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

Percentage Accurate: 81.8% → 89.4%

Time: 9.9s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+32} \lor \neg \left(z \leq 1.42 \cdot 10^{-15}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.16e+32) (not (<= z 1.42e-15)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.16e+32) || !(z <= 1.42e-15)) {
		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 <= (-1.16d+32)) .or. (.not. (z <= 1.42d-15))) 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 <= -1.16e+32) || !(z <= 1.42e-15)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.16e+32) or not (z <= 1.42e-15):
		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 <= -1.16e+32) || !(z <= 1.42e-15))
		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 <= -1.16e+32) || ~((z <= 1.42e-15)))
		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, -1.16e+32], N[Not[LessEqual[z, 1.42e-15]], $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 < -1.16e32 or 1.42e-15 < z

   1. Initial program 70.9%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in y around 0 94.0%

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

   if -1.16e32 < z < 1.42e-15

   1. Initial program 89.2%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 94.7%

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

      1. *-commutative 94.7%

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

      2. associate-*l/ 94.7%

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

   7. Simplified 94.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+32} \lor \neg \left(z \leq 1.42 \cdot 10^{-15}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{t\_1 - y \cdot t} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* 2.0 z))))
   (if (<= (- x (/ (* (* y 2.0) z) (- t_1 (* y t)))) 2e+303)
     (+ 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 ((x - (((y * 2.0) * z) / (t_1 - (y * t)))) <= 2e+303) {
		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 ((x - (((y * 2.0d0) * z) / (t_1 - (y * t)))) <= 2d+303) 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 ((x - (((y * 2.0) * z) / (t_1 - (y * t)))) <= 2e+303) {
		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 (x - (((y * 2.0) * z) / (t_1 - (y * t)))) <= 2e+303:
		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(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(t_1 - Float64(y * t)))) <= 2e+303)
		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 ((x - (((y * 2.0) * z) / (t_1 - (y * t)))) <= 2e+303)
		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[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(t$95$1 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], 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 x (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))) < 2e303

   1. Initial program 95.5%

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

   3. Simplified 96.1%

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

   if 2e303 < (-.f64 x (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))))

   1. Initial program 5.0%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in y around 0 83.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+32} \lor \neg \left(z \leq 4.5 \cdot 10^{-20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.16e+32) (not (<= z 4.5e-20)))
   (- x (/ y z))
   (+ x (* z (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.16e+32) || !(z <= 4.5e-20)) {
		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 <= (-1.16d+32)) .or. (.not. (z <= 4.5d-20))) 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 <= -1.16e+32) || !(z <= 4.5e-20)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.16e+32) or not (z <= 4.5e-20):
		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 <= -1.16e+32) || !(z <= 4.5e-20))
		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 <= -1.16e+32) || ~((z <= 4.5e-20)))
		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, -1.16e+32], N[Not[LessEqual[z, 4.5e-20]], $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 < -1.16e32 or 4.5000000000000001e-20 < z

   1. Initial program 70.9%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in y around 0 94.0%

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

   if -1.16e32 < z < 4.5000000000000001e-20

   1. Initial program 89.2%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in z around 0 82.8%

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

      1. associate-*r/ 82.8%

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

      2. *-commutative 82.8%

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

      3. associate-/r* 89.1%

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

   7. Simplified 89.1%

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

      1. fma-undefine 89.2%

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

      2. associate-/l/ 82.8%

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

      3. *-commutative 82.8%

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

      4. associate-/l* 82.8%

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

   9. Applied egg-rr 82.8%

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

       1. associate-*r* 82.8%

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

       2. clear-num 82.8%

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

       3. un-div-inv 82.8%

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

       4. *-commutative 82.8%

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

       5. associate-/l* 82.2%

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

   11. Applied egg-rr 82.2%

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

   12. Taylor expanded in y around 0 94.7%

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

       1. associate-*r/ 94.7%

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

       2. remove-double-neg 94.7%

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

       3. neg-mul-1 94.7%

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

       4. distribute-lft-neg-in 94.7%

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

       5. metadata-eval 94.7%

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

       6. *-inverses 94.7%

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

       7. associate-*l/ 91.9%

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

       8. associate-*r/ 85.4%

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

       9. associate-*l* 85.4%

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

       10. *-commutative 85.4%

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

       11. associate-*l/ 78.3%

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

       12. associate-*r/ 78.2%

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

       13. *-commutative 78.2%

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

       14. associate-*l/ 78.3%

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

       15. associate-*r/ 80.3%

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

       16. associate-*l/ 94.6%

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

       17. *-inverses 94.6%

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

       18. associate-*r/ 94.6%

           \[\leadsto z \cdot \color{blue}{\frac{1 \cdot 2}{t}} + x \]

       19. metadata-eval 94.6%

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

   14. Simplified 94.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+32} \lor \neg \left(z \leq 4.5 \cdot 10^{-20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+23} \lor \neg \left(z \leq 5.4 \cdot 10^{-14}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.7e+23) (not (<= z 5.4e-14))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.7e+23) || !(z <= 5.4e-14)) {
		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 <= (-3.7d+23)) .or. (.not. (z <= 5.4d-14))) 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 <= -3.7e+23) || !(z <= 5.4e-14)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.7e+23) or not (z <= 5.4e-14):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.7e+23) || !(z <= 5.4e-14))
		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 <= -3.7e+23) || ~((z <= 5.4e-14)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.7e+23], N[Not[LessEqual[z, 5.4e-14]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.70000000000000013e23 or 5.3999999999999997e-14 < z

   1. Initial program 70.8%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in y around 0 91.7%

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

   if -3.70000000000000013e23 < z < 5.3999999999999997e-14

   1. Initial program 89.6%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around inf 73.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+23} \lor \neg \left(z \leq 5.4 \cdot 10^{-14}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% 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 81.0%

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

3. Simplified 88.1%

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

5. Taylor expanded in x around inf 71.5%

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

6. Final simplification 71.5%

   \[\leadsto x \]
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 2024039 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))