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

Percentage Accurate: 81.8% → 95.7%

Time: 20.9s

Alternatives: 5

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: 95.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Simplified 87.2%

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

   1. clear-num 87.1%

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

   2. un-div-inv 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*l* 87.2%

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

   5. pow2 87.2%

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

6. Applied egg-rr 87.2%

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

7. Taylor expanded in y around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. neg-mul-1 95.9%

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

   3. *-commutative 95.9%

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

   4. distribute-rgt-neg-in 95.9%

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

9. Applied egg-rr 95.9%

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

10. Final simplification 95.9%

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

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+68} \lor \neg \left(z \leq 1.45 \cdot 10^{-97}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.85e+68) (not (<= z 1.45e-97)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.85e+68) || !(z <= 1.45e-97)) {
		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 <= (-1.85d+68)) .or. (.not. (z <= 1.45d-97))) 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 <= -1.85e+68) || !(z <= 1.45e-97)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.85e+68) or not (z <= 1.45e-97):
		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 <= -1.85e+68) || !(z <= 1.45e-97))
		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 <= -1.85e+68) || ~((z <= 1.45e-97)))
		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, -1.85e+68], N[Not[LessEqual[z, 1.45e-97]], $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 < -1.84999999999999999e68 or 1.45e-97 < z

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. +-commutative 65.4%

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

      3. associate-/l* 82.1%

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

      4. distribute-rgt-neg-in 82.1%

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

      5. fma-define 82.2%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in y around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   7. Simplified 86.9%

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

   if -1.84999999999999999e68 < z < 1.45e-97

   1. Initial program 90.7%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around inf 91.6%

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

      1. associate-*r/ 91.6%

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

      2. *-commutative 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+68} \lor \neg \left(z \leq 1.45 \cdot 10^{-97}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+66} \lor \neg \left(z \leq 1.45 \cdot 10^{-97}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7e+66) (not (<= z 1.45e-97)))
   (- x (/ y z))
   (- x (* z (/ -2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7e+66) || !(z <= 1.45e-97)) {
		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 <= (-7d+66)) .or. (.not. (z <= 1.45d-97))) 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 <= -7e+66) || !(z <= 1.45e-97)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (z * (-2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7e+66) or not (z <= 1.45e-97):
		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 <= -7e+66) || !(z <= 1.45e-97))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(z * Float64(-2.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -7e+66) || ~((z <= 1.45e-97)))
		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, -7e+66], N[Not[LessEqual[z, 1.45e-97]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.9999999999999994e66 or 1.45e-97 < z

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. +-commutative 65.4%

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

      3. associate-/l* 82.1%

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

      4. distribute-rgt-neg-in 82.1%

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

      5. fma-define 82.2%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in y around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   7. Simplified 86.9%

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

   if -6.9999999999999994e66 < z < 1.45e-97

   1. Initial program 90.7%

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

   3. Simplified 91.5%

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

      1. clear-num 91.4%

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

      2. un-div-inv 91.5%

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

      3. *-commutative 91.5%

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

      4. associate-*l* 91.5%

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

      5. pow2 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in y around inf 91.6%

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

      1. associate-*r/ 91.6%

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

      2. associate-*l/ 91.5%

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

      3. *-commutative 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+66} \lor \neg \left(z \leq 1.45 \cdot 10^{-97}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3e-178) x (if (<= t 7.5e-57) (- x (/ y z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e-178) {
		tmp = x;
	} else if (t <= 7.5e-57) {
		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 (t <= (-3d-178)) then
        tmp = x
    else if (t <= 7.5d-57) 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 (t <= -3e-178) {
		tmp = x;
	} else if (t <= 7.5e-57) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3e-178:
		tmp = x
	elif t <= 7.5e-57:
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3e-178)
		tmp = x;
	elseif (t <= 7.5e-57)
		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 (t <= -3e-178)
		tmp = x;
	elseif (t <= 7.5e-57)
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3e-178], x, If[LessEqual[t, 7.5e-57], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9999999999999999e-178 or 7.49999999999999973e-57 < t

   1. Initial program 82.2%

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

      1. sub-neg 82.2%

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

      2. +-commutative 82.2%

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

      3. associate-/l* 90.6%

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

      4. distribute-rgt-neg-in 90.6%

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

      5. fma-define 90.6%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in y around 0 84.1%

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

   if -2.9999999999999999e-178 < t < 7.49999999999999973e-57

   1. Initial program 71.1%

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

      1. sub-neg 71.1%

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

      2. +-commutative 71.1%

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

      3. associate-/l* 78.4%

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

      4. distribute-rgt-neg-in 78.4%

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

      5. fma-define 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in y around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unsub-neg 72.3%

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

   7. Simplified 72.3%

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

Alternative 5: 75.8% 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 79.0%

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

   1. sub-neg 79.0%

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

   2. +-commutative 79.0%

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

   3. associate-/l* 87.2%

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

   4. distribute-rgt-neg-in 87.2%

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

   5. fma-define 87.2%

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

3. Simplified 90.5%

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

5. Taylor expanded in y around 0 74.3%

   \[\leadsto \color{blue}{x} \]
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 2024143 
(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)))))