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

Percentage Accurate: 81.8% → 97.7%

Time: 9.7s

Alternatives: 4

Speedup: 1.1×

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: 81.8% 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: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 / (t - (2.0d0 * (z * (z / y))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

3. Simplified 89.0%

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

5. Taylor expanded in y around inf 91.0%

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

   1. pow1 91.0%

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

   2. *-commutative 91.0%

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

   3. associate-*l* 91.0%

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

   4. fmm-def 91.0%

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

7. Applied egg-rr 91.0%

   \[\leadsto x - \color{blue}{{\left(2 \cdot \left(y \cdot \frac{z}{y \cdot \mathsf{fma}\left(2, \frac{{z}^{2}}{y}, -t\right)}\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 91.0%

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

   2. associate-*r/ 81.1%

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

   3. times-frac 95.5%

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

   4. *-inverses 95.5%

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

   5. *-lft-identity 95.5%

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

   6. fmm-undef 95.5%

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

9. Simplified 95.5%

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

    1. div-inv 95.5%

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

    2. unpow2 95.5%

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

    3. associate-*l* 97.4%

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

    4. div-inv 97.5%

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

11. Applied egg-rr 97.5%

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

12. Final simplification 97.5%

    \[\leadsto x + 2 \cdot \frac{z}{t - 2 \cdot \left(z \cdot \frac{z}{y}\right)} \]
13. Add Preprocessing

Alternative 2: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-53} \lor \neg \left(z \leq 1.2 \cdot 10^{+80}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9.2e-53) (not (<= z 1.2e+80)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.2e-53) || !(z <= 1.2e+80)) {
		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 <= (-9.2d-53)) .or. (.not. (z <= 1.2d+80))) 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 <= -9.2e-53) || !(z <= 1.2e+80)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -9.2e-53) or not (z <= 1.2e+80):
		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 <= -9.2e-53) || !(z <= 1.2e+80))
		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 <= -9.2e-53) || ~((z <= 1.2e+80)))
		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, -9.2e-53], N[Not[LessEqual[z, 1.2e+80]], $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 < -9.2000000000000005e-53 or 1.1999999999999999e80 < z

   1. Initial program 71.9%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 91.5%

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

      1. mul-1-neg 91.5%

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

      2. unsub-neg 91.5%

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

   7. Simplified 91.5%

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

   if -9.2000000000000005e-53 < z < 1.1999999999999999e80

   1. Initial program 91.0%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in y around inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. *-commutative 90.5%

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

   7. Simplified 90.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-53} \lor \neg \left(z \leq 1.2 \cdot 10^{+80}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-53} \lor \neg \left(z \leq 1.6 \cdot 10^{+80}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.4e-53) (not (<= z 1.6e+80))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.4e-53) || !(z <= 1.6e+80)) {
		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 <= (-4.4d-53)) .or. (.not. (z <= 1.6d+80))) 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 <= -4.4e-53) || !(z <= 1.6e+80)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.4e-53) or not (z <= 1.6e+80):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.4e-53) || !(z <= 1.6e+80))
		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 <= -4.4e-53) || ~((z <= 1.6e+80)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.4e-53], N[Not[LessEqual[z, 1.6e+80]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -4.40000000000000037e-53 or 1.59999999999999995e80 < z

   1. Initial program 71.9%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 91.5%

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

      1. mul-1-neg 91.5%

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

      2. unsub-neg 91.5%

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

   7. Simplified 91.5%

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

   if -4.40000000000000037e-53 < z < 1.59999999999999995e80

   1. Initial program 91.0%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in y around 0 72.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-53} \lor \neg \left(z \leq 1.6 \cdot 10^{+80}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 81.1%

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

3. Simplified 90.6%

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

5. Taylor expanded in y around 0 73.8%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 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 2024170 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (/ 1 (- (/ z y) (/ (/ t 2) z)))))

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