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

Percentage Accurate: 82.8% → 94.3%

Time: 9.0s

Alternatives: 5

Speedup: 1.0×

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: 82.8% 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: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{t\_1 - y \cdot t} \leq 10^{+100}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* 2.0 z))))
   (if (<= (/ (* (* y 2.0) z) (- t_1 (* y t))) 1e+100)
     (+ x (* z (/ (* y 2.0) (- (* 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))) <= 1e+100) {
		tmp = x + (z * ((y * 2.0) / ((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))) <= 1d+100) then
        tmp = x + (z * ((y * 2.0d0) / ((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))) <= 1e+100) {
		tmp = x + (z * ((y * 2.0) / ((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))) <= 1e+100:
		tmp = x + (z * ((y * 2.0) / ((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))) <= 1e+100)
		tmp = Float64(x + Float64(z * Float64(Float64(y * 2.0) / 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))) <= 1e+100)
		tmp = x + (z * ((y * 2.0) / ((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], 1e+100], N[(x + N[(z * N[(N[(y * 2.0), $MachinePrecision] / 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))) < 1.00000000000000002e100

   1. Initial program 93.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{\left(y \cdot 2\right) \cdot z}}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{1}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{y \cdot 2}}{\color{blue}{z}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{1}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{y \cdot 2}} \cdot \color{blue}{z}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{y \cdot 2}{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot z\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{y \cdot 2}{\left(z \cdot 2\right) \cdot z - y \cdot t}\right), \color{blue}{z}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 2\right), \left(\left(z \cdot 2\right) \cdot z - y \cdot t\right)\right), z\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(\left(z \cdot 2\right) \cdot z - y \cdot t\right)\right), z\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\left(\left(z \cdot 2\right) \cdot z\right), \left(y \cdot t\right)\right)\right), z\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\left(z \cdot \left(z \cdot 2\right)\right), \left(y \cdot t\right)\right)\right), z\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot 2\right)\right), \left(y \cdot t\right)\right)\right), z\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(2 \cdot z\right)\right), \left(y \cdot t\right)\right)\right), z\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(2, z\right)\right), \left(y \cdot t\right)\right)\right), z\right)\right) \]

      13. *-lowering-*.f64 96.5%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(2, z\right)\right), \mathsf{*.f64}\left(y, t\right)\right)\right), z\right)\right) \]

   4. Applied egg-rr 96.5%

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

   if 1.00000000000000002e100 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 5.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\frac{y}{z}\right)\right) \]

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      4. /-lowering-/.f64 85.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 85.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 10^{+100}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+126}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.8e+125)
     t_1
     (if (<= z 4e+126)
       (+ x (* (* y 2.0) (/ z (- (* y t) (* z (* 2.0 z))))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.8e+125) {
		tmp = t_1;
	} else if (z <= 4e+126) {
		tmp = x + ((y * 2.0) * (z / ((y * t) - (z * (2.0 * z)))));
	} else {
		tmp = t_1;
	}
	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 = x - (y / z)
    if (z <= (-1.8d+125)) then
        tmp = t_1
    else if (z <= 4d+126) then
        tmp = x + ((y * 2.0d0) * (z / ((y * t) - (z * (2.0d0 * z)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.8e+125) {
		tmp = t_1;
	} else if (z <= 4e+126) {
		tmp = x + ((y * 2.0) * (z / ((y * t) - (z * (2.0 * z)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1.8e+125:
		tmp = t_1
	elif z <= 4e+126:
		tmp = x + ((y * 2.0) * (z / ((y * t) - (z * (2.0 * z)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -1.8e+125)
		tmp = t_1;
	elseif (z <= 4e+126)
		tmp = Float64(x + Float64(Float64(y * 2.0) * Float64(z / Float64(Float64(y * t) - Float64(z * Float64(2.0 * z))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -1.8e+125)
		tmp = t_1;
	elseif (z <= 4e+126)
		tmp = x + ((y * 2.0) * (z / ((y * t) - (z * (2.0 * z)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e+125], t$95$1, If[LessEqual[z, 4e+126], N[(x + N[(N[(y * 2.0), $MachinePrecision] * N[(z / N[(N[(y * t), $MachinePrecision] - N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8000000000000002e125 or 3.9999999999999997e126 < z

   1. Initial program 62.2%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\frac{y}{z}\right)\right) \]

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      4. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 98.8%

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

   if -1.8000000000000002e125 < z < 3.9999999999999997e126

   1. Initial program 88.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(y \cdot 2\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \color{blue}{\left(y \cdot 2\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\right), \color{blue}{\left(y \cdot 2\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\left(z \cdot 2\right) \cdot z - y \cdot t\right)\right), \left(\color{blue}{y} \cdot 2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(\left(\left(z \cdot 2\right) \cdot z\right), \left(y \cdot t\right)\right)\right), \left(y \cdot 2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(\left(z \cdot \left(z \cdot 2\right)\right), \left(y \cdot t\right)\right)\right), \left(y \cdot 2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot 2\right)\right), \left(y \cdot t\right)\right)\right), \left(y \cdot 2\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(2 \cdot z\right)\right), \left(y \cdot t\right)\right)\right), \left(y \cdot 2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(2, z\right)\right), \left(y \cdot t\right)\right)\right), \left(y \cdot 2\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(2, z\right)\right), \mathsf{*.f64}\left(y, t\right)\right)\right), \left(y \cdot 2\right)\right)\right) \]

      11. *-lowering-*.f64 91.5%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(2, z\right)\right), \mathsf{*.f64}\left(y, t\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right)\right) \]

   4. Applied egg-rr 91.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+125}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+126}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+48}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1e-26) t_1 (if (<= z 8e+48) (- x (/ (* z -2.0) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1e-26) {
		tmp = t_1;
	} else if (z <= 8e+48) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = t_1;
	}
	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 = x - (y / z)
    if (z <= (-1d-26)) then
        tmp = t_1
    else if (z <= 8d+48) then
        tmp = x - ((z * (-2.0d0)) / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1e-26) {
		tmp = t_1;
	} else if (z <= 8e+48) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1e-26:
		tmp = t_1
	elif z <= 8e+48:
		tmp = x - ((z * -2.0) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -1e-26)
		tmp = t_1;
	elseif (z <= 8e+48)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -1e-26)
		tmp = t_1;
	elseif (z <= 8e+48)
		tmp = x - ((z * -2.0) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-26], t$95$1, If[LessEqual[z, 8e+48], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e-26 or 8.00000000000000035e48 < z

   1. Initial program 71.2%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\frac{y}{z}\right)\right) \]

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      4. /-lowering-/.f64 92.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 92.7%

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

   if -1e-26 < z < 8.00000000000000035e48

   1. Initial program 88.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(-2 \cdot \frac{z}{t}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{-2 \cdot z}{\color{blue}{t}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(-2 \cdot z\right), \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 87.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, z\right), t\right)\right) \]

   5. Simplified 87.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-26}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+48}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -130000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-84}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -130000000000.0) t_1 (if (<= z 1.8e-84) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -130000000000.0) {
		tmp = t_1;
	} else if (z <= 1.8e-84) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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 = x - (y / z)
    if (z <= (-130000000000.0d0)) then
        tmp = t_1
    else if (z <= 1.8d-84) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -130000000000.0) {
		tmp = t_1;
	} else if (z <= 1.8e-84) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -130000000000.0:
		tmp = t_1
	elif z <= 1.8e-84:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -130000000000.0)
		tmp = t_1;
	elseif (z <= 1.8e-84)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -130000000000.0)
		tmp = t_1;
	elseif (z <= 1.8e-84)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -130000000000.0], t$95$1, If[LessEqual[z, 1.8e-84], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e11 or 1.80000000000000002e-84 < z

   1. Initial program 73.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\frac{y}{z}\right)\right) \]

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      4. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 85.9%

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

   if -1.3e11 < z < 1.80000000000000002e-84

   1. Initial program 88.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 80.5%

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

Alternative 5: 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 79.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 75.8%

   \[\leadsto \color{blue}{x} \]
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 2024160 
(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)))))