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

Percentage Accurate: 82.1% → 94.9%

Time: 9.7s

Alternatives: 7

Speedup: 0.6×

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: 82.1% 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: 94.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, 2 \cdot \frac{-0.5}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+68}:\\ \;\;\;\;x + \frac{\frac{2}{y} \cdot \left(y \cdot z\right)}{t - \frac{z \cdot \left(2 \cdot z\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.1e+116)
   (fma y (* 2.0 (/ -0.5 z)) x)
   (if (<= z 1.8e+68)
     (+ x (/ (* (/ 2.0 y) (* y z)) (- t (/ (* z (* 2.0 z)) y))))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.1e+116) {
		tmp = fma(y, (2.0 * (-0.5 / z)), x);
	} else if (z <= 1.8e+68) {
		tmp = x + (((2.0 / y) * (y * z)) / (t - ((z * (2.0 * z)) / y)));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.1e+116)
		tmp = fma(y, Float64(2.0 * Float64(-0.5 / z)), x);
	elseif (z <= 1.8e+68)
		tmp = Float64(x + Float64(Float64(Float64(2.0 / y) * Float64(y * z)) / Float64(t - Float64(Float64(z * Float64(2.0 * z)) / y))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6.1e+116], N[(y * N[(2.0 * N[(-0.5 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.8e+68], N[(x + N[(N[(N[(2.0 / y), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.10000000000000018e116

   1. Initial program 77.0%

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

      1. sub-neg 77.0%

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

      2. +-commutative 77.0%

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

      3. associate-/l* 90.4%

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

      4. associate-*l* 90.4%

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

      5. distribute-rgt-neg-in 90.4%

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

      6. fma-define 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in z around inf 94.5%

      \[\leadsto \mathsf{fma}\left(y, 2 \cdot \color{blue}{\frac{-0.5}{z}}, x\right) \]

   if -6.10000000000000018e116 < z < 1.7999999999999999e68

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf 92.9%

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

      1. associate-*r/ 92.9%

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

   5. Simplified 92.9%

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

      1. unpow2 92.9%

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

      2. associate-*r* 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*r* 92.9%

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

      3. times-frac 92.8%

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

      4. *-commutative 92.8%

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

      5. associate-*l* 92.8%

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

      6. *-un-lft-identity 92.8%

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

      7. times-frac 92.8%

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

      8. metadata-eval 92.8%

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

      9. pow2 92.8%

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

   9. Applied egg-rr 92.8%

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

       1. associate-*r/ 96.2%

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

       2. associate-*r/ 96.2%

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

   11. Simplified 96.2%

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

       1. unpow2 92.9%

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

       2. associate-*r* 92.9%

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

   13. Applied egg-rr 96.2%

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

   if 1.7999999999999999e68 < z

   1. Initial program 67.5%

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

   3. Taylor expanded in y around 0 96.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, 2 \cdot \frac{-0.5}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+68}:\\ \;\;\;\;x + \frac{\frac{2}{y} \cdot \left(y \cdot z\right)}{t - \frac{z \cdot \left(2 \cdot z\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(y, 2 \cdot \frac{z}{\mathsf{fma}\left(-2, z \cdot z, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t))) 2e+170)
   (fma y (* 2.0 (/ z (fma -2.0 (* z z) (* 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))) <= 2e+170) {
		tmp = fma(y, (2.0 * (z / fma(-2.0, (z * z), (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))) <= 2e+170)
		tmp = fma(y, Float64(2.0 * Float64(z / fma(-2.0, Float64(z * z), 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], 2e+170], N[(y * N[(2.0 * N[(z / N[(-2.0 * N[(z * z), $MachinePrecision] + N[(y * t), $MachinePrecision]), $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))) < 2.00000000000000007e170

   1. Initial program 95.7%

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

      1. sub-neg 95.7%

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

      2. +-commutative 95.7%

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

      3. associate-/l* 97.0%

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

      4. associate-*l* 97.0%

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

      5. distribute-rgt-neg-in 97.0%

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

      6. fma-define 97.0%

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

   3. Simplified 97.0%

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

   if 2.00000000000000007e170 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.6%

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

   3. Taylor expanded in y around 0 82.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(y, 2 \cdot \frac{z}{\mathsf{fma}\left(-2, z \cdot z, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot 2\right) \cdot z\\ t_2 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;\frac{t\_1}{t\_2 - y \cdot t} \leq 2 \cdot 10^{+170}:\\ \;\;\;\;x + \frac{t\_1}{y \cdot t - t\_2}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* y 2.0) z)) (t_2 (* z (* 2.0 z))))
   (if (<= (/ t_1 (- t_2 (* y t))) 2e+170)
     (+ x (/ t_1 (- (* y t) t_2)))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * 2.0) * z;
	double t_2 = z * (2.0 * z);
	double tmp;
	if ((t_1 / (t_2 - (y * t))) <= 2e+170) {
		tmp = x + (t_1 / ((y * t) - t_2));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (y * 2.0d0) * z
    t_2 = z * (2.0d0 * z)
    if ((t_1 / (t_2 - (y * t))) <= 2d+170) then
        tmp = x + (t_1 / ((y * t) - t_2))
    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 = (y * 2.0) * z;
	double t_2 = z * (2.0 * z);
	double tmp;
	if ((t_1 / (t_2 - (y * t))) <= 2e+170) {
		tmp = x + (t_1 / ((y * t) - t_2));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * 2.0) * z
	t_2 = z * (2.0 * z)
	tmp = 0
	if (t_1 / (t_2 - (y * t))) <= 2e+170:
		tmp = x + (t_1 / ((y * t) - t_2))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * 2.0) * z)
	t_2 = Float64(z * Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t_1 / Float64(t_2 - Float64(y * t))) <= 2e+170)
		tmp = Float64(x + Float64(t_1 / Float64(Float64(y * t) - t_2)));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * 2.0) * z;
	t_2 = z * (2.0 * z);
	tmp = 0.0;
	if ((t_1 / (t_2 - (y * t))) <= 2e+170)
		tmp = x + (t_1 / ((y * t) - t_2));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(t$95$2 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+170], N[(x + N[(t$95$1 / N[(N[(y * t), $MachinePrecision] - t$95$2), $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))) < 2.00000000000000007e170

   1. Initial program 95.7%

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

   if 2.00000000000000007e170 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.6%

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

   3. Taylor expanded in y around 0 82.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 2 \cdot 10^{+170}:\\ \;\;\;\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+115} \lor \neg \left(z \leq 2.9 \cdot 10^{+69}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{2}{y} \cdot \left(y \cdot z\right)}{t - \frac{z \cdot \left(2 \cdot z\right)}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.5e+115) (not (<= z 2.9e+69)))
   (- x (/ y z))
   (+ x (/ (* (/ 2.0 y) (* y z)) (- t (/ (* z (* 2.0 z)) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e+115) || !(z <= 2.9e+69)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (((2.0 / y) * (y * z)) / (t - ((z * (2.0 * z)) / y)));
	}
	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 <= (-2.5d+115)) .or. (.not. (z <= 2.9d+69))) then
        tmp = x - (y / z)
    else
        tmp = x + (((2.0d0 / y) * (y * z)) / (t - ((z * (2.0d0 * z)) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e+115) || !(z <= 2.9e+69)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (((2.0 / y) * (y * z)) / (t - ((z * (2.0 * z)) / y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.5e+115) or not (z <= 2.9e+69):
		tmp = x - (y / z)
	else:
		tmp = x + (((2.0 / y) * (y * z)) / (t - ((z * (2.0 * z)) / y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.5e+115) || !(z <= 2.9e+69))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x + Float64(Float64(Float64(2.0 / y) * Float64(y * z)) / Float64(t - Float64(Float64(z * Float64(2.0 * z)) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.5e+115) || ~((z <= 2.9e+69)))
		tmp = x - (y / z);
	else
		tmp = x + (((2.0 / y) * (y * z)) / (t - ((z * (2.0 * z)) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.5e+115], N[Not[LessEqual[z, 2.9e+69]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(2.0 / y), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.50000000000000004e115 or 2.8999999999999998e69 < z

   1. Initial program 70.7%

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

   3. Taylor expanded in y around 0 95.9%

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

   if -2.50000000000000004e115 < z < 2.8999999999999998e69

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf 92.9%

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

      1. associate-*r/ 92.9%

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

   5. Simplified 92.9%

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

      1. unpow2 92.9%

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

      2. associate-*r* 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*r* 92.9%

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

      3. times-frac 92.8%

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

      4. *-commutative 92.8%

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

      5. associate-*l* 92.8%

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

      6. *-un-lft-identity 92.8%

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

      7. times-frac 92.8%

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

      8. metadata-eval 92.8%

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

      9. pow2 92.8%

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

   9. Applied egg-rr 92.8%

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

       1. associate-*r/ 96.2%

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

       2. associate-*r/ 96.2%

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

   11. Simplified 96.2%

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

       1. unpow2 92.9%

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

       2. associate-*r* 92.9%

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

   13. Applied egg-rr 96.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+115} \lor \neg \left(z \leq 2.9 \cdot 10^{+69}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{2}{y} \cdot \left(y \cdot z\right)}{t - \frac{z \cdot \left(2 \cdot z\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ t_2 := x - -2 \cdot \frac{z}{t}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-73}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))) (t_2 (- x (* -2.0 (/ z t)))))
   (if (<= z -8.5e+17)
     t_1
     (if (<= z 1.4e-73)
       t_2
       (if (<= z 2.4e+28) x (if (<= z 1.05e+60) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double t_2 = x - (-2.0 * (z / t));
	double tmp;
	if (z <= -8.5e+17) {
		tmp = t_1;
	} else if (z <= 1.4e-73) {
		tmp = t_2;
	} else if (z <= 2.4e+28) {
		tmp = x;
	} else if (z <= 1.05e+60) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = x - (y / z)
    t_2 = x - ((-2.0d0) * (z / t))
    if (z <= (-8.5d+17)) then
        tmp = t_1
    else if (z <= 1.4d-73) then
        tmp = t_2
    else if (z <= 2.4d+28) then
        tmp = x
    else if (z <= 1.05d+60) then
        tmp = t_2
    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 t_2 = x - (-2.0 * (z / t));
	double tmp;
	if (z <= -8.5e+17) {
		tmp = t_1;
	} else if (z <= 1.4e-73) {
		tmp = t_2;
	} else if (z <= 2.4e+28) {
		tmp = x;
	} else if (z <= 1.05e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	t_2 = x - (-2.0 * (z / t))
	tmp = 0
	if z <= -8.5e+17:
		tmp = t_1
	elif z <= 1.4e-73:
		tmp = t_2
	elif z <= 2.4e+28:
		tmp = x
	elif z <= 1.05e+60:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	t_2 = Float64(x - Float64(-2.0 * Float64(z / t)))
	tmp = 0.0
	if (z <= -8.5e+17)
		tmp = t_1;
	elseif (z <= 1.4e-73)
		tmp = t_2;
	elseif (z <= 2.4e+28)
		tmp = x;
	elseif (z <= 1.05e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	t_2 = x - (-2.0 * (z / t));
	tmp = 0.0;
	if (z <= -8.5e+17)
		tmp = t_1;
	elseif (z <= 1.4e-73)
		tmp = t_2;
	elseif (z <= 2.4e+28)
		tmp = x;
	elseif (z <= 1.05e+60)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x - N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.5e+17], t$95$1, If[LessEqual[z, 1.4e-73], t$95$2, If[LessEqual[z, 2.4e+28], x, If[LessEqual[z, 1.05e+60], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.5e17 or 1.0500000000000001e60 < z

   1. Initial program 74.1%

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

   3. Taylor expanded in y around 0 93.1%

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

   if -8.5e17 < z < 1.40000000000000006e-73 or 2.39999999999999981e28 < z < 1.0500000000000001e60

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf 93.4%

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

   if 1.40000000000000006e-73 < z < 2.39999999999999981e28

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf 87.5%

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

Alternative 6: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+23} \lor \neg \left(z \leq 1.35 \cdot 10^{+60}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e+23) (not (<= z 1.35e+60))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+23) || !(z <= 1.35e+60)) {
		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 <= (-1.25d+23)) .or. (.not. (z <= 1.35d+60))) 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 <= -1.25e+23) || !(z <= 1.35e+60)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e+23) or not (z <= 1.35e+60):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.25e+23) || !(z <= 1.35e+60))
		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 <= -1.25e+23) || ~((z <= 1.35e+60)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.25e+23], N[Not[LessEqual[z, 1.35e+60]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e23 or 1.35e60 < z

   1. Initial program 73.6%

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

   3. Taylor expanded in y around 0 93.9%

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

   if -1.25e23 < z < 1.35e60

   1. Initial program 93.5%

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

   3. Taylor expanded in x around inf 79.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+23} \lor \neg \left(z \leq 1.35 \cdot 10^{+60}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% 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 85.3%

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

3. Taylor expanded in x around inf 80.3%

   \[\leadsto \color{blue}{x} \]
4. 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 2024096 
(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)))))