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

Percentage Accurate: 81.2% → 99.9%

Time: 5.8s

Alternatives: 4

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: 81.2% 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: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. sub-neg 76.3%

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

   2. associate-/l* 86.0%

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

   3. *-commutative 86.0%

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

   4. associate-/l* 86.4%

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

   5. distribute-neg-frac 86.4%

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

   6. metadata-eval 86.4%

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

   7. associate-/l/ 76.7%

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

   8. div-sub 69.6%

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

   9. times-frac 86.6%

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

   10. *-inverses 86.6%

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

   11. *-rgt-identity 86.6%

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

   12. *-commutative 86.6%

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

   13. associate-*l/ 86.6%

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

   14. *-commutative 86.6%

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

   15. times-frac 99.9%

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

   16. *-inverses 99.9%

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

   17. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 89.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-21} \lor \neg \left(z \leq 0.031\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.5e-21) (not (<= z 0.031)))
   (- x (/ y z))
   (- x (/ z (* t -0.5)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.5e-21) || !(z <= 0.031)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (z / (t * -0.5));
	}
	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 <= (-4.5d-21)) .or. (.not. (z <= 0.031d0))) then
        tmp = x - (y / z)
    else
        tmp = x - (z / (t * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.5e-21) || !(z <= 0.031)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (z / (t * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.5e-21) or not (z <= 0.031):
		tmp = x - (y / z)
	else:
		tmp = x - (z / (t * -0.5))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.5e-21) || !(z <= 0.031))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(z / Float64(t * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.5e-21) || ~((z <= 0.031)))
		tmp = x - (y / z);
	else
		tmp = x - (z / (t * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.5e-21], N[Not[LessEqual[z, 0.031]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(z / N[(t * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.49999999999999968e-21 or 0.031 < z

   1. Initial program 65.0%

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

      1. sub-neg 65.0%

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

      2. associate-/l* 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-/l* 87.1%

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

      5. distribute-neg-frac 87.1%

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

      6. metadata-eval 87.1%

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

      7. associate-/l/ 65.0%

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

      8. div-sub 64.9%

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

      9. times-frac 88.1%

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

      10. *-inverses 88.1%

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

      11. *-rgt-identity 88.1%

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

      12. *-commutative 88.1%

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

      13. associate-*l/ 88.1%

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

      14. *-commutative 88.1%

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

      15. times-frac 99.9%

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

      16. *-inverses 99.9%

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

      17. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 89.7%

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

      1. +-commutative 89.7%

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

      2. mul-1-neg 89.7%

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

      3. sub-neg 89.7%

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

   6. Simplified 89.7%

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

   if -4.49999999999999968e-21 < z < 0.031

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-/l* 86.5%

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

      3. div-sub 86.5%

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

      4. sub-neg 86.5%

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

      5. *-commutative 86.5%

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

      6. associate-*l* 86.5%

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

      7. *-commutative 86.5%

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

      8. times-frac 86.5%

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

      9. metadata-eval 86.5%

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

      10. *-lft-identity 86.5%

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

      11. associate-*r/ 92.6%

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

      12. fma-def 92.5%

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

      13. associate-/r* 93.3%

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

      14. distribute-neg-frac 93.3%

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

      15. *-commutative 93.3%

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

      16. associate-/l* 100.0%

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

      17. *-inverses 100.0%

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

      18. /-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 91.1%

      \[\leadsto x - \frac{z}{\color{blue}{-0.5 \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto x - \frac{z}{\color{blue}{t \cdot -0.5}} \]

   6. Simplified 91.1%

      \[\leadsto x - \frac{z}{\color{blue}{t \cdot -0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-21} \lor \neg \left(z \leq 0.031\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t \cdot -0.5}\\ \end{array} \]

Alternative 3: 81.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-19} \lor \neg \left(z \leq 0.036\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.1e-19) (not (<= z 0.036))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.1e-19) || !(z <= 0.036)) {
		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 <= (-2.1d-19)) .or. (.not. (z <= 0.036d0))) 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 <= -2.1e-19) || !(z <= 0.036)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.1e-19) or not (z <= 0.036):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.1e-19) || !(z <= 0.036))
		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 <= -2.1e-19) || ~((z <= 0.036)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.1e-19], N[Not[LessEqual[z, 0.036]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e-19 or 0.0359999999999999973 < z

   1. Initial program 65.0%

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

      1. sub-neg 65.0%

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

      2. associate-/l* 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-/l* 87.1%

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

      5. distribute-neg-frac 87.1%

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

      6. metadata-eval 87.1%

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

      7. associate-/l/ 65.0%

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

      8. div-sub 64.9%

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

      9. times-frac 88.1%

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

      10. *-inverses 88.1%

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

      11. *-rgt-identity 88.1%

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

      12. *-commutative 88.1%

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

      13. associate-*l/ 88.1%

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

      14. *-commutative 88.1%

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

      15. times-frac 99.9%

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

      16. *-inverses 99.9%

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

      17. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 89.7%

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

      1. +-commutative 89.7%

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

      2. mul-1-neg 89.7%

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

      3. sub-neg 89.7%

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

   6. Simplified 89.7%

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

   if -2.0999999999999999e-19 < z < 0.0359999999999999973

   1. Initial program 86.0%

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

      1. sub-neg 86.0%

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

      2. associate-/l* 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-/l* 85.8%

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

      5. distribute-neg-frac 85.8%

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

      6. metadata-eval 85.8%

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

      7. associate-/l/ 86.7%

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

      8. div-sub 73.7%

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

      9. times-frac 85.3%

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

      10. *-inverses 85.3%

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

      11. *-rgt-identity 85.3%

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

      12. *-commutative 85.3%

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

      13. associate-*l/ 85.3%

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

      14. *-commutative 85.3%

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

      15. times-frac 99.9%

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

      16. *-inverses 99.9%

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

      17. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 75.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-19} \lor \neg \left(z \leq 0.036\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 74.1% 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 76.3%

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

   1. sub-neg 76.3%

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

   2. associate-/l* 86.0%

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

   3. *-commutative 86.0%

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

   4. associate-/l* 86.4%

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

   5. distribute-neg-frac 86.4%

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

   6. metadata-eval 86.4%

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

   7. associate-/l/ 76.7%

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

   8. div-sub 69.6%

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

   9. times-frac 86.6%

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

   10. *-inverses 86.6%

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

   11. *-rgt-identity 86.6%

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

   12. *-commutative 86.6%

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

   13. associate-*l/ 86.6%

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

   14. *-commutative 86.6%

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

   15. times-frac 99.9%

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

   16. *-inverses 99.9%

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

   17. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 75.1%

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

5. Final simplification 75.1%

   \[\leadsto x \]

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 2023196 
(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)))))