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

Percentage Accurate: 82.0% → 93.8%

Time: 9.0s

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: 82.0% 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: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{t\_1 - y \cdot t} \leq 5 \cdot 10^{+188}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* 2.0 z))))
   (if (<= (/ (* (* y 2.0) z) (- t_1 (* y t))) 5e+188)
     (+ 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))) <= 5e+188) {
		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))) <= 5d+188) 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))) <= 5e+188) {
		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))) <= 5e+188:
		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))) <= 5e+188)
		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))) <= 5e+188)
		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], 5e+188], 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))) < 5.0000000000000001e188

   1. Initial program 94.8%

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

   3. Simplified 95.6%

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

   if 5.0000000000000001e188 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.4%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in y around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. unsub-neg 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 5 \cdot 10^{+188}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+48} \lor \neg \left(z \leq 8.2 \cdot 10^{-22}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.15e+48) (not (<= z 8.2e-22)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.15e+48) || !(z <= 8.2e-22)) {
		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 <= (-1.15d+48)) .or. (.not. (z <= 8.2d-22))) 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 <= -1.15e+48) || !(z <= 8.2e-22)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.15e+48) or not (z <= 8.2e-22):
		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 <= -1.15e+48) || !(z <= 8.2e-22))
		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 <= -1.15e+48) || ~((z <= 8.2e-22)))
		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, -1.15e+48], N[Not[LessEqual[z, 8.2e-22]], $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 < -1.15e48 or 8.1999999999999999e-22 < z

   1. Initial program 61.1%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in y around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. unsub-neg 91.6%

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

   7. Simplified 91.6%

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

   if -1.15e48 < z < 8.1999999999999999e-22

   1. Initial program 90.4%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in y around inf 88.3%

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

      1. associate-*r/ 88.3%

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

      2. *-commutative 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+48} \lor \neg \left(z \leq 8.2 \cdot 10^{-22}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-39} \lor \neg \left(z \leq 1.66 \cdot 10^{-22}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.1e-39) (not (<= z 1.66e-22))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.1e-39) || !(z <= 1.66e-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 <= (-1.1d-39)) .or. (.not. (z <= 1.66d-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 <= -1.1e-39) || !(z <= 1.66e-22)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.1e-39) or not (z <= 1.66e-22):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.1e-39) || !(z <= 1.66e-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 <= -1.1e-39) || ~((z <= 1.66e-22)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.1e-39], N[Not[LessEqual[z, 1.66e-22]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e-39 or 1.65999999999999997e-22 < z

   1. Initial program 64.9%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in y around 0 88.6%

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

      1. mul-1-neg 88.6%

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

      2. unsub-neg 88.6%

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

   7. Simplified 88.6%

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

   if -1.1e-39 < z < 1.65999999999999997e-22

   1. Initial program 91.2%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 77.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-39} \lor \neg \left(z \leq 1.66 \cdot 10^{-22}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-244}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-222}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.4e-244) x (if (<= x 1.2e-222) (* 2.0 (/ z t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.4e-244) {
		tmp = x;
	} else if (x <= 1.2e-222) {
		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 <= (-4.4d-244)) then
        tmp = x
    else if (x <= 1.2d-222) 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 <= -4.4e-244) {
		tmp = x;
	} else if (x <= 1.2e-222) {
		tmp = 2.0 * (z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.4e-244:
		tmp = x
	elif x <= 1.2e-222:
		tmp = 2.0 * (z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.4e-244)
		tmp = x;
	elseif (x <= 1.2e-222)
		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 <= -4.4e-244)
		tmp = x;
	elseif (x <= 1.2e-222)
		tmp = 2.0 * (z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.4e-244], x, If[LessEqual[x, 1.2e-222], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.39999999999999969e-244 or 1.19999999999999997e-222 < x

   1. Initial program 78.2%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in y around 0 81.2%

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

   if -4.39999999999999969e-244 < x < 1.19999999999999997e-222

   1. Initial program 53.9%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in y around inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. *-commutative 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in x around 0 54.1%

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

Alternative 5: 75.2% 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 75.6%

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

3. Simplified 87.3%

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

5. Taylor expanded in y around 0 74.7%

   \[\leadsto \color{blue}{x} \]
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 2024165 
(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)))))