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

Percentage Accurate: 81.1% → 99.9%

Time: 9.2s

Alternatives: 6

Speedup: 1.3×

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: 81.1% 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: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. associate-/l* 87.5%

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

   2. associate-*l* 87.5%

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

3. Simplified 87.5%

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

5. Taylor expanded in z around 0 94.4%

   \[\leadsto x - \frac{y \cdot 2}{\color{blue}{-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z}} \]
6. 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. *-commutative 94.4%

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

   3. mul-1-neg 94.4%

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

   4. associate-*r/ 96.2%

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

   5. sub-neg 96.2%

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

   6. *-commutative 96.2%

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

   7. associate-*l/ 94.4%

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

   8. associate-*r/ 98.1%

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

7. Simplified 98.1%

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

8. Taylor expanded in y around 0 94.4%

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

   1. associate-/l* 96.4%

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

10. Simplified 96.4%

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

    1. expm1-log1p-u 88.8%

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

    2. expm1-udef 75.3%

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

    3. *-commutative 75.3%

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

    4. *-un-lft-identity 75.3%

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

    5. times-frac 75.3%

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

    6. metadata-eval 75.3%

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

    7. *-commutative 75.3%

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

    8. div-inv 75.4%

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

    9. clear-num 75.4%

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

12. Applied egg-rr 75.4%

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

    1. expm1-def 88.8%

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

    2. expm1-log1p 96.2%

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

    3. associate-*r/ 96.2%

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

    4. associate-/l* 96.1%

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

    5. div-sub 96.1%

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

    6. associate-*r/ 94.3%

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

    7. associate-*l/ 98.0%

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

    8. associate-/l* 99.9%

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

    9. *-inverses 99.9%

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

    10. /-rgt-identity 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 2: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -46000000 \lor \neg \left(z \leq 2.2 \cdot 10^{+24}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{2}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -46000000.0) (not (<= z 2.2e+24)))
   (- x (/ y z))
   (+ x (/ 2.0 (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -46000000.0) || !(z <= 2.2e+24)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (2.0 / (t / z));
	}
	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 <= (-46000000.0d0)) .or. (.not. (z <= 2.2d+24))) then
        tmp = x - (y / z)
    else
        tmp = x + (2.0d0 / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -46000000.0) || !(z <= 2.2e+24)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (2.0 / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -46000000.0) or not (z <= 2.2e+24):
		tmp = x - (y / z)
	else:
		tmp = x + (2.0 / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -46000000.0) || !(z <= 2.2e+24))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x + Float64(2.0 / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -46000000.0) || ~((z <= 2.2e+24)))
		tmp = x - (y / z);
	else
		tmp = x + (2.0 / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -46000000.0], N[Not[LessEqual[z, 2.2e+24]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6e7 or 2.20000000000000002e24 < z

   1. Initial program 70.8%

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

      1. associate-/l* 79.7%

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

      2. associate-*l* 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in y around 0 91.0%

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

   if -4.6e7 < z < 2.20000000000000002e24

   1. Initial program 95.6%

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

      1. associate-/l* 95.3%

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

      2. associate-*l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. associate-*r/ 87.7%

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

   7. Simplified 87.7%

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

      1. sub-neg 87.7%

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

      2. associate-/l* 87.6%

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

      3. distribute-neg-frac 87.6%

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

      4. metadata-eval 87.6%

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

   9. Applied egg-rr 87.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -46000000 \lor \neg \left(z \leq 2.2 \cdot 10^{+24}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{2}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2100000000 \lor \neg \left(z \leq 3.8 \cdot 10^{+24}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2100000000.0) (not (<= z 3.8e+24)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2100000000.0) || !(z <= 3.8e+24)) {
		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 <= (-2100000000.0d0)) .or. (.not. (z <= 3.8d+24))) 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 <= -2100000000.0) || !(z <= 3.8e+24)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2100000000.0) or not (z <= 3.8e+24):
		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 <= -2100000000.0) || !(z <= 3.8e+24))
		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 <= -2100000000.0) || ~((z <= 3.8e+24)))
		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, -2100000000.0], N[Not[LessEqual[z, 3.8e+24]], $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 < -2.1e9 or 3.80000000000000015e24 < z

   1. Initial program 70.8%

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

      1. associate-/l* 79.7%

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

      2. associate-*l* 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in y around 0 91.0%

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

   if -2.1e9 < z < 3.80000000000000015e24

   1. Initial program 95.6%

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

      1. associate-/l* 95.3%

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

      2. associate-*l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. associate-*r/ 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2100000000 \lor \neg \left(z \leq 3.8 \cdot 10^{+24}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+35} \lor \neg \left(z \leq 7.4 \cdot 10^{-26}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4e+35) (not (<= z 7.4e-26))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4e+35) || !(z <= 7.4e-26)) {
		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 <= (-4d+35)) .or. (.not. (z <= 7.4d-26))) 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 <= -4e+35) || !(z <= 7.4e-26)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4e+35) or not (z <= 7.4e-26):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4e+35) || !(z <= 7.4e-26))
		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 <= -4e+35) || ~((z <= 7.4e-26)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4e+35], N[Not[LessEqual[z, 7.4e-26]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.9999999999999999e35 or 7.3999999999999997e-26 < z

   1. Initial program 71.0%

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

      1. associate-/l* 79.9%

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

      2. associate-*l* 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in y around 0 90.4%

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

   if -3.9999999999999999e35 < z < 7.3999999999999997e-26

   1. Initial program 95.6%

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

      1. associate-/l* 95.3%

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

      2. associate-*l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in y around 0 45.0%

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

   6. Taylor expanded in x around inf 76.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+35} \lor \neg \left(z \leq 7.4 \cdot 10^{-26}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-239}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-209}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.95e-239) x (if (<= x 9.2e-209) (/ (- y) z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.95e-239) {
		tmp = x;
	} else if (x <= 9.2e-209) {
		tmp = -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 (x <= (-1.95d-239)) then
        tmp = x
    else if (x <= 9.2d-209) then
        tmp = -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 (x <= -1.95e-239) {
		tmp = x;
	} else if (x <= 9.2e-209) {
		tmp = -y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.95e-239:
		tmp = x
	elif x <= 9.2e-209:
		tmp = -y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.95e-239)
		tmp = x;
	elseif (x <= 9.2e-209)
		tmp = Float64(Float64(-y) / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.95e-239)
		tmp = x;
	elseif (x <= 9.2e-209)
		tmp = -y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.95e-239], x, If[LessEqual[x, 9.2e-209], N[((-y) / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e-239 or 9.1999999999999999e-209 < x

   1. Initial program 85.5%

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

      1. associate-/l* 91.4%

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

      2. associate-*l* 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in y around 0 68.3%

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

   6. Taylor expanded in x around inf 81.3%

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

   if -1.95e-239 < x < 9.1999999999999999e-209

   1. Initial program 71.4%

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

      1. associate-/l* 67.7%

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

      2. associate-*l* 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in y around 0 65.5%

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

   6. Taylor expanded in x around 0 51.6%

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

      1. mul-1-neg 51.6%

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

      2. distribute-neg-frac 51.6%

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

   8. Simplified 51.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-239}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-209}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% 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 83.2%

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

   1. associate-/l* 87.5%

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

   2. associate-*l* 87.5%

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

3. Simplified 87.5%

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

5. Taylor expanded in y around 0 67.8%

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

6. Taylor expanded in x around inf 72.4%

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

7. Final simplification 72.4%

   \[\leadsto x \]
8. 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 2024017 
(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)))))