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

Percentage Accurate: 81.7% → 99.8%

Time: 8.9s

Alternatives: 5

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: 81.7% 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: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. sub-neg 81.6%

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

   2. associate-/l* 88.9%

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

   3. *-commutative 88.9%

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

   4. associate-/l* 88.8%

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

   5. distribute-neg-frac 88.8%

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

   6. metadata-eval 88.8%

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

   7. associate-/l/ 81.6%

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

   8. div-sub 72.2%

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

   9. times-frac 91.2%

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

   10. *-inverses 91.2%

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

   11. *-rgt-identity 91.2%

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

   12. *-commutative 91.2%

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

   13. associate-*l/ 91.1%

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

   14. *-commutative 91.1%

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

   15. times-frac 99.8%

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

   16. *-inverses 99.8%

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

   17. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

   1. fma-neg 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.8% 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 81.6%

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

   1. sub-neg 81.6%

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

   2. associate-/l* 88.9%

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

   3. *-commutative 88.9%

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

   4. associate-/l* 88.8%

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

   5. distribute-neg-frac 88.8%

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

   6. metadata-eval 88.8%

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

   7. associate-/l/ 81.6%

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

   8. div-sub 72.2%

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

   9. times-frac 91.2%

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

   10. *-inverses 91.2%

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

   11. *-rgt-identity 91.2%

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

   12. *-commutative 91.2%

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

   13. associate-*l/ 91.1%

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

   14. *-commutative 91.1%

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

   15. times-frac 99.8%

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

   16. *-inverses 99.8%

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

   17. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 89.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-10} \lor \neg \left(z \leq 4.2 \cdot 10^{-38}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.4e-10) (not (<= z 4.2e-38)))
   (- x (/ y z))
   (- x (* -2.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.4e-10) || !(z <= 4.2e-38)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (-2.0 * (z / 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 <= (-1.4d-10)) .or. (.not. (z <= 4.2d-38))) then
        tmp = x - (y / z)
    else
        tmp = x - ((-2.0d0) * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.4e-10) || !(z <= 4.2e-38)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (-2.0 * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.4e-10) or not (z <= 4.2e-38):
		tmp = x - (y / z)
	else:
		tmp = x - (-2.0 * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.4e-10) || !(z <= 4.2e-38))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(-2.0 * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.4e-10) || ~((z <= 4.2e-38)))
		tmp = x - (y / z);
	else
		tmp = x - (-2.0 * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.4e-10], N[Not[LessEqual[z, 4.2e-38]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.40000000000000008e-10 or 4.20000000000000026e-38 < z

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. associate-/l* 88.9%

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

      3. *-commutative 88.9%

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

      4. associate-/l* 88.9%

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

      5. distribute-neg-frac 88.9%

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

      6. metadata-eval 88.9%

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

      7. associate-/l/ 75.7%

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

      8. div-sub 75.7%

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

      9. times-frac 94.5%

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

      10. *-inverses 94.5%

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

      11. *-rgt-identity 94.5%

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

      12. *-commutative 94.5%

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

      13. associate-*l/ 94.5%

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

      14. *-commutative 94.5%

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

      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 92.6%

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

      1. +-commutative 92.6%

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

      2. mul-1-neg 92.6%

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

      3. sub-neg 92.6%

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

   6. Simplified 92.6%

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

   if -1.40000000000000008e-10 < z < 4.20000000000000026e-38

   1. Initial program 89.6%

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

      1. associate-/l* 88.9%

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

      2. *-commutative 88.9%

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

      3. associate-*r/ 88.9%

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

      4. div-sub 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-/l* 95.2%

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

      7. associate-/r* 95.2%

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

      8. *-inverses 95.2%

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

      9. metadata-eval 95.2%

         \[\leadsto x - 2 \cdot \frac{y}{\frac{z}{\color{blue}{0.5}} - \frac{y \cdot t}{z}} \]

      10. *-commutative 95.2%

          \[\leadsto x - 2 \cdot \frac{y}{\frac{z}{0.5} - \frac{\color{blue}{t \cdot y}}{z}} \]

      11. associate-*l/ 96.2%

          \[\leadsto x - 2 \cdot \frac{y}{\frac{z}{0.5} - \color{blue}{\frac{t}{z} \cdot y}} \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{x - 2 \cdot \frac{y}{\frac{z}{0.5} - \frac{t}{z} \cdot y}} \]

   4. Taylor expanded in y around inf 88.6%

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

      1. *-commutative 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-10} \lor \neg \left(z \leq 4.2 \cdot 10^{-38}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \end{array} \]

Alternative 4: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-21} \lor \neg \left(z \leq 125\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.6e-21) (not (<= z 125.0))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.6e-21) || !(z <= 125.0)) {
		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 <= (-1.6d-21)) .or. (.not. (z <= 125.0d0))) 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 <= -1.6e-21) || !(z <= 125.0)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.6e-21) or not (z <= 125.0):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.6e-21) || !(z <= 125.0))
		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 <= -1.6e-21) || ~((z <= 125.0)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.6e-21], N[Not[LessEqual[z, 125.0]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.6000000000000001e-21 or 125 < z

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. associate-/l* 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-/l* 88.2%

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

      5. distribute-neg-frac 88.2%

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

      6. metadata-eval 88.2%

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

      7. associate-/l/ 75.7%

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

      8. div-sub 75.6%

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

      9. times-frac 94.5%

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

      10. *-inverses 94.5%

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

      11. *-rgt-identity 94.5%

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

      12. *-commutative 94.5%

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

      13. associate-*l/ 94.4%

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

      14. *-commutative 94.4%

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

      15. times-frac 99.8%

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

      16. *-inverses 99.8%

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

      17. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 90.6%

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

      1. +-commutative 90.6%

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

      2. mul-1-neg 90.6%

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

      3. sub-neg 90.6%

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

   6. Simplified 90.6%

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

   if -1.6000000000000001e-21 < z < 125

   1. Initial program 89.6%

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

      1. sub-neg 89.6%

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

      2. associate-/l* 89.8%

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

      3. *-commutative 89.8%

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

      4. associate-/l* 89.7%

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

      5. distribute-neg-frac 89.7%

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

      6. metadata-eval 89.7%

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

      7. associate-/l/ 89.6%

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

      8. div-sub 67.5%

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

      9. times-frac 86.7%

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

      10. *-inverses 86.7%

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

      11. *-rgt-identity 86.7%

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

      12. *-commutative 86.7%

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

      13. associate-*l/ 86.7%

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

      14. *-commutative 86.7%

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

      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 77.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-21} \lor \neg \left(z \leq 125\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 75.2% 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 81.6%

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

   1. sub-neg 81.6%

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

   2. associate-/l* 88.9%

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

   3. *-commutative 88.9%

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

   4. associate-/l* 88.8%

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

   5. distribute-neg-frac 88.8%

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

   6. metadata-eval 88.8%

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

   7. associate-/l/ 81.6%

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

   8. div-sub 72.2%

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

   9. times-frac 91.2%

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

   10. *-inverses 91.2%

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

   11. *-rgt-identity 91.2%

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

   12. *-commutative 91.2%

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

   13. associate-*l/ 91.1%

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

   14. *-commutative 91.1%

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

   15. times-frac 99.8%

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

   16. *-inverses 99.8%

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

   17. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 77.1%

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

5. Final simplification 77.1%

   \[\leadsto x \]

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 2023200 
(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)))))