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

Percentage Accurate: 81.0% → 94.6%

Time: 11.9s

Alternatives: 6

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: 81.0% 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: 94.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 2, \frac{z}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t))) INFINITY)
   (fma (* y 2.0) (/ z (fma z (* z -2.0) (* y t))) x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y * 2.0) * z) / ((z * (2.0 * z)) - (y * t))) <= ((double) INFINITY)) {
		tmp = fma((y * 2.0), (z / fma(z, (z * -2.0), (y * t))), x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(z * Float64(2.0 * z)) - Float64(y * t))) <= Inf)
		tmp = fma(Float64(y * 2.0), Float64(z / fma(z, Float64(z * -2.0), Float64(y * t))), x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))) < +inf.0

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. +-commutative 95.2%

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

      3. associate-/l* 97.3%

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

      4. distribute-rgt-neg-in 97.3%

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

      5. fma-define 97.3%

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

   3. Simplified 97.3%

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

   if +inf.0 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.0%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in y around 0 72.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 2, \frac{z}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{t\_1 - y \cdot t} \leq \infty:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* 2.0 z))))
   (if (<= (/ (* (* y 2.0) z) (- t_1 (* y t))) INFINITY)
     (+ x (* (* y 2.0) (/ z (- (* y t) t_1))))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (2.0 * z);
	double tmp;
	if ((((y * 2.0) * z) / (t_1 - (y * t))) <= ((double) INFINITY)) {
		tmp = x + ((y * 2.0) * (z / ((y * t) - t_1)));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (2.0 * z);
	double tmp;
	if ((((y * 2.0) * z) / (t_1 - (y * t))) <= Double.POSITIVE_INFINITY) {
		tmp = x + ((y * 2.0) * (z / ((y * t) - t_1)));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (2.0 * z)
	tmp = 0
	if (((y * 2.0) * z) / (t_1 - (y * t))) <= math.inf:
		tmp = x + ((y * 2.0) * (z / ((y * t) - t_1)))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(2.0 * z))
	tmp = 0.0
	if (Float64(Float64(Float64(y * 2.0) * z) / Float64(t_1 - Float64(y * t))) <= Inf)
		tmp = Float64(x + Float64(Float64(y * 2.0) * Float64(z / Float64(Float64(y * t) - t_1))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (2.0 * z);
	tmp = 0.0;
	if ((((y * 2.0) * z) / (t_1 - (y * t))) <= Inf)
		tmp = x + ((y * 2.0) * (z / ((y * t) - t_1)));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(t$95$1 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x + N[(N[(y * 2.0), $MachinePrecision] * N[(z / N[(N[(y * t), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))) < +inf.0

   1. Initial program 95.2%

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

   3. Simplified 97.3%

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

   if +inf.0 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.0%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in y around 0 72.0%

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

4. Final simplification 94.6%

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

Alternative 3: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-108} \lor \neg \left(x \leq -3 \cdot 10^{-280}\right) \land x \leq 1.45 \cdot 10^{-229}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4e-94)
   x
   (if (or (<= x -5.4e-108) (and (not (<= x -3e-280)) (<= x 1.45e-229)))
     (* 2.0 (/ z t))
     x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4e-94) {
		tmp = x;
	} else if ((x <= -5.4e-108) || (!(x <= -3e-280) && (x <= 1.45e-229))) {
		tmp = 2.0 * (z / t);
	} 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 <= (-4d-94)) then
        tmp = x
    else if ((x <= (-5.4d-108)) .or. (.not. (x <= (-3d-280))) .and. (x <= 1.45d-229)) then
        tmp = 2.0d0 * (z / t)
    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 <= -4e-94) {
		tmp = x;
	} else if ((x <= -5.4e-108) || (!(x <= -3e-280) && (x <= 1.45e-229))) {
		tmp = 2.0 * (z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4e-94:
		tmp = x
	elif (x <= -5.4e-108) or (not (x <= -3e-280) and (x <= 1.45e-229)):
		tmp = 2.0 * (z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4e-94)
		tmp = x;
	elseif ((x <= -5.4e-108) || (!(x <= -3e-280) && (x <= 1.45e-229)))
		tmp = Float64(2.0 * Float64(z / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4e-94)
		tmp = x;
	elseif ((x <= -5.4e-108) || (~((x <= -3e-280)) && (x <= 1.45e-229)))
		tmp = 2.0 * (z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4e-94], x, If[Or[LessEqual[x, -5.4e-108], And[N[Not[LessEqual[x, -3e-280]], $MachinePrecision], LessEqual[x, 1.45e-229]]], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999998e-94 or -5.4000000000000001e-108 < x < -2.99999999999999987e-280 or 1.45e-229 < x

   1. Initial program 86.0%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in x around inf 80.9%

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

   if -3.9999999999999998e-94 < x < -5.4000000000000001e-108 or -2.99999999999999987e-280 < x < 1.45e-229

   1. Initial program 79.5%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in y around inf 76.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in x around 0 70.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-108} \lor \neg \left(x \leq -3 \cdot 10^{-280}\right) \land x \leq 1.45 \cdot 10^{-229}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-11} \lor \neg \left(z \leq 7.5 \cdot 10^{-57}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.25e-11) (not (<= z 7.5e-57)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.25e-11) || !(z <= 7.5e-57)) {
		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 <= (-2.25d-11)) .or. (.not. (z <= 7.5d-57))) 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 <= -2.25e-11) || !(z <= 7.5e-57)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.25e-11) or not (z <= 7.5e-57):
		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 <= -2.25e-11) || !(z <= 7.5e-57))
		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 <= -2.25e-11) || ~((z <= 7.5e-57)))
		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, -2.25e-11], N[Not[LessEqual[z, 7.5e-57]], $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 < -2.25e-11 or 7.49999999999999973e-57 < z

   1. Initial program 75.8%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y around 0 87.0%

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

   if -2.25e-11 < z < 7.49999999999999973e-57

   1. Initial program 94.5%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in y around inf 91.6%

      \[\leadsto x - \color{blue}{-2 \cdot \frac{z}{t}} \]
   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 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-11} \lor \neg \left(z \leq 7.5 \cdot 10^{-57}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+15} \lor \neg \left(z \leq 1.6 \cdot 10^{-22}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.2e+15) (not (<= z 1.6e-22))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.2e+15) || !(z <= 1.6e-22)) {
		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 <= (-8.2d+15)) .or. (.not. (z <= 1.6d-22))) 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 <= -8.2e+15) || !(z <= 1.6e-22)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.2e+15) or not (z <= 1.6e-22):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.2e+15) || !(z <= 1.6e-22))
		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 <= -8.2e+15) || ~((z <= 1.6e-22)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.2e+15], N[Not[LessEqual[z, 1.6e-22]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e15 or 1.59999999999999994e-22 < z

   1. Initial program 74.4%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in y around 0 89.0%

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

   if -8.2e15 < z < 1.59999999999999994e-22

   1. Initial program 94.2%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around inf 75.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+15} \lor \neg \left(z \leq 1.6 \cdot 10^{-22}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

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 85.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in x around inf 73.1%

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

6. Final simplification 73.1%

   \[\leadsto x \]
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 2024095 
(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)))))