		tmp = (1.0 + (((2.0 * t) / (1.0 + t)) * (t + t))) / (2.0 + (4.0 * (t * t)))
	else:
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.58)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.56)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(2.0 * t) / Float64(1.0 + t)) * Float64(t + t))) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.58)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.56)
		tmp = (1.0 + (((2.0 * t) / (1.0 + t)) * (t + t))) / (2.0 + (4.0 * (t * t)));
	else
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.58], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.56], N[(N[(1.0 + N[(N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.57999999999999996

   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 inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   4. Simplified 98.6%

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

   if -0.57999999999999996 < 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 98.9%

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

      1. unpow2 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in t around 0 99.0%

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

      1. count-2 99.0%

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

   7. Simplified 99.0%

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

   if 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 98.6%

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

      1. associate--l+ 98.6%

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

      2. associate-*r/ 98.6%

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

      3. metadata-eval 98.6%

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

      4. unpow2 98.6%

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

      5. associate-*r/ 98.6%

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

      6. metadata-eval 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

      3. unpow2 98.6%

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

      4. associate-*r/ 98.6%

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

      5. metadata-eval 98.6%

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

      6. *-rgt-identity 98.6%

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

      7. unpow2 98.6%

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

      8. *-inverses 98.6%

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

      9. times-frac 98.6%

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

      10. unpow2 98.6%

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

      11. *-lft-identity 98.6%

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

      12. *-inverses 50.8%

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

      13. times-frac 50.8%

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

      14. unpow2 50.8%

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

      15. unpow2 50.8%

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

      16. div-sub 50.8%

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

      17. unpow2 50.8%

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

   7. Simplified 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \left(t + t\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]

Alternative 4: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.78)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.235)
     (+ (* t t) 0.5)
     (+
      0.8333333333333334
      (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.78d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.235d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.78:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.235:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.235)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.78)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.235)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.78], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.235], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.78000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   4. Simplified 98.6%

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

   if -0.78000000000000003 < t < 0.23499999999999999

   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 98.9%

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

      1. +-commutative 98.9%

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

      2. unpow2 98.9%

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

   4. Simplified 98.9%

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

   if 0.23499999999999999 < 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 98.6%

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

      1. associate--l+ 98.6%

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

      2. associate-*r/ 98.6%

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

      3. metadata-eval 98.6%

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

      4. unpow2 98.6%

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

      5. associate-*r/ 98.6%

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

      6. metadata-eval 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

      3. unpow2 98.6%

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

      4. associate-*r/ 98.6%

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

      5. metadata-eval 98.6%

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

      6. *-rgt-identity 98.6%

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

      7. unpow2 98.6%

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

      8. *-inverses 98.6%

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

      9. times-frac 98.6%

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

      10. unpow2 98.6%

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

      11. *-lft-identity 98.6%

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

      12. *-inverses 50.8%

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

      13. times-frac 50.8%

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

      14. unpow2 50.8%

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

      15. unpow2 50.8%

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

      16. div-sub 50.8%

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

      17. unpow2 50.8%

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

   7. Simplified 98.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]

Alternative 5: 99.2% 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 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   4. Simplified 98.4%

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

   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 98.9%

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

      1. +-commutative 98.9%

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

      2. unpow2 98.9%

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

   4. Simplified 98.9%

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

4. Final simplification 98.7%

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

Alternative 6: 98.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.92)
   0.8333333333333334
   (if (<= t 0.56) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		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 <= (-0.92d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.56d0) 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 <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.92:
		tmp = 0.8333333333333334
	elif t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		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 <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.92], 0.8333333333333334, If[LessEqual[t, 0.56], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.92000000000000004 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 97.5%

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

   if -0.92000000000000004 < 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 98.9%

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

      1. +-commutative 98.9%

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

      2. unpow2 98.9%

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

   4. Simplified 98.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 7: 98.4% 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. Taylor expanded in t around inf 97.5%

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

   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. Taylor expanded in t around 0 98.5%

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

4. Final simplification 98.0%

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

Alternative 8: 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 61.2%

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

3. Final simplification 61.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023282 
(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))))))