Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 6

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, 1.7× 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%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{6 + \left(\color{blue}{0} + \log \left(e^{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}\right)\right)} \]

   5. add-log-exp 100.0%

      \[\leadsto 1 - \frac{1}{6 + \left(0 + \color{blue}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}\right)} \]

   6. associate-*l/ 100.0%

      \[\leadsto 1 - \frac{1}{6 + \left(0 + \color{blue}{\frac{2 \cdot \left(\frac{2}{1 + t} + -4\right)}{1 + t}}\right)} \]

   7. distribute-lft-in 100.0%

      \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{\color{blue}{2 \cdot \frac{2}{1 + t} + 2 \cdot -4}}{1 + t}\right)} \]

   8. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{-8}}{1 + t}\right)} \]

   9. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{4 \cdot -2}}{1 + t}\right)} \]

   10. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \color{blue}{\left(-2 \cdot -2\right)} \cdot -2}{1 + t}\right)} \]

   11. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \left(\color{blue}{\left(-2\right)} \cdot -2\right) \cdot -2}{1 + t}\right)} \]

   12. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \left(\left(-2\right) \cdot \color{blue}{\left(-2\right)}\right) \cdot -2}{1 + t}\right)} \]

   13. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{2 \cdot \frac{2}{1 + t} + \left(\left(-2\right) \cdot \left(-2\right)\right) \cdot \color{blue}{\left(-2\right)}}{1 + t}\right)} \]

   14. fma-def 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{\color{blue}{\mathsf{fma}\left(2, \frac{2}{1 + t}, \left(\left(-2\right) \cdot \left(-2\right)\right) \cdot \left(-2\right)\right)}}{1 + t}\right)} \]

   15. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \left(\color{blue}{-2} \cdot \left(-2\right)\right) \cdot \left(-2\right)\right)}{1 + t}\right)} \]

   16. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \left(-2 \cdot \color{blue}{-2}\right) \cdot \left(-2\right)\right)}{1 + t}\right)} \]

   17. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \color{blue}{4} \cdot \left(-2\right)\right)}{1 + t}\right)} \]

   18. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, 4 \cdot \color{blue}{-2}\right)}{1 + t}\right)} \]

   19. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{6 + \left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, \color{blue}{-8}\right)}{1 + t}\right)} \]

5. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{6 + \color{blue}{\left(0 + \frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto 1 - \frac{1}{6 + \color{blue}{\frac{\mathsf{fma}\left(2, \frac{2}{1 + t}, -8\right)}{1 + t}}} \]

   2. fma-udef 100.0%

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

   3. associate-*r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. +-commutative 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.82)
   (+
    1.0
    (-
     (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)
     0.16666666666666666))
   (if (<= t 0.56)
     (+ (* t t) 0.5)
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.82) {
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.82d0)) then
        tmp = 1.0d0 + ((((0.037037037037037035d0 / t) - 0.2222222222222222d0) / t) - 0.16666666666666666d0)
    else if (t <= 0.56d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.82) {
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.82:
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666)
	elif t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.82)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666));
	elseif (t <= 0.56)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.82)
		tmp = 1.0 + ((((0.037037037037037035 / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	elseif (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.82], N[(1.0 + N[(N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.56], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.819999999999999951

   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%

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

   4. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

   9. Simplified 100.0%

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

   if -0.819999999999999951 < t < 0.56000000000000005

   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%

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

   4. Taylor expanded in t around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. unpow2 99.1%

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

   6. Simplified 99.1%

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

      1. associate--r- 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

   if 0.56000000000000005 < 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%

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

   4. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.8) (not (<= t 0.56)))
   (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.56)) {
		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 <= (-0.8d0)) .or. (.not. (t <= 0.56d0))) 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 <= -0.8) || !(t <= 0.56)) {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.8) or not (t <= 0.56):
		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 <= -0.8) || !(t <= 0.56))
		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 <= -0.8) || ~((t <= 0.56)))
		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, -0.8], N[Not[LessEqual[t, 0.56]], $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 < -0.80000000000000004 or 0.56000000000000005 < 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%

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

   4. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   if -0.80000000000000004 < t < 0.56000000000000005

   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%

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

   4. Taylor expanded in t around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. unpow2 99.1%

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

   6. Simplified 99.1%

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

      1. associate--r- 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 4: 98.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.56) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		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 <= (-0.9d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.56d0) 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 <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.9:
		tmp = 0.8333333333333334
	elif t <= 0.56:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		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 <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.9], 0.8333333333333334, If[LessEqual[t, 0.56], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.56000000000000005 < 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%

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

   4. Taylor expanded in t around inf 99.1%

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

   if -0.900000000000000022 < t < 0.56000000000000005

   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%

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

   4. Taylor expanded in t around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. unpow2 99.1%

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

   6. Simplified 99.1%

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

      1. associate--r- 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 5: 98.5% 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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 99.1%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 1 - \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

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

Alternative 6: 58.4% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0
end function

Java

public static double code(double t) {
	return 0.8333333333333334;
}

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334;
end

Wolfram

code[t_] := 0.8333333333333334

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%

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

4. Taylor expanded in t around inf 62.5%

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

5. Final simplification 62.5%

   \[\leadsto 0.8333333333333334 \]

Reproduce

herbie shell --seed 2023271 
(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)))))))))