Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 12.9s

Alternatives: 6

Speedup: N/A×

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, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

      \[\leadsto 1 - \frac{1}{\color{blue}{\left(2 + \left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}}} \]

   2. lift-+.f64 N/A

      \[\leadsto 1 - \frac{1}{\color{blue}{\left(2 + \left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right)\right)} + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}} \]

   3. associate-+l+ N/A

      \[\leadsto 1 - \frac{1}{\color{blue}{2 + \left(\left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)}} \]

   4. +-commutative N/A

      \[\leadsto 1 - \frac{1}{\color{blue}{\left(\left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right) + 2}} \]

   5. lower-+.f64 N/A

      \[\leadsto 1 - \frac{1}{\color{blue}{\left(\left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right) + 2}} \]

6. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

      \[\leadsto 1 - \frac{1}{\color{blue}{2 \cdot 3} - \frac{4 - \left(\frac{4}{t + 1} + -4\right)}{t + 1}} \]

   7. lower--.f64 N/A

      \[\leadsto 1 - \frac{1}{\color{blue}{2 \cdot 3 - \frac{4 - \left(\frac{4}{t + 1} + -4\right)}{t + 1}}} \]

   8. metadata-eval 100.0

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. associate--r+ N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

       \[\leadsto 1 - \frac{1}{6 - \frac{\color{blue}{{2}^{3}} - \frac{4}{t + 1}}{t + 1}} \]

   15. lower--.f64 N/A

       \[\leadsto 1 - \frac{1}{6 - \frac{\color{blue}{{2}^{3} - \frac{4}{t + 1}}}{t + 1}} \]

   16. metadata-eval 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

   \[\leadsto 1 + \frac{1}{\frac{\frac{4}{1 + t} - 8}{-1 - t} - 6} \]
10. Add Preprocessing

Alternative 2: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 5e-17)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))
   0.5))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / (1.0d0 + (1.0d0 / t))) <= 5d-17) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) + (-0.2222222222222222d0)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 5e-17)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17)
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-17], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 4.9999999999999999e-17

   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. Add Preprocessing

   3. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right)} \]

      3. distribute-neg-frac N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{5}{6} + \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}}{t} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{5}{6} + \frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}\right)\right)}{t} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{5}{6} + \frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}}{t} \]

      7. +-commutative N/A

         \[\leadsto \frac{5}{6} + \frac{\color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}}{t} \]

      8. sub-neg N/A

         \[\leadsto \frac{5}{6} + \frac{\color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}}{t}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}} \]

   if 4.9999999999999999e-17 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 5e-17)
   (+
    0.8333333333333334
    (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t)))
   0.5))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17) {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 5e-17)
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-17], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 4.9999999999999999e-17

   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. Add Preprocessing

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{\left(2 + \left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}}} \]

      2. lift-+.f64 N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{\left(2 + \left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right)\right)} + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}} \]

      3. associate-+l+ N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{2 + \left(\left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{\left(\left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right) + 2}} \]

      5. lower-+.f64 N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{\left(\left(4 + \left(\mathsf{neg}\left(\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right)\right)\right) + \frac{\frac{4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)} - 4}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}\right) + 2}} \]

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   9. Applied rewrites 99.0%

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

   if 4.9999999999999999e-17 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 5e-17)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / (1.0d0 + (1.0d0 / t))) <= 5d-17) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 5e-17)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 5e-17)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-17], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 4.9999999999999999e-17

   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. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{5}{6} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{5}{6} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{5}{6} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{9}}}{t}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{t}} \]

      7. metadata-eval 98.4

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

   5. Applied rewrites 98.4%

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

   if 4.9999999999999999e-17 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 1:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 1.0) 0.8333333333333334 0.5))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 1.0) {
		tmp = 0.8333333333333334;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / (1.0d0 + (1.0d0 / t))) <= 1.0d0) then
        tmp = 0.8333333333333334d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 1.0) {
		tmp = 0.8333333333333334;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + (1.0 / t))) <= 1.0:
		tmp = 0.8333333333333334
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 1.0)
		tmp = 0.8333333333333334;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 1.0)
		tmp = 0.8333333333333334;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], 0.8333333333333334, 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.2%

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

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 59.2% accurate, 101.0× 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. Add Preprocessing

3. Taylor expanded in t around 0

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

5. Applied rewrites 61.1%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))))))))