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

Percentage Accurate: 82.0% → 98.4%

Time: 10.4s

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: 82.0% 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, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((2.0 * y) / ((y * (t / z)) - (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 + ((2.0d0 * y) / ((y * (t / z)) - (2.0d0 * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 90.9%

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

   1. clear-num 91.0%

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

   2. un-div-inv 91.0%

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

   3. *-commutative 91.0%

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

   4. *-commutative 91.0%

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

   5. associate-*l* 91.0%

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

   6. pow2 91.0%

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

6. Applied egg-rr 91.0%

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

7. Taylor expanded in z around 0 96.4%

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

   1. +-commutative 96.4%

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

   2. mul-1-neg 96.4%

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

   3. *-commutative 96.4%

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

   4. associate-*r/ 98.1%

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

   5. unsub-neg 98.1%

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

   6. *-commutative 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

    \[\leadsto x + \frac{2 \cdot y}{y \cdot \frac{t}{z} - 2 \cdot z} \]
11. Add Preprocessing

Alternative 2: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-11} \lor \neg \left(z \leq 1.15 \cdot 10^{-34}\right) \land \left(z \leq 1040000000000 \lor \neg \left(z \leq 7.8 \cdot 10^{+29}\right)\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.2e-11)
         (and (not (<= z 1.15e-34))
              (or (<= z 1040000000000.0) (not (<= z 7.8e+29)))))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.2e-11) || (!(z <= 1.15e-34) && ((z <= 1040000000000.0) || !(z <= 7.8e+29)))) {
		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 <= (-8.2d-11)) .or. (.not. (z <= 1.15d-34)) .and. (z <= 1040000000000.0d0) .or. (.not. (z <= 7.8d+29))) 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 <= -8.2e-11) || (!(z <= 1.15e-34) && ((z <= 1040000000000.0) || !(z <= 7.8e+29)))) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.2e-11) or (not (z <= 1.15e-34) and ((z <= 1040000000000.0) or not (z <= 7.8e+29))):
		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 <= -8.2e-11) || (!(z <= 1.15e-34) && ((z <= 1040000000000.0) || !(z <= 7.8e+29))))
		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 <= -8.2e-11) || (~((z <= 1.15e-34)) && ((z <= 1040000000000.0) || ~((z <= 7.8e+29)))))
		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, -8.2e-11], And[N[Not[LessEqual[z, 1.15e-34]], $MachinePrecision], Or[LessEqual[z, 1040000000000.0], N[Not[LessEqual[z, 7.8e+29]], $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 < -8.2000000000000001e-11 or 1.15000000000000006e-34 < z < 1.04e12 or 7.79999999999999937e29 < z

   1. Initial program 76.6%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in y around 0 90.3%

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

   if -8.2000000000000001e-11 < z < 1.15000000000000006e-34 or 1.04e12 < z < 7.79999999999999937e29

   1. Initial program 92.1%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in y around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.4%

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

   7. Simplified 94.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-11} \lor \neg \left(z \leq 1.15 \cdot 10^{-34}\right) \land \left(z \leq 1040000000000 \lor \neg \left(z \leq 7.8 \cdot 10^{+29}\right)\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -205 \lor \neg \left(z \leq 3.2 \cdot 10^{-42}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -205.0) (not (<= z 3.2e-42))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -205.0) || !(z <= 3.2e-42)) {
		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 <= (-205.0d0)) .or. (.not. (z <= 3.2d-42))) 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 <= -205.0) || !(z <= 3.2e-42)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -205.0) or not (z <= 3.2e-42):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -205.0) || !(z <= 3.2e-42))
		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 <= -205.0) || ~((z <= 3.2e-42)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -205.0], N[Not[LessEqual[z, 3.2e-42]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -205 or 3.20000000000000025e-42 < z

   1. Initial program 77.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in y around 0 86.0%

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

   if -205 < z < 3.20000000000000025e-42

   1. Initial program 91.6%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in x around inf 72.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -205 \lor \neg \left(z \leq 3.2 \cdot 10^{-42}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 84.2%

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

3. Simplified 90.9%

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

5. Taylor expanded in x around inf 72.6%

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

6. Final simplification 72.6%

   \[\leadsto x \]
7. Add Preprocessing

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 2024047 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

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