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

Percentage Accurate: 81.9% → 95.0%

Time: 9.4s

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: 81.9% 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: 95.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 2 \cdot 10^{+207}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - 2 \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t))) 2e+207)
   (+ x (* z (/ (* y 2.0) (- (* y t) (* 2.0 (pow z 2.0))))))
   (- 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))) <= 2e+207) {
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * pow(z, 2.0)))));
	} else {
		tmp = x - (y / 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 ((((y * 2.0d0) * z) / ((z * (2.0d0 * z)) - (y * t))) <= 2d+207) then
        tmp = x + (z * ((y * 2.0d0) / ((y * t) - (2.0d0 * (z ** 2.0d0)))))
    else
        tmp = x - (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y * 2.0) * z) / ((z * (2.0 * z)) - (y * t))) <= 2e+207) {
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * Math.pow(z, 2.0)))));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((y * 2.0) * z) / ((z * (2.0 * z)) - (y * t))) <= 2e+207:
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * math.pow(z, 2.0)))))
	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))) <= 2e+207)
		tmp = Float64(x + Float64(z * Float64(Float64(y * 2.0) / Float64(Float64(y * t) - Float64(2.0 * (z ^ 2.0))))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((y * 2.0) * z) / ((z * (2.0 * z)) - (y * t))) <= 2e+207)
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * (z ^ 2.0)))));
	else
		tmp = x - (y / z);
	end
	tmp_2 = 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], 2e+207], N[(x + N[(z * N[(N[(y * 2.0), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(2.0 * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))) < 2.0000000000000001e207

   1. Initial program 94.6%

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

   3. Simplified 96.3%

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

      1. clear-num 96.3%

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

      2. un-div-inv 96.4%

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

      3. *-commutative 96.4%

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

      4. *-commutative 96.4%

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

      5. associate-*l* 96.3%

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

      6. pow2 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. associate-/r/ 96.3%

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

      2. *-commutative 96.3%

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

      3. *-commutative 96.3%

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

      4. *-commutative 96.3%

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

   8. Simplified 96.3%

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

   if 2.0000000000000001e207 < (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))

   1. Initial program 2.8%

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

   3. Simplified 43.2%

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

   5. Taylor expanded in y around 0 92.2%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 2 \cdot 10^{+207}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - 2 \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{t\_1 - y \cdot t} \leq 2 \cdot 10^{+207}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* 2.0 z))))
   (if (<= (/ (* (* y 2.0) z) (- t_1 (* y t))) 2e+207)
     (+ 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))) <= 2e+207) {
		tmp = x + ((y * 2.0) * (z / ((y * t) - t_1)));
	} else {
		tmp = x - (y / 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) :: t_1
    real(8) :: tmp
    t_1 = z * (2.0d0 * z)
    if ((((y * 2.0d0) * z) / (t_1 - (y * t))) <= 2d+207) then
        tmp = x + ((y * 2.0d0) * (z / ((y * t) - t_1)))
    else
        tmp = x - (y / z)
    end if
    code = tmp
end function

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))) <= 2e+207) {
		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))) <= 2e+207:
		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))) <= 2e+207)
		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))) <= 2e+207)
		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], 2e+207], 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 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))) < 2.0000000000000001e207

   1. Initial program 94.6%

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

   3. Simplified 96.3%

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

   if 2.0000000000000001e207 < (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))

   1. Initial program 2.8%

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

   3. Simplified 43.2%

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

   5. Taylor expanded in y around 0 92.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 2 \cdot 10^{+207}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \lor \neg \left(z \leq 6.4 \cdot 10^{+69}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.9) (not (<= z 6.4e+69)))
   (- x (/ y z))
   (- x (* z (/ -2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.9) || !(z <= 6.4e+69)) {
		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 <= (-3.9d0)) .or. (.not. (z <= 6.4d+69))) 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 <= -3.9) || !(z <= 6.4e+69)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (z * (-2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.9) or not (z <= 6.4e+69):
		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 <= -3.9) || !(z <= 6.4e+69))
		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 <= -3.9) || ~((z <= 6.4e+69)))
		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, -3.9], N[Not[LessEqual[z, 6.4e+69]], $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 < -3.89999999999999991 or 6.3999999999999997e69 < z

   1. Initial program 67.0%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in y around 0 95.5%

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

   if -3.89999999999999991 < z < 6.3999999999999997e69

   1. Initial program 91.9%

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

   3. Simplified 93.9%

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

      1. clear-num 93.9%

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

      2. un-div-inv 94.0%

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

      3. *-commutative 94.0%

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

      4. *-commutative 94.0%

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

      5. associate-*l* 93.9%

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

      6. pow2 93.9%

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

   6. Applied egg-rr 93.9%

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

      1. associate-/r/ 93.9%

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

      2. *-commutative 93.9%

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

      3. *-commutative 93.9%

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

      4. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   9. Taylor expanded in y around inf 85.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \lor \neg \left(z \leq 6.4 \cdot 10^{+69}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.125 \lor \neg \left(z \leq 6.4 \cdot 10^{+69}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.125) (not (<= z 6.4e+69)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.125) || !(z <= 6.4e+69)) {
		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 <= (-0.125d0)) .or. (.not. (z <= 6.4d+69))) 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 <= -0.125) || !(z <= 6.4e+69)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -0.125) or not (z <= 6.4e+69):
		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 <= -0.125) || !(z <= 6.4e+69))
		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 <= -0.125) || ~((z <= 6.4e+69)))
		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, -0.125], N[Not[LessEqual[z, 6.4e+69]], $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 < -0.125 or 6.3999999999999997e69 < z

   1. Initial program 67.0%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in y around 0 95.5%

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

   if -0.125 < z < 6.3999999999999997e69

   1. Initial program 91.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in y around inf 85.9%

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

      1. associate-*r/ 85.9%

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

      2. *-commutative 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.125 \lor \neg \left(z \leq 6.4 \cdot 10^{+69}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+32} \lor \neg \left(z \leq 2.1 \cdot 10^{-91}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.6e+32) (not (<= z 2.1e-91))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.6e+32) || !(z <= 2.1e-91)) {
		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 <= (-1.6d+32)) .or. (.not. (z <= 2.1d-91))) 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 <= -1.6e+32) || !(z <= 2.1e-91)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.6e+32) or not (z <= 2.1e-91):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.6e+32) || !(z <= 2.1e-91))
		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 <= -1.6e+32) || ~((z <= 2.1e-91)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.6e+32], N[Not[LessEqual[z, 2.1e-91]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.5999999999999999e32 or 2.0999999999999999e-91 < z

   1. Initial program 72.7%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in y around 0 87.0%

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

   if -1.5999999999999999e32 < z < 2.0999999999999999e-91

   1. Initial program 92.2%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around inf 77.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+32} \lor \neg \left(z \leq 2.1 \cdot 10^{-91}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% 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 81.7%

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

3. Simplified 88.8%

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

5. Taylor expanded in x around inf 72.5%

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

6. Final simplification 72.5%

   \[\leadsto x \]
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 2024054 
(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)))))