Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 11.0s

Alternatives: 11

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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double t) {
	double t_1 = (t * 2.0) / (1.0 + t);
	return (1.0 + (1.0 + (Math.pow(t_1, 2.0) + -1.0))) / (2.0 + (t_1 * t_1));
}

Python

def code(t):
	t_1 = (t * 2.0) / (1.0 + t)
	return (1.0 + (1.0 + (math.pow(t_1, 2.0) + -1.0))) / (2.0 + (t_1 * t_1))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   2. expm1-udef 98.9%

      \[\leadsto \frac{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)} - 1\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. log1p-udef 98.9%

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

   4. add-exp-log 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. pow2 100.0%

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

   8. *-un-lft-identity 100.0%

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

   9. times-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate--l+ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 - \frac{8}{t}\\ t_2 := \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 100000000:\\ \;\;\;\;\frac{1 + t_2}{2 + t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 4.0 (/ 8.0 t)))
        (t_2 (/ (* t (* t 4.0)) (* (+ 1.0 t) (+ 1.0 t)))))
   (if (<= t -5e+154)
     0.8333333333333334
     (if (<= t 100000000.0)
       (/ (+ 1.0 t_2) (+ 2.0 t_2))
       (/ (+ 1.0 t_1) (+ 2.0 t_1))))))

C

double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double t_2 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	double tmp;
	if (t <= -5e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 100000000.0) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 4.0d0 - (8.0d0 / t)
    t_2 = (t * (t * 4.0d0)) / ((1.0d0 + t) * (1.0d0 + t))
    if (t <= (-5d+154)) then
        tmp = 0.8333333333333334d0
    else if (t <= 100000000.0d0) then
        tmp = (1.0d0 + t_2) / (2.0d0 + t_2)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double t_2 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	double tmp;
	if (t <= -5e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 100000000.0) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = 4.0 - (8.0 / t)
	t_2 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t))
	tmp = 0
	if t <= -5e+154:
		tmp = 0.8333333333333334
	elif t <= 100000000.0:
		tmp = (1.0 + t_2) / (2.0 + t_2)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(4.0 - Float64(8.0 / t))
	t_2 = Float64(Float64(t * Float64(t * 4.0)) / Float64(Float64(1.0 + t) * Float64(1.0 + t)))
	tmp = 0.0
	if (t <= -5e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 100000000.0)
		tmp = Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 4.0 - (8.0 / t);
	t_2 = (t * (t * 4.0)) / ((1.0 + t) * (1.0 + t));
	tmp = 0.0;
	if (t <= -5e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 100000000.0)
		tmp = (1.0 + t_2) / (2.0 + t_2);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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, -5e+154], 0.8333333333333334, If[LessEqual[t, 100000000.0], N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.00000000000000004e154

   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 -5.00000000000000004e154 < 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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \color{blue}{\left(4 - \frac{8}{t}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 100000000:\\ \;\;\;\;\frac{1 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \left(4 - \frac{8}{t}\right)}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t 2.0) (+ 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 = (t * 2.0) / (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 = (t * 2.0d0) / (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 = (t * 2.0) / (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 = (t * 2.0) / (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(t * 2.0) / 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 = (t * 2.0) / (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[(t * 2.0), $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%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 - \frac{8}{t}\\ t_2 := \frac{t \cdot \left(t \cdot 4\right)}{1 + t \cdot 2}\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 2.65:\\ \;\;\;\;\frac{1 + t_2}{2 + t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 4.0 (/ 8.0 t))) (t_2 (/ (* t (* t 4.0)) (+ 1.0 (* t 2.0)))))
   (if (<= t -0.6)
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (if (<= t 2.65)
       (/ (+ 1.0 t_2) (+ 2.0 t_2))
       (/ (+ 1.0 t_1) (+ 2.0 t_1))))))

C

double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double t_2 = (t * (t * 4.0)) / (1.0 + (t * 2.0));
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 2.65) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 4.0d0 - (8.0d0 / t)
    t_2 = (t * (t * 4.0d0)) / (1.0d0 + (t * 2.0d0))
    if (t <= (-0.6d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 2.65d0) then
        tmp = (1.0d0 + t_2) / (2.0d0 + t_2)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double t_2 = (t * (t * 4.0)) / (1.0 + (t * 2.0));
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 2.65) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = 4.0 - (8.0 / t)
	t_2 = (t * (t * 4.0)) / (1.0 + (t * 2.0))
	tmp = 0
	if t <= -0.6:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 2.65:
		tmp = (1.0 + t_2) / (2.0 + t_2)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(4.0 - Float64(8.0 / t))
	t_2 = Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + Float64(t * 2.0)))
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 2.65)
		tmp = Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 4.0 - (8.0 / t);
	t_2 = (t * (t * 4.0)) / (1.0 + (t * 2.0));
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 2.65)
		tmp = (1.0 + t_2) / (2.0 + t_2);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.6], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.65], N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   4. Simplified 98.9%

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

   if -0.599999999999999978 < t < 2.64999999999999991

   1. Initial program 100.0%

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

      1. times-frac 100.0%

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

      2. sqr-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. swap-sqr 100.0%

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

      6. *-commutative 100.0%

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

      7. sqr-neg 100.0%

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

      8. associate-*r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.3%

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

   5. Taylor expanded in t around 0 99.3%

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

   if 2.64999999999999991 < 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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \color{blue}{\left(4 - \frac{8}{t}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 2.65:\\ \;\;\;\;\frac{1 + \frac{t \cdot \left(t \cdot 4\right)}{1 + t \cdot 2}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{1 + t \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \left(4 - \frac{8}{t}\right)}\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot 4\right)\\ t_2 := 4 - \frac{8}{t}\\ \mathbf{if}\;t \leq -0.385:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1.72:\\ \;\;\;\;\frac{1 + \frac{t_1}{1 + t \cdot 2}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_2}{2 + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (* t 4.0))) (t_2 (- 4.0 (/ 8.0 t))))
   (if (<= t -0.385)
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (if (<= t 1.72)
       (/ (+ 1.0 (/ t_1 (+ 1.0 (* t 2.0)))) (+ 2.0 t_1))
       (/ (+ 1.0 t_2) (+ 2.0 t_2))))))

C

double code(double t) {
	double t_1 = t * (t * 4.0);
	double t_2 = 4.0 - (8.0 / t);
	double tmp;
	if (t <= -0.385) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 1.72) {
		tmp = (1.0 + (t_1 / (1.0 + (t * 2.0)))) / (2.0 + t_1);
	} else {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (t * 4.0d0)
    t_2 = 4.0d0 - (8.0d0 / t)
    if (t <= (-0.385d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 1.72d0) then
        tmp = (1.0d0 + (t_1 / (1.0d0 + (t * 2.0d0)))) / (2.0d0 + t_1)
    else
        tmp = (1.0d0 + t_2) / (2.0d0 + t_2)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (t * 4.0);
	double t_2 = 4.0 - (8.0 / t);
	double tmp;
	if (t <= -0.385) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 1.72) {
		tmp = (1.0 + (t_1 / (1.0 + (t * 2.0)))) / (2.0 + t_1);
	} else {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (t * 4.0)
	t_2 = 4.0 - (8.0 / t)
	tmp = 0
	if t <= -0.385:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 1.72:
		tmp = (1.0 + (t_1 / (1.0 + (t * 2.0)))) / (2.0 + t_1)
	else:
		tmp = (1.0 + t_2) / (2.0 + t_2)
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(t * 4.0))
	t_2 = Float64(4.0 - Float64(8.0 / t))
	tmp = 0.0
	if (t <= -0.385)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 1.72)
		tmp = Float64(Float64(1.0 + Float64(t_1 / Float64(1.0 + Float64(t * 2.0)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (t * 4.0);
	t_2 = 4.0 - (8.0 / t);
	tmp = 0.0;
	if (t <= -0.385)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 1.72)
		tmp = (1.0 + (t_1 / (1.0 + (t * 2.0)))) / (2.0 + t_1);
	else
		tmp = (1.0 + t_2) / (2.0 + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.385], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.72], N[(N[(1.0 + N[(t$95$1 / N[(1.0 + N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.38500000000000001

   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 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   4. Simplified 98.9%

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

   if -0.38500000000000001 < t < 1.71999999999999997

   1. Initial program 100.0%

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

      1. times-frac 100.0%

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

      2. sqr-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. swap-sqr 100.0%

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

      6. *-commutative 100.0%

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

      7. sqr-neg 100.0%

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

      8. associate-*r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.0%

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

   5. Taylor expanded in t around 0 99.0%

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

   if 1.71999999999999997 < 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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \color{blue}{\left(4 - \frac{8}{t}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.385:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1.72:\\ \;\;\;\;\frac{1 + \frac{t \cdot \left(t \cdot 4\right)}{1 + t \cdot 2}}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \left(4 - \frac{8}{t}\right)}\\ \end{array} \]

Alternative 6: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 - \frac{8}{t}\\ \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1.65:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 4.0 (/ 8.0 t))))
   (if (<= t -0.78)
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (if (<= t 1.65) (+ 0.5 (* t t)) (/ (+ 1.0 t_1) (+ 2.0 t_1))))))

C

double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 1.65) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 4.0d0 - (8.0d0 / t)
    if (t <= (-0.78d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 1.65d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 1.65) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = 4.0 - (8.0 / t)
	tmp = 0
	if t <= -0.78:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 1.65:
		tmp = 0.5 + (t * t)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(4.0 - Float64(8.0 / t))
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 1.65)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 4.0 - (8.0 / t);
	tmp = 0.0;
	if (t <= -0.78)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 1.65)
		tmp = 0.5 + (t * t);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.78], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $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 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   4. Simplified 98.9%

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

   if -0.78000000000000003 < t < 1.6499999999999999

   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 99.0%

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

      1. unpow2 99.0%

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

   4. Simplified 99.0%

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

   if 1.6499999999999999 < 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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \color{blue}{\left(4 - \frac{8}{t}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1.65:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \left(4 - \frac{8}{t}\right)}\\ \end{array} \]

Alternative 7: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 - \frac{8}{t}\\ t_2 := t \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 2.05:\\ \;\;\;\;\frac{1 + t_2}{2 + t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 4.0 (/ 8.0 t))) (t_2 (* t (* t 4.0))))
   (if (<= t -0.58)
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (if (<= t 2.05)
       (/ (+ 1.0 t_2) (+ 2.0 t_2))
       (/ (+ 1.0 t_1) (+ 2.0 t_1))))))

C

double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double t_2 = t * (t * 4.0);
	double tmp;
	if (t <= -0.58) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 2.05) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 4.0d0 - (8.0d0 / t)
    t_2 = t * (t * 4.0d0)
    if (t <= (-0.58d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 2.05d0) then
        tmp = (1.0d0 + t_2) / (2.0d0 + t_2)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 4.0 - (8.0 / t);
	double t_2 = t * (t * 4.0);
	double tmp;
	if (t <= -0.58) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 2.05) {
		tmp = (1.0 + t_2) / (2.0 + t_2);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = 4.0 - (8.0 / t)
	t_2 = t * (t * 4.0)
	tmp = 0
	if t <= -0.58:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 2.05:
		tmp = (1.0 + t_2) / (2.0 + t_2)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(4.0 - Float64(8.0 / t))
	t_2 = Float64(t * Float64(t * 4.0))
	tmp = 0.0
	if (t <= -0.58)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 2.05)
		tmp = Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 4.0 - (8.0 / t);
	t_2 = t * (t * 4.0);
	tmp = 0.0;
	if (t <= -0.58)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 2.05)
		tmp = (1.0 + t_2) / (2.0 + t_2);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.58], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05], N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $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 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   4. Simplified 98.9%

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

   if -0.57999999999999996 < t < 2.0499999999999998

   1. Initial program 100.0%

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

      1. times-frac 100.0%

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

      2. sqr-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. swap-sqr 100.0%

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

      6. *-commutative 100.0%

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

      7. sqr-neg 100.0%

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

      8. associate-*r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 99.0%

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

   5. Taylor expanded in t around 0 99.0%

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

   if 2.0499999999999998 < 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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \color{blue}{\left(4 - \frac{8}{t}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 2.05:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \left(4 - \frac{8}{t}\right)}\\ \end{array} \]

Alternative 8: 99.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5 + (t * t);
	}
	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 = 0.5d0 + (t * t)
    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 = 0.5 + (t * t);
	}
	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 = 0.5 + (t * t)
	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(0.5 + Float64(t * t));
	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 = 0.5 + (t * t);
	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[(0.5 + N[(t * t), $MachinePrecision]), $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 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   4. Simplified 99.4%

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

   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 99.0%

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

      1. unpow2 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 99.2%

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

Alternative 9: 98.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * t);
	} 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 = 0.5d0 + (t * t)
    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 = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.9:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = 0.5 + (t * t)
	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(0.5 + Float64(t * t));
	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 = 0.5 + (t * t);
	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[(0.5 + N[(t * t), $MachinePrecision]), $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 99.1%

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

   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 99.0%

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

      1. unpow2 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 99.1%

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

Alternative 10: 98.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 99.1%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.9%

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

Alternative 11: 59.5% 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 61.1%

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

3. Final simplification 61.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023279 
(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))))))