Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 13.1s

Alternatives: 8

Speedup: 1.6×

Specification

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 = 2.0 - ((2.0 / t) / (1.0 + (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 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (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 = 2.0 - ((2.0 / t) / (1.0 + (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 = 2.0 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 = 2.0 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 = 2.0 - ((2.0 / t) / (1.0 + (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 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (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 = 2.0 - ((2.0 / t) / (1.0 + (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 = 2.0 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 = 2.0 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ \frac{5 - t_1 \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + t_1 \cdot \left(t_1 - 4\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))))
   (/
    (- 5.0 (* t_1 (- (/ -2.0 (+ 1.0 t)) -4.0)))
    (+ 6.0 (* t_1 (- t_1 4.0))))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return (5.0 - (t_1 * ((-2.0 / (1.0 + t)) - -4.0))) / (6.0 + (t_1 * (t_1 - 4.0)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 / (1.0d0 + t)
    code = (5.0d0 - (t_1 * (((-2.0d0) / (1.0d0 + t)) - (-4.0d0)))) / (6.0d0 + (t_1 * (t_1 - 4.0d0)))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return (5.0 - (t_1 * ((-2.0 / (1.0 + t)) - -4.0))) / (6.0 + (t_1 * (t_1 - 4.0)));
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	return (5.0 - (t_1 * ((-2.0 / (1.0 + t)) - -4.0))) / (6.0 + (t_1 * (t_1 - 4.0)))

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	return Float64(Float64(5.0 - Float64(t_1 * Float64(Float64(-2.0 / Float64(1.0 + t)) - -4.0))) / Float64(6.0 + Float64(t_1 * Float64(t_1 - 4.0))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	tmp = (5.0 - (t_1 * ((-2.0 / (1.0 + t)) - -4.0))) / (6.0 + (t_1 * (t_1 - 4.0)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 - N[(t$95$1 * N[(N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(t$95$1 * N[(t$95$1 - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

4. Final simplification 100.0%

   \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

Alternative 2: 98.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{5 - \frac{\frac{-4}{1 + t} + 8}{t}}{6 + \frac{-8}{t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/ (- 5.0 (/ (+ (/ -4.0 (+ 1.0 t)) 8.0) t)) (+ 6.0 (/ -8.0 t))))

C

double code(double t) {
	return (5.0 - (((-4.0 / (1.0 + t)) + 8.0) / t)) / (6.0 + (-8.0 / t));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (5.0d0 - ((((-4.0d0) / (1.0d0 + t)) + 8.0d0) / t)) / (6.0d0 + ((-8.0d0) / t))
end function

Java

public static double code(double t) {
	return (5.0 - (((-4.0 / (1.0 + t)) + 8.0) / t)) / (6.0 + (-8.0 / t));
}

Python

def code(t):
	return (5.0 - (((-4.0 / (1.0 + t)) + 8.0) / t)) / (6.0 + (-8.0 / t))

Julia

function code(t)
	return Float64(Float64(5.0 - Float64(Float64(Float64(-4.0 / Float64(1.0 + t)) + 8.0) / t)) / Float64(6.0 + Float64(-8.0 / t)))
end

MATLAB

function tmp = code(t)
	tmp = (5.0 - (((-4.0 / (1.0 + t)) + 8.0) / t)) / (6.0 + (-8.0 / t));
end

Wolfram

code[t_] := N[(N[(5.0 - N[(N[(N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + 8.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

4. Taylor expanded in t around inf 44.8%

   \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

5. Taylor expanded in t around inf 99.0%

   \[\leadsto \frac{5 - \color{blue}{\frac{2}{t}} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{-8}{t}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 69.0%

      \[\leadsto \frac{5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{t} \cdot \left(\frac{-2}{1 + t} - -4\right)\right)\right)}}{6 + \frac{-8}{t}} \]

   2. expm1-udef 69.0%

      \[\leadsto \frac{5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{t} \cdot \left(\frac{-2}{1 + t} - -4\right)\right)} - 1\right)}}{6 + \frac{-8}{t}} \]

   3. sub-neg 69.0%

      \[\leadsto \frac{5 - \left(e^{\mathsf{log1p}\left(\frac{2}{t} \cdot \color{blue}{\left(\frac{-2}{1 + t} + \left(--4\right)\right)}\right)} - 1\right)}{6 + \frac{-8}{t}} \]

   4. +-commutative 69.0%

      \[\leadsto \frac{5 - \left(e^{\mathsf{log1p}\left(\frac{2}{t} \cdot \left(\frac{-2}{\color{blue}{t + 1}} + \left(--4\right)\right)\right)} - 1\right)}{6 + \frac{-8}{t}} \]

   5. metadata-eval 69.0%

      \[\leadsto \frac{5 - \left(e^{\mathsf{log1p}\left(\frac{2}{t} \cdot \left(\frac{-2}{t + 1} + \color{blue}{4}\right)\right)} - 1\right)}{6 + \frac{-8}{t}} \]

7. Applied egg-rr 69.0%

   \[\leadsto \frac{5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{t} \cdot \left(\frac{-2}{t + 1} + 4\right)\right)} - 1\right)}}{6 + \frac{-8}{t}} \]
8. Step-by-step derivation

   1. expm1-def 69.0%

      \[\leadsto \frac{5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{t} \cdot \left(\frac{-2}{t + 1} + 4\right)\right)\right)}}{6 + \frac{-8}{t}} \]

   2. expm1-log1p 99.0%

      \[\leadsto \frac{5 - \color{blue}{\frac{2}{t} \cdot \left(\frac{-2}{t + 1} + 4\right)}}{6 + \frac{-8}{t}} \]

   3. associate-*l/ 99.0%

      \[\leadsto \frac{5 - \color{blue}{\frac{2 \cdot \left(\frac{-2}{t + 1} + 4\right)}{t}}}{6 + \frac{-8}{t}} \]

   4. +-commutative 99.0%

      \[\leadsto \frac{5 - \frac{2 \cdot \color{blue}{\left(4 + \frac{-2}{t + 1}\right)}}{t}}{6 + \frac{-8}{t}} \]

   5. distribute-lft-in 99.0%

      \[\leadsto \frac{5 - \frac{\color{blue}{2 \cdot 4 + 2 \cdot \frac{-2}{t + 1}}}{t}}{6 + \frac{-8}{t}} \]

   6. metadata-eval 99.0%

      \[\leadsto \frac{5 - \frac{\color{blue}{8} + 2 \cdot \frac{-2}{t + 1}}{t}}{6 + \frac{-8}{t}} \]

   7. associate-*r/ 99.0%

      \[\leadsto \frac{5 - \frac{8 + \color{blue}{\frac{2 \cdot -2}{t + 1}}}{t}}{6 + \frac{-8}{t}} \]

   8. metadata-eval 99.0%

      \[\leadsto \frac{5 - \frac{8 + \frac{\color{blue}{-4}}{t + 1}}{t}}{6 + \frac{-8}{t}} \]

9. Simplified 99.0%

   \[\leadsto \frac{5 - \color{blue}{\frac{8 + \frac{-4}{t + 1}}{t}}}{6 + \frac{-8}{t}} \]

10. Final simplification 99.0%

    \[\leadsto \frac{5 - \frac{\frac{-4}{1 + t} + 8}{t}}{6 + \frac{-8}{t}} \]

Alternative 3: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 + \frac{0.037037037037037035}{t \cdot t}\right) - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.78)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.33)
     (+ (* t t) 0.5)
     (-
      (+ 0.8333333333333334 (/ 0.037037037037037035 (* t t)))
      (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.33) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / 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 <= 0.33d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = (0.8333333333333334d0 + (0.037037037037037035d0 / (t * t))) - (0.2222222222222222d0 / 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 <= 0.33) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.78:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.33:
		tmp = (t * t) + 0.5
	else:
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.33)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(Float64(0.8333333333333334 + Float64(0.037037037037037035 / Float64(t * t))) - Float64(0.2222222222222222 / 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 <= 0.33)
		tmp = (t * t) + 0.5;
	else
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / 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, 0.33], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(0.8333333333333334 + N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.78000000000000003

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.78000000000000003 < t < 0.330000000000000016

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around 0 100.0%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + -4 \cdot {t}^{2}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{{t}^{2} \cdot -4}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. unpow2 100.0%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{\left(t \cdot t\right)} \cdot -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. Simplified 100.0%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + \left(t \cdot t\right) \cdot -4\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   7. Taylor expanded in t around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if 0.330000000000000016 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 97.7%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\color{blue}{\frac{12 \cdot 1}{{t}^{2}}} - 8 \cdot \frac{1}{t}\right)} \]

      2. metadata-eval 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{\color{blue}{12}}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)} \]

      3. unpow2 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{12}{\color{blue}{t \cdot t}} - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{12}{t \cdot t} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      5. metadata-eval 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{12}{t \cdot t} - \frac{\color{blue}{8}}{t}\right)} \]

   6. Simplified 97.7%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\left(\frac{12}{t \cdot t} - \frac{8}{t}\right)}} \]

   7. Taylor expanded in t around inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

      3. unpow2 98.0%

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

      4. associate-*r/ 98.0%

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

      5. metadata-eval 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 + \frac{0.037037037037037035}{t \cdot t}\right) - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 4: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.56:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 + \frac{0.037037037037037035}{t \cdot t}\right) - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.56)
   (/ (- 5.0 (/ 8.0 t)) (+ 6.0 (/ -8.0 t)))
   (if (<= t 0.33)
     (+ (* t t) 0.5)
     (-
      (+ 0.8333333333333334 (/ 0.037037037037037035 (* t t)))
      (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.56) {
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	} else if (t <= 0.33) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.56d0)) then
        tmp = (5.0d0 - (8.0d0 / t)) / (6.0d0 + ((-8.0d0) / t))
    else if (t <= 0.33d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = (0.8333333333333334d0 + (0.037037037037037035d0 / (t * t))) - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.56) {
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	} else if (t <= 0.33) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.56:
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t))
	elif t <= 0.33:
		tmp = (t * t) + 0.5
	else:
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.56)
		tmp = Float64(Float64(5.0 - Float64(8.0 / t)) / Float64(6.0 + Float64(-8.0 / t)));
	elseif (t <= 0.33)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(Float64(0.8333333333333334 + Float64(0.037037037037037035 / Float64(t * t))) - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.56)
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	elseif (t <= 0.33)
		tmp = (t * t) + 0.5;
	else
		tmp = (0.8333333333333334 + (0.037037037037037035 / (t * t))) - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.56], N[(N[(5.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.33], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(0.8333333333333334 + N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 - \color{blue}{\frac{8}{t}}}{6 + \frac{-8}{t}} \]

   if -0.56000000000000005 < t < 0.330000000000000016

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around 0 100.0%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + -4 \cdot {t}^{2}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{{t}^{2} \cdot -4}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. unpow2 100.0%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{\left(t \cdot t\right)} \cdot -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. Simplified 100.0%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + \left(t \cdot t\right) \cdot -4\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   7. Taylor expanded in t around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if 0.330000000000000016 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 97.7%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\left(12 \cdot \frac{1}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\color{blue}{\frac{12 \cdot 1}{{t}^{2}}} - 8 \cdot \frac{1}{t}\right)} \]

      2. metadata-eval 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{\color{blue}{12}}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)} \]

      3. unpow2 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{12}{\color{blue}{t \cdot t}} - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{12}{t \cdot t} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      5. metadata-eval 97.7%

         \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \left(\frac{12}{t \cdot t} - \frac{\color{blue}{8}}{t}\right)} \]

   6. Simplified 97.7%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\left(\frac{12}{t \cdot t} - \frac{8}{t}\right)}} \]

   7. Taylor expanded in t around inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

      3. unpow2 98.0%

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

      4. associate-*r/ 98.0%

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

      5. metadata-eval 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.56:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 + \frac{0.037037037037037035}{t \cdot t}\right) - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 99.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.55)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.55)) {
		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 <= 0.55d0))) 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 <= 0.55)) {
		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 <= 0.55):
		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 <= 0.55))
		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 <= 0.55)))
		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, 0.55]], $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 0.55000000000000004 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 99.4%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

   5. Taylor expanded in t around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   if -0.78000000000000003 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around 0 99.4%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + -4 \cdot {t}^{2}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{{t}^{2} \cdot -4}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. unpow2 99.4%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{\left(t \cdot t\right)} \cdot -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. Simplified 99.4%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + \left(t \cdot t\right) \cdot -4\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   7. Taylor expanded in t around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 6: 98.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.91:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.91)
   0.8333333333333334
   (if (<= t 0.58) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.91) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.91d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.91) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.91:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.91)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.91)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.91], 0.8333333333333334, If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.910000000000000031 or 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 99.4%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

   5. Taylor expanded in t around inf 98.8%

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

   if -0.910000000000000031 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around 0 99.4%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + -4 \cdot {t}^{2}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{{t}^{2} \cdot -4}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. unpow2 99.4%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{\left(t \cdot t\right)} \cdot -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. Simplified 99.4%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + \left(t \cdot t\right) \cdot -4\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   7. Taylor expanded in t around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.91:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 7: 98.5% accurate, 10.0× 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%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 99.4%

      \[\leadsto \frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \color{blue}{\frac{-8}{t}}} \]

   5. Taylor expanded in t around inf 98.8%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around 0 99.4%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + -4 \cdot {t}^{2}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{{t}^{2} \cdot -4}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

      2. unpow2 99.4%

         \[\leadsto \frac{5 - \left(4 + \color{blue}{\left(t \cdot t\right)} \cdot -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   6. Simplified 99.4%

      \[\leadsto \frac{5 - \color{blue}{\left(4 + \left(t \cdot t\right) \cdot -4\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   7. Taylor expanded in t around 0 99.5%

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

4. Final simplification 99.2%

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

Alternative 8: 59.0% accurate, 51.0× 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%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 - \frac{2}{1 + t} \cdot \left(\frac{-2}{1 + t} - -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

4. Taylor expanded in t around 0 58.5%

   \[\leadsto \frac{5 - \color{blue}{\left(4 + -4 \cdot {t}^{2}\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]
5. Step-by-step derivation

   1. *-commutative 58.5%

      \[\leadsto \frac{5 - \left(4 + \color{blue}{{t}^{2} \cdot -4}\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

   2. unpow2 58.5%

      \[\leadsto \frac{5 - \left(4 + \color{blue}{\left(t \cdot t\right)} \cdot -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

6. Simplified 58.5%

   \[\leadsto \frac{5 - \color{blue}{\left(4 + \left(t \cdot t\right) \cdot -4\right)}}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)} \]

7. Taylor expanded in t around 0 65.2%

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

8. Final simplification 65.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))