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

Percentage Accurate: 82.1% → 98.2%

Time: 9.4s

Alternatives: 6

Speedup: 1.5×

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: 82.1% 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: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. associate-/l* 87.6%

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

   2. associate-*l* 87.6%

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

3. Simplified 87.6%

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

4. Taylor expanded in z around 0 94.4%

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

   1. +-commutative 94.4%

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

   2. mul-1-neg 94.4%

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

   3. *-commutative 94.4%

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

   4. associate-*r/ 98.5%

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

   5. unsub-neg 98.5%

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

   6. *-commutative 98.5%

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

6. Simplified 98.5%

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

   1. clear-num 98.5%

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

   2. un-div-inv 98.6%

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

8. Applied egg-rr 98.6%

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

9. Final simplification 98.6%

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

Alternative 2: 96.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. associate-/l* 87.6%

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

   2. associate-*l* 87.6%

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

3. Simplified 87.6%

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

4. Taylor expanded in z around 0 94.4%

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

   1. +-commutative 94.4%

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

   2. mul-1-neg 94.4%

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

   3. *-commutative 94.4%

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

   4. associate-*r/ 98.5%

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

   5. unsub-neg 98.5%

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

   6. *-commutative 98.5%

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

6. Simplified 98.5%

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

   1. expm1-log1p-u 89.2%

      \[\leadsto x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}}\right)\right)} \]

   2. expm1-udef 78.1%

      \[\leadsto x - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right)} \]

   3. *-commutative 78.1%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{2 \cdot y}}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right) \]

   4. *-un-lft-identity 78.1%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\frac{2 \cdot y}{\color{blue}{1 \cdot \left(z \cdot 2 - y \cdot \frac{t}{z}\right)}}\right)} - 1\right) \]

   5. times-frac 78.1%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{1} \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}}\right)} - 1\right) \]

   6. metadata-eval 78.1%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\color{blue}{2} \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right) \]

8. Applied egg-rr 78.1%

   \[\leadsto x - \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 89.2%

      \[\leadsto x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}\right)\right)} \]

   2. expm1-log1p 98.5%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*r/ 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-/r/ 97.0%

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

10. Simplified 97.0%

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

11. Final simplification 97.0%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. associate-/l* 87.6%

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

   2. associate-*l* 87.6%

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

3. Simplified 87.6%

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

4. Taylor expanded in z around 0 94.4%

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

   1. +-commutative 94.4%

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

   2. mul-1-neg 94.4%

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

   3. *-commutative 94.4%

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

   4. associate-*r/ 98.5%

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

   5. unsub-neg 98.5%

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

   6. *-commutative 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 88.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+37} \lor \neg \left(z \leq 2.55 \cdot 10^{-36}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.3e+37) (not (<= z 2.55e-36)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.3e+37) || !(z <= 2.55e-36)) {
		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 <= (-3.3d+37)) .or. (.not. (z <= 2.55d-36))) 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 <= -3.3e+37) || !(z <= 2.55e-36)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.3e+37) or not (z <= 2.55e-36):
		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 <= -3.3e+37) || !(z <= 2.55e-36))
		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 <= -3.3e+37) || ~((z <= 2.55e-36)))
		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, -3.3e+37], N[Not[LessEqual[z, 2.55e-36]], $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 < -3.3000000000000001e37 or 2.54999999999999987e-36 < z

   1. Initial program 69.8%

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

      1. sub-neg 69.8%

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

      2. associate-*l* 69.8%

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

      3. *-commutative 69.8%

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

      4. associate-*l/ 82.6%

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

      5. distribute-rgt-neg-in 82.6%

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

      6. *-commutative 82.6%

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

      7. associate-*l* 82.6%

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

      8. distribute-rgt-neg-in 82.6%

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

      9. metadata-eval 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in y around 0 92.4%

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

      1. mul-1-neg 92.4%

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

      2. sub-neg 92.4%

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

   6. Simplified 92.4%

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

   if -3.3000000000000001e37 < z < 2.54999999999999987e-36

   1. Initial program 88.4%

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

      1. sub-neg 88.4%

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

      2. associate-*l* 88.4%

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

      3. *-commutative 88.4%

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

      4. associate-*l/ 91.6%

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

      5. distribute-rgt-neg-in 91.6%

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

      6. *-commutative 91.6%

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

      7. associate-*l* 91.6%

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

      8. distribute-rgt-neg-in 91.6%

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

      9. metadata-eval 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in y around inf 90.4%

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

      1. metadata-eval 90.4%

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

      2. cancel-sign-sub-inv 90.4%

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

      3. associate-*r/ 90.4%

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

      4. *-commutative 90.4%

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

   6. Simplified 90.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+37} \lor \neg \left(z \leq 2.55 \cdot 10^{-36}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]

Alternative 5: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-74} \lor \neg \left(z \leq 2.4 \cdot 10^{-36}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.65e-74) (not (<= z 2.4e-36))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.65e-74) || !(z <= 2.4e-36)) {
		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 <= (-1.65d-74)) .or. (.not. (z <= 2.4d-36))) 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 <= -1.65e-74) || !(z <= 2.4e-36)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.65e-74) or not (z <= 2.4e-36):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.65e-74) || !(z <= 2.4e-36))
		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 <= -1.65e-74) || ~((z <= 2.4e-36)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.65e-74], N[Not[LessEqual[z, 2.4e-36]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.64999999999999998e-74 or 2.4e-36 < z

   1. Initial program 73.4%

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

      1. sub-neg 73.4%

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

      2. associate-*l* 73.4%

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

      3. *-commutative 73.4%

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

      4. associate-*l/ 84.7%

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

      5. distribute-rgt-neg-in 84.7%

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

      6. *-commutative 84.7%

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

      7. associate-*l* 84.7%

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

      8. distribute-rgt-neg-in 84.7%

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

      9. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in y around 0 89.9%

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

      1. mul-1-neg 89.9%

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

      2. sub-neg 89.9%

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

   6. Simplified 89.9%

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

   if -1.64999999999999998e-74 < z < 2.4e-36

   1. Initial program 86.7%

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

      1. sub-neg 86.7%

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

      2. associate-*l* 86.7%

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

      3. *-commutative 86.7%

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

      4. associate-*l/ 90.4%

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

      5. distribute-rgt-neg-in 90.4%

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

      6. *-commutative 90.4%

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

      7. associate-*l* 90.4%

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

      8. distribute-rgt-neg-in 90.4%

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

      9. metadata-eval 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around inf 77.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-74} \lor \neg \left(z \leq 2.4 \cdot 10^{-36}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 75.5% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. sub-neg 79.4%

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

   2. associate-*l* 79.4%

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

   3. *-commutative 79.4%

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

   4. associate-*l/ 87.3%

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

   5. distribute-rgt-neg-in 87.3%

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

   6. *-commutative 87.3%

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

   7. associate-*l* 87.3%

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

   8. distribute-rgt-neg-in 87.3%

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

   9. metadata-eval 87.3%

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

3. Simplified 87.3%

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

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 2023320 
(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)))))