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

Percentage Accurate: 82.3% → 94.5%

Time: 9.3s

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: 82.3% 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: 94.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 2, \frac{z}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t))) 5e+280)
   (fma (* y 2.0) (/ z (fma z (* z -2.0) (* y t))) x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y * 2.0) * z) / ((z * (2.0 * z)) - (y * t))) <= 5e+280) {
		tmp = fma((y * 2.0), (z / fma(z, (z * -2.0), (y * t))), x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(z * Float64(2.0 * z)) - Float64(y * t))) <= 5e+280)
		tmp = fma(Float64(y * 2.0), Float64(z / fma(z, Float64(z * -2.0), Float64(y * t))), x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+280], N[(N[(y * 2.0), $MachinePrecision] * N[(z / N[(z * N[(z * -2.0), $MachinePrecision] + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))) < 5.0000000000000002e280

   1. Initial program 97.6%

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

      1. sub-neg 97.6%

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

      2. +-commutative 97.6%

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

      3. associate-/l* 98.2%

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

      4. distribute-rgt-neg-in 98.2%

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

      5. fma-define 98.2%

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

   3. Simplified 98.2%

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

   if 5.0000000000000002e280 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.3%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in y around 0 84.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 2, \frac{z}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(2 \cdot z\right) - y \cdot t\\ \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{t\_1} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot \frac{z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z (* 2.0 z)) (* y t))))
   (if (<= (/ (* (* y 2.0) z) t_1) 5e+280)
     (- x (* (* y 2.0) (/ z t_1)))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * (2.0 * z)) - (y * t);
	double tmp;
	if ((((y * 2.0) * z) / t_1) <= 5e+280) {
		tmp = x - ((y * 2.0) * (z / 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)) - (y * t)
    if ((((y * 2.0d0) * z) / t_1) <= 5d+280) then
        tmp = x - ((y * 2.0d0) * (z / 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)) - (y * t);
	double tmp;
	if ((((y * 2.0) * z) / t_1) <= 5e+280) {
		tmp = x - ((y * 2.0) * (z / t_1));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * (2.0 * z)) - (y * t)
	tmp = 0
	if (((y * 2.0) * z) / t_1) <= 5e+280:
		tmp = x - ((y * 2.0) * (z / t_1))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * Float64(2.0 * z)) - Float64(y * t))
	tmp = 0.0
	if (Float64(Float64(Float64(y * 2.0) * z) / t_1) <= 5e+280)
		tmp = Float64(x - Float64(Float64(y * 2.0) * Float64(z / 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)) - (y * t);
	tmp = 0.0;
	if ((((y * 2.0) * z) / t_1) <= 5e+280)
		tmp = x - ((y * 2.0) * (z / t_1));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision], 5e+280], N[(x - N[(N[(y * 2.0), $MachinePrecision] * N[(z / t$95$1), $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))) < 5.0000000000000002e280

   1. Initial program 97.6%

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

   3. Simplified 98.2%

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

   if 5.0000000000000002e280 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.3%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in y around 0 84.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot \frac{z}{z \cdot \left(2 \cdot z\right) - y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.044 \lor \neg \left(z \leq 3 \cdot 10^{-14}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.044) (not (<= z 3e-14)))
   (- x (/ y z))
   (+ x (* z (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.044) || !(z <= 3e-14)) {
		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 <= (-0.044d0)) .or. (.not. (z <= 3d-14))) 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 <= -0.044) || !(z <= 3e-14)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -0.044) or not (z <= 3e-14):
		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 <= -0.044) || !(z <= 3e-14))
		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 <= -0.044) || ~((z <= 3e-14)))
		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, -0.044], N[Not[LessEqual[z, 3e-14]], $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 < -0.043999999999999997 or 2.9999999999999998e-14 < z

   1. Initial program 74.6%

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

   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 y around 0 93.8%

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

   if -0.043999999999999997 < z < 2.9999999999999998e-14

   1. Initial program 90.3%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around inf 90.2%

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

      1. associate-*r/ 90.2%

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

      2. *-commutative 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in x around 0 90.2%

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

      1. cancel-sign-sub-inv 90.2%

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

      2. metadata-eval 90.2%

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

      3. associate-*r/ 90.2%

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

      4. *-commutative 90.2%

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

      5. associate-/l* 90.2%

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

   10. Simplified 90.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.044 \lor \neg \left(z \leq 3 \cdot 10^{-14}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.029 \lor \neg \left(z \leq 4.8 \cdot 10^{-16}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.029) (not (<= z 4.8e-16)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.029) || !(z <= 4.8e-16)) {
		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 <= (-0.029d0)) .or. (.not. (z <= 4.8d-16))) 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 <= -0.029) || !(z <= 4.8e-16)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -0.029) or not (z <= 4.8e-16):
		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 <= -0.029) || !(z <= 4.8e-16))
		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 <= -0.029) || ~((z <= 4.8e-16)))
		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, -0.029], N[Not[LessEqual[z, 4.8e-16]], $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 < -0.0290000000000000015 or 4.8000000000000001e-16 < z

   1. Initial program 74.6%

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

   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 y around 0 93.8%

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

   if -0.0290000000000000015 < z < 4.8000000000000001e-16

   1. Initial program 90.3%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around inf 90.2%

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

      1. associate-*r/ 90.2%

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

      2. *-commutative 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.029 \lor \neg \left(z \leq 4.8 \cdot 10^{-16}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1100000 \lor \neg \left(z \leq 2.3 \cdot 10^{-15}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1100000.0) (not (<= z 2.3e-15))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1100000.0) || !(z <= 2.3e-15)) {
		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 <= (-1100000.0d0)) .or. (.not. (z <= 2.3d-15))) 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 <= -1100000.0) || !(z <= 2.3e-15)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1100000.0) or not (z <= 2.3e-15):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1100000.0) || !(z <= 2.3e-15))
		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 <= -1100000.0) || ~((z <= 2.3e-15)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1100000.0], N[Not[LessEqual[z, 2.3e-15]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e6 or 2.2999999999999999e-15 < z

   1. Initial program 74.1%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y around 0 93.6%

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

   if -1.1e6 < z < 2.2999999999999999e-15

   1. Initial program 90.6%

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

   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 x around inf 73.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1100000 \lor \neg \left(z \leq 2.3 \cdot 10^{-15}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% 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 82.0%

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

3. Simplified 89.7%

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

5. Taylor expanded in x around inf 75.6%

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

6. Final simplification 75.6%

   \[\leadsto x \]
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 2024059 
(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)))))