Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 7

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (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 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (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 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{6 + \left(8 + \frac{-4}{1 + t}\right) \cdot \frac{1}{-1 - t}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. frac-2neg 100.0%

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

   2. div-inv 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. +-commutative 100.0%

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

   7. distribute-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-neg-frac 100.0%

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

   2. metadata-eval 100.0%

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

   3. unsub-neg 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.82) (not (<= t 0.33)))
   (+
    1.0
    (-
     (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)
     0.16666666666666666))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.82) || !(t <= 0.33)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.82d0)) .or. (.not. (t <= 0.33d0))) then
        tmp = 1.0d0 + ((((0.037037037037037035d0 / t) - 0.2222222222222222d0) / t) - 0.16666666666666666d0)
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.82) || !(t <= 0.33)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.82) or not (t <= 0.33):
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666)
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.82) || !(t <= 0.33))
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.82) || ~((t <= 0.33)))
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.82], N[Not[LessEqual[t, 0.33]], $MachinePrecision]], N[(1.0 + N[(N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.330000000000000016 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

      2. metadata-eval 100.0%

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

      3. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 99.5%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto 1 - \left(\left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) \]

      2. metadata-eval 99.5%

         \[\leadsto 1 - \left(\left(0.16666666666666666 + \frac{\color{blue}{0.2222222222222222}}{t}\right) - 0.037037037037037035 \cdot \frac{1}{{t}^{2}}\right) \]

      3. associate-*r/ 99.5%

         \[\leadsto 1 - \left(\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right) - \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}}\right) \]

      4. metadata-eval 99.5%

         \[\leadsto 1 - \left(\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right) - \frac{\color{blue}{0.037037037037037035}}{{t}^{2}}\right) \]

      5. unpow2 99.5%

         \[\leadsto 1 - \left(\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right) - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right) \]

   10. Simplified 99.5%

       \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right) - \frac{0.037037037037037035}{t \cdot t}\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 99.5%

          \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035}{t \cdot t}\right)\right)} \]

       2. +-commutative 99.5%

          \[\leadsto 1 - \color{blue}{\left(\left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035}{t \cdot t}\right) + 0.16666666666666666\right)} \]

       3. associate-/r* 99.5%

          \[\leadsto 1 - \left(\left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right) + 0.16666666666666666\right) \]

       4. sub-div 99.5%

          \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}} + 0.16666666666666666\right) \]

   12. Applied egg-rr 99.5%

       \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right)} \]

   if -0.819999999999999951 < t < 0.330000000000000016

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. unpow2 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in t around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 4: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.55)))
   (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.55)) {
		tmp = 1.0 - (0.16666666666666666 + (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 = 1.0d0 - (0.16666666666666666d0 + (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 = 1.0 - (0.16666666666666666 + (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 = 1.0 - (0.16666666666666666 + (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(1.0 - Float64(0.16666666666666666 + 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 = 1.0 - (0.16666666666666666 + (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[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $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%

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

   2. Taylor expanded in t around inf 99.3%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.3%

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) \]

      2. metadata-eval 99.3%

         \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.2222222222222222}}{t}\right) \]

   4. Simplified 99.3%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]

   if -0.78000000000000003 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. unpow2 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in t around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 5: 98.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.41:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.41)
   0.8333333333333334
   (if (<= t 0.6) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.41) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.6) {
		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 <= (-0.41d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.6d0) 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 <= -0.41) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.6) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.41:
		tmp = 0.8333333333333334
	elif t <= 0.6:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.41)
		tmp = 0.8333333333333334;
	elseif (t <= 0.6)
		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 <= -0.41)
		tmp = 0.8333333333333334;
	elseif (t <= 0.6)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.41], 0.8333333333333334, If[LessEqual[t, 0.6], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.409999999999999976 or 0.599999999999999978 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 98.5%

      \[\leadsto 1 - \color{blue}{0.16666666666666666} \]

   if -0.409999999999999976 < t < 0.599999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. unpow2 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in t around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.41:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 6: 98.7% accurate, 5.7× 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%

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

   2. Taylor expanded in t around inf 98.5%

      \[\leadsto 1 - \color{blue}{0.16666666666666666} \]

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. unpow2 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in t around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in t around 0 97.5%

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

4. Final simplification 98.1%

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

Alternative 7: 59.6% accurate, 29.0× 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%

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

2. Taylor expanded in t around 0 55.4%

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

   1. *-commutative 55.4%

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

   2. unpow2 55.4%

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

4. Simplified 55.4%

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

5. Taylor expanded in t around 0 45.9%

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

   1. +-commutative 45.9%

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

   2. unpow2 45.9%

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

7. Simplified 45.9%

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

8. Taylor expanded in t around 0 54.3%

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

9. Final simplification 54.3%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023192 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))