Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Percentage Accurate: 81.3% → 98.4%

Time: 9.0s

Alternatives: 6

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))

C

double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Python

def code(x, y, z, t):
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))

C

double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Python

def code(x, y, z, t):
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{z \cdot -2 + \frac{y}{\frac{z}{t}}}, 2, x\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ y (+ (* z -2.0) (/ y (/ z t)))) 2.0 x))

C

double code(double x, double y, double z, double t) {
	return fma((y / ((z * -2.0) + (y / (z / t)))), 2.0, x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(y / Float64(Float64(z * -2.0) + Float64(y / Float64(z / t)))), 2.0, x)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y / N[(N[(z * -2.0), $MachinePrecision] + N[(y / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + x), $MachinePrecision]

Derivation

1. Initial program 84.0%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z \cdot -2 + \frac{y}{\frac{z}{t}}}, 2, x\right)} \]

4. Final simplification 98.1%

   \[\leadsto \mathsf{fma}\left(\frac{y}{z \cdot -2 + \frac{y}{\frac{z}{t}}}, 2, x\right) \]

Alternative 2: 87.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+95} \lor \neg \left(z \leq 1.25 \cdot 10^{-51} \lor \neg \left(z \leq 5.2 \cdot 10^{+25}\right) \land z \leq 3.9 \cdot 10^{+44}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.5e+95)
         (not (or (<= z 1.25e-51) (and (not (<= z 5.2e+25)) (<= z 3.9e+44)))))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.5e+95) || !((z <= 1.25e-51) || (!(z <= 5.2e+25) && (z <= 3.9e+44)))) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-7.5d+95)) .or. (.not. (z <= 1.25d-51) .or. (.not. (z <= 5.2d+25)) .and. (z <= 3.9d+44))) then
        tmp = x - (y / z)
    else
        tmp = x - ((z * (-2.0d0)) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.5e+95) || !((z <= 1.25e-51) || (!(z <= 5.2e+25) && (z <= 3.9e+44)))) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7.5e+95) or not ((z <= 1.25e-51) or (not (z <= 5.2e+25) and (z <= 3.9e+44))):
		tmp = x - (y / z)
	else:
		tmp = x - ((z * -2.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.5e+95) || !((z <= 1.25e-51) || (!(z <= 5.2e+25) && (z <= 3.9e+44))))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -7.5e+95) || ~(((z <= 1.25e-51) || (~((z <= 5.2e+25)) && (z <= 3.9e+44)))))
		tmp = x - (y / z);
	else
		tmp = x - ((z * -2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7.5e+95], N[Not[Or[LessEqual[z, 1.25e-51], And[N[Not[LessEqual[z, 5.2e+25]], $MachinePrecision], LessEqual[z, 3.9e+44]]]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5000000000000001e95 or 1.25000000000000001e-51 < z < 5.1999999999999997e25 or 3.9000000000000003e44 < z

   1. Initial program 76.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Step-by-step derivation

      1. associate-/l* 89.6%

         \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}} \]

      2. associate-*l* 89.6%

         \[\leadsto x - \frac{y \cdot 2}{\frac{\color{blue}{z \cdot \left(2 \cdot z\right)} - y \cdot t}{z}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]

   4. Taylor expanded in y around 0 92.3%

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   if -7.5000000000000001e95 < z < 1.25000000000000001e-51 or 5.1999999999999997e25 < z < 3.9000000000000003e44

   1. Initial program 90.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Step-by-step derivation

      1. associate-/l* 93.3%

         \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}} \]

      2. associate-*l* 93.3%

         \[\leadsto x - \frac{y \cdot 2}{\frac{\color{blue}{z \cdot \left(2 \cdot z\right)} - y \cdot t}{z}} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]

   4. Taylor expanded in y around inf 89.0%

      \[\leadsto x - \color{blue}{-2 \cdot \frac{z}{t}} \]
   5. Step-by-step derivation

      1. associate-*r/ 89.0%

         \[\leadsto x - \color{blue}{\frac{-2 \cdot z}{t}} \]

      2. *-commutative 89.0%

         \[\leadsto x - \frac{\color{blue}{z \cdot -2}}{t} \]

   6. Simplified 89.0%

      \[\leadsto x - \color{blue}{\frac{z \cdot -2}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+95} \lor \neg \left(z \leq 1.25 \cdot 10^{-51} \lor \neg \left(z \leq 5.2 \cdot 10^{+25}\right) \land z \leq 3.9 \cdot 10^{+44}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x - y \cdot \frac{2}{z \cdot 2 - y \cdot \frac{t}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (* y (/ 2.0 (- (* z 2.0) (* y (/ t z)))))))

C

double code(double x, double y, double z, double t) {
	return x - (y * (2.0 / ((z * 2.0) - (y * (t / z)))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (y * (2.0d0 / ((z * 2.0d0) - (y * (t / z)))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (y * (2.0 / ((z * 2.0) - (y * (t / z)))));
}

Python

def code(x, y, z, t):
	return x - (y * (2.0 / ((z * 2.0) - (y * (t / z)))))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y * Float64(2.0 / Float64(Float64(z * 2.0) - Float64(y * Float64(t / z))))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (y * (2.0 / ((z * 2.0) - (y * (t / z)))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(y * N[(2.0 / N[(N[(z * 2.0), $MachinePrecision] - N[(y * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.0%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
2. Step-by-step derivation

   1. associate-/l* 91.6%

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}} \]

   2. associate-*l* 91.6%

      \[\leadsto x - \frac{y \cdot 2}{\frac{\color{blue}{z \cdot \left(2 \cdot z\right)} - y \cdot t}{z}} \]

3. Simplified 91.6%

   \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]

4. Taylor expanded in z around 0 96.2%

   \[\leadsto x - \frac{y \cdot 2}{\color{blue}{-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z}} \]
5. Step-by-step derivation

   1. +-commutative 96.2%

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{2 \cdot z + -1 \cdot \frac{t \cdot y}{z}}} \]

   2. mul-1-neg 96.2%

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

   3. *-commutative 96.2%

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

   4. associate-*r/ 98.0%

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

   5. sub-neg 98.0%

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{2 \cdot z - y \cdot \frac{t}{z}}} \]

   6. *-commutative 98.0%

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{z \cdot 2} - y \cdot \frac{t}{z}} \]

6. Simplified 98.0%

   \[\leadsto x - \frac{y \cdot 2}{\color{blue}{z \cdot 2 - y \cdot \frac{t}{z}}} \]
7. Step-by-step derivation

   1. associate-*r/ 96.2%

      \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{y \cdot t}{z}}} \]

   2. *-commutative 96.2%

      \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \frac{\color{blue}{t \cdot y}}{z}} \]

   3. associate-/l* 94.7%

      \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{t}{\frac{z}{y}}}} \]

8. Applied egg-rr 94.7%

   \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{t}{\frac{z}{y}}}} \]
9. Step-by-step derivation

   1. associate-/r/ 98.0%

      \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{t}{z} \cdot y}} \]

   2. *-commutative 98.0%

      \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{y \cdot \frac{t}{z}}} \]

   3. *-un-lft-identity 98.0%

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

   4. times-frac 98.0%

      \[\leadsto x - \color{blue}{\frac{y}{1} \cdot \frac{2}{z \cdot 2 - y \cdot \frac{t}{z}}} \]

   5. /-rgt-identity 98.0%

      \[\leadsto x - \color{blue}{y} \cdot \frac{2}{z \cdot 2 - y \cdot \frac{t}{z}} \]

   6. *-commutative 98.0%

      \[\leadsto x - y \cdot \frac{2}{\color{blue}{2 \cdot z} - y \cdot \frac{t}{z}} \]

10. Applied egg-rr 98.0%

    \[\leadsto x - \color{blue}{y \cdot \frac{2}{2 \cdot z - y \cdot \frac{t}{z}}} \]

11. Final simplification 98.0%

    \[\leadsto x - y \cdot \frac{2}{z \cdot 2 - y \cdot \frac{t}{z}} \]

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x - \frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* y 2.0) (- (* z 2.0) (* y (/ t z))))))

C

double code(double x, double y, double z, double t) {
	return x - ((y * 2.0) / ((z * 2.0) - (y * (t / z))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - ((y * 2.0d0) / ((z * 2.0d0) - (y * (t / z))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - ((y * 2.0) / ((z * 2.0) - (y * (t / z))));
}

Python

def code(x, y, z, t):
	return x - ((y * 2.0) / ((z * 2.0) - (y * (t / z))))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(y * 2.0) / Float64(Float64(z * 2.0) - Float64(y * Float64(t / z)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - ((y * 2.0) / ((z * 2.0) - (y * (t / z))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(y * 2.0), $MachinePrecision] / N[(N[(z * 2.0), $MachinePrecision] - N[(y * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.0%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
2. Step-by-step derivation

   1. associate-/l* 91.6%

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}} \]

   2. associate-*l* 91.6%

      \[\leadsto x - \frac{y \cdot 2}{\frac{\color{blue}{z \cdot \left(2 \cdot z\right)} - y \cdot t}{z}} \]

3. Simplified 91.6%

   \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]

4. Taylor expanded in z around 0 96.2%

   \[\leadsto x - \frac{y \cdot 2}{\color{blue}{-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z}} \]
5. Step-by-step derivation

   1. +-commutative 96.2%

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{2 \cdot z + -1 \cdot \frac{t \cdot y}{z}}} \]

   2. mul-1-neg 96.2%

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

   3. *-commutative 96.2%

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

   4. associate-*r/ 98.0%

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

   5. sub-neg 98.0%

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{2 \cdot z - y \cdot \frac{t}{z}}} \]

   6. *-commutative 98.0%

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{z \cdot 2} - y \cdot \frac{t}{z}} \]

6. Simplified 98.0%

   \[\leadsto x - \frac{y \cdot 2}{\color{blue}{z \cdot 2 - y \cdot \frac{t}{z}}} \]

7. Final simplification 98.0%

   \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}} \]

Alternative 5: 82.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -260000000 \lor \neg \left(z \leq 1.9 \cdot 10^{-102}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -260000000.0) (not (<= z 1.9e-102))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -260000000.0) || !(z <= 1.9e-102)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-260000000.0d0)) .or. (.not. (z <= 1.9d-102))) then
        tmp = x - (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -260000000.0) || !(z <= 1.9e-102)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -260000000.0) or not (z <= 1.9e-102):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -260000000.0) || !(z <= 1.9e-102))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -260000000.0) || ~((z <= 1.9e-102)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -260000000.0], N[Not[LessEqual[z, 1.9e-102]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e8 or 1.90000000000000013e-102 < z

   1. Initial program 77.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Step-by-step derivation

      1. associate-/l* 89.7%

         \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}} \]

      2. associate-*l* 89.7%

         \[\leadsto x - \frac{y \cdot 2}{\frac{\color{blue}{z \cdot \left(2 \cdot z\right)} - y \cdot t}{z}} \]

   3. Simplified 89.7%

      \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]

   4. Taylor expanded in y around 0 83.0%

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   if -2.6e8 < z < 1.90000000000000013e-102

   1. Initial program 94.5%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Step-by-step derivation

      1. associate-/l* 94.5%

         \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}} \]

      2. associate-*l* 94.5%

         \[\leadsto x - \frac{y \cdot 2}{\frac{\color{blue}{z \cdot \left(2 \cdot z\right)} - y \cdot t}{z}} \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]

   4. Taylor expanded in x around inf 75.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -260000000 \lor \neg \left(z \leq 1.9 \cdot 10^{-102}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 75.5% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 x)

C

double code(double x, double y, double z, double t) {
	return x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

public static double code(double x, double y, double z, double t) {
	return x;
}

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 84.0%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
2. Step-by-step derivation

   1. associate-/l* 91.6%

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}} \]

   2. associate-*l* 91.6%

      \[\leadsto x - \frac{y \cdot 2}{\frac{\color{blue}{z \cdot \left(2 \cdot z\right)} - y \cdot t}{z}} \]

3. Simplified 91.6%

   \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]

4. Taylor expanded in x around inf 71.6%

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

5. Final simplification 71.6%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z)))))

C

double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (1.0d0 / ((z / y) - ((t / 2.0d0) / z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Python

def code(x, y, z, t):
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(1.0 / Float64(Float64(z / y) - Float64(Float64(t / 2.0) / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (1.0 / ((z / y) - ((t / 2.0) / z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(1.0 / N[(N[(z / y), $MachinePrecision] - N[(N[(t / 2.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023309 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))