Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 8

Speedup: 1.4×

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ t_2 := 2 - t\_1\\ 1 + \frac{2 + t\_2 \cdot \left(t\_1 - 2\right)}{{t\_2}^{4} - 4} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))) (t_2 (- 2.0 t_1)))
   (+ 1.0 (/ (+ 2.0 (* t_2 (- t_1 2.0))) (- (pow t_2 4.0) 4.0)))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double t_2 = 2.0 - t_1;
	return 1.0 + ((2.0 + (t_2 * (t_1 - 2.0))) / (pow(t_2, 4.0) - 4.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 / (1.0d0 + t)
    t_2 = 2.0d0 - t_1
    code = 1.0d0 + ((2.0d0 + (t_2 * (t_1 - 2.0d0))) / ((t_2 ** 4.0d0) - 4.0d0))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double t_2 = 2.0 - t_1;
	return 1.0 + ((2.0 + (t_2 * (t_1 - 2.0))) / (Math.pow(t_2, 4.0) - 4.0));
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	t_2 = 2.0 - t_1
	return 1.0 + ((2.0 + (t_2 * (t_1 - 2.0))) / (math.pow(t_2, 4.0) - 4.0))

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	t_2 = Float64(2.0 - t_1)
	return Float64(1.0 + Float64(Float64(2.0 + Float64(t_2 * Float64(t_1 - 2.0))) / Float64((t_2 ^ 4.0) - 4.0)))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	t_2 = 2.0 - t_1;
	tmp = 1.0 + ((2.0 + (t_2 * (t_1 - 2.0))) / ((t_2 ^ 4.0) - 4.0));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 - t$95$1), $MachinePrecision]}, N[(1.0 + N[(N[(2.0 + N[(t$95$2 * N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$2, 4.0], $MachinePrecision] - 4.0), $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
3. Step-by-step derivation

   1. flip-+ 100.0%

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

   2. associate-/r/ 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

6. Simplified 100.0%

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

   1. unpow2 100.0%

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

   2. +-commutative 100.0%

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

   3. +-commutative 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto 1 + \frac{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(\frac{2}{1 + t} - 2\right)}{{\left(2 - \frac{2}{1 + t}\right)}^{4} - 4} \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.42)
   0.8333333333333334
   (if (<= t 0.45)
     (+ 0.5 (* (* t t) (+ 1.0 (* t -2.0))))
     (+
      0.8333333333333334
      (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.45) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)));
	} 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 <= (-0.42d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.45d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (-2.0d0))))
    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 <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.45) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)));
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.42:
		tmp = 0.8333333333333334
	elif t <= 0.45:
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)))
	else:
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.45)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * -2.0))));
	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 <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.45)
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)));
	else
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.42], 0.8333333333333334, If[LessEqual[t, 0.45], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.419999999999999984

   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 100.0%

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

   if -0.419999999999999984 < t < 0.450000000000000011

   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 99.6%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto 0.5 + {t}^{2} \cdot \left(1 + \color{blue}{t \cdot -2}\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot -2\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + t \cdot -2\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + t \cdot -2\right) \]

   if 0.450000000000000011 < 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 inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. unpow2 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. div-sub 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

      18. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

Alternative 3: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ 1 + \frac{1}{\left(2 - t\_1\right) \cdot \left(t\_1 - 2\right) - 2} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(1.0 / N[(N[(N[(2.0 - t$95$1), $MachinePrecision] * N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $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
3. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/l/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. associate-/l/ 100.0%

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

   5. unsub-neg 100.0%

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

   6. associate-/r* 100.0%

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

   7. distribute-lft-in 100.0%

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

   8. rgt-mult-inverse 100.0%

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

   9. *-rgt-identity 100.0%

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

6. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate-/l/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. associate-/l/ 100.0%

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

   5. unsub-neg 100.0%

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

   6. associate-/r* 100.0%

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

   7. distribute-lft-in 100.0%

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

   8. rgt-mult-inverse 100.0%

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

   9. *-rgt-identity 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 4: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.24:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.42)
   0.8333333333333334
   (if (<= t 0.24)
     (+ 0.5 (* t t))
     (+
      0.8333333333333334
      (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.24) {
		tmp = 0.5 + (t * t);
	} 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 <= (-0.42d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.24d0) then
        tmp = 0.5d0 + (t * t)
    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 <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.24) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.42:
		tmp = 0.8333333333333334
	elif t <= 0.24:
		tmp = 0.5 + (t * t)
	else:
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.24)
		tmp = Float64(0.5 + Float64(t * t));
	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 <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.24)
		tmp = 0.5 + (t * t);
	else
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.42], 0.8333333333333334, If[LessEqual[t, 0.24], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.419999999999999984

   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 100.0%

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

   if -0.419999999999999984 < t < 0.23999999999999999

   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 99.6%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto 0.5 + {t}^{2} \cdot \left(1 + \color{blue}{t \cdot -2}\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot -2\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + t \cdot -2\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + t \cdot -2\right) \]

   8. Taylor expanded in t around 0 99.1%

      \[\leadsto 0.5 + \left(t \cdot t\right) \cdot \color{blue}{1} \]

   if 0.23999999999999999 < 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 inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. unpow2 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. div-sub 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

      18. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.24:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.42)
   0.8333333333333334
   (if (<= t 0.58)
     (+ 0.5 (* t t))
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * t);
	} 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 <= (-0.42d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.42:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = 0.5 + (t * t)
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = 0.5 + (t * t);
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.42], 0.8333333333333334, If[LessEqual[t, 0.58], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.419999999999999984

   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 100.0%

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

   if -0.419999999999999984 < t < 0.57999999999999996

   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 99.6%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto 0.5 + {t}^{2} \cdot \left(1 + \color{blue}{t \cdot -2}\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot -2\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + t \cdot -2\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + t \cdot -2\right) \]

   8. Taylor expanded in t around 0 99.1%

      \[\leadsto 0.5 + \left(t \cdot t\right) \cdot \color{blue}{1} \]

   if 0.57999999999999996 < 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 inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34)
   0.8333333333333334
   (if (<= t 0.65) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = 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 <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.65d0) then
        tmp = 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 <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 0.65:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = 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 <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 0.65], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.340000000000000024

   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 100.0%

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

   if -0.340000000000000024 < t < 0.650000000000000022

   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 98.3%

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

   if 0.650000000000000022 < 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 inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

Alternative 7: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		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 <= (-0.34d0)) 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 <= -0.34) {
		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 <= -0.34:
		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 <= -0.34)
		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 <= -0.34)
		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, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 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. Add Preprocessing

   3. Taylor expanded in t around inf 99.7%

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

   if -0.340000000000000024 < 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 0 98.3%

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

Alternative 8: 58.8% 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. Add Preprocessing

3. Taylor expanded in t around 0 58.4%

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

Reproduce

herbie shell --seed 2024163 
(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)))))))))