Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 33.0s

Alternatives: 5

Speedup: 1.9×

• could not determine a ground truth

  1. t = -6.345523484422262e+49

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: 100.0% 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, 1.0× 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]]]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{4}{1 + t} + -8}{1 + t}\\ \frac{5 + t_1}{t_1 + 6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t))))
   (/ (+ 5.0 t_1) (+ t_1 6.0))))

C

double code(double t) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (5.0 + t_1) / (t_1 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((4.0d0 / (1.0d0 + t)) + (-8.0d0)) / (1.0d0 + t)
    code = (5.0d0 + t_1) / (t_1 + 6.0d0)
end function

Java

public static double code(double t) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (5.0 + t_1) / (t_1 + 6.0);
}

Python

def code(t):
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)
	return (5.0 + t_1) / (t_1 + 6.0)

Julia

function code(t)
	t_1 = Float64(Float64(Float64(4.0 / Float64(1.0 + t)) + -8.0) / Float64(1.0 + t))
	return Float64(Float64(5.0 + t_1) / Float64(t_1 + 6.0))
end

MATLAB

function tmp = code(t)
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	tmp = (5.0 + t_1) / (t_1 + 6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(t$95$1 + 6.0), $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%

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

4. Final simplification 100.0%

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

Alternative 3: 97.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.24:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t 0.24)
   (+ (* t t) 0.5)
   (+
    0.8333333333333334
    (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t))))

C

double code(double t) {
	double tmp;
	if (t <= 0.24) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 0.24d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= 0.24) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= 0.24:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= 0.24)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= 0.24)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, 0.24], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.23999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

   4. Simplified 97.0%

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

   if 0.23999999999999999 < 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%

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

   4. Taylor expanded in t around inf 91.8%

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

      1. +-commutative 91.8%

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

      2. associate--l+ 91.8%

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

      3. associate-*r/ 91.8%

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

      4. metadata-eval 91.8%

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

      5. unpow2 91.8%

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

      6. associate-*r/ 91.8%

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

      7. metadata-eval 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in t around inf 92.0%

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

   8. Taylor expanded in t around inf 94.1%

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

      1. +-commutative 94.1%

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

      2. associate--l+ 94.1%

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

      3. associate-*r/ 94.1%

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

      4. metadata-eval 94.1%

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

      5. unpow2 94.1%

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

      6. associate-*r/ 94.1%

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

      7. metadata-eval 94.1%

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

   10. Simplified 94.1%

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

       1. associate-/r* 94.1%

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

       2. sub-div 94.1%

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

   12. Applied egg-rr 94.1%

       \[\leadsto 0.8333333333333334 + \color{blue}{\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.24:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \]

Alternative 4: 96.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t 0.56)
   (+ (* t t) 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334 - (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 = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

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

Python

def code(t):
	tmp = 0
	if t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= 0.56)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 0.56000000000000005

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

   4. Simplified 97.0%

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

   if 0.56000000000000005 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. metadata-eval 88.1%

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

   4. Simplified 88.1%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 95.1% 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%

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

2. Taylor expanded in t around 0 95.2%

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

3. Final simplification 95.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023278 
(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))))))))