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

Average Error: 12.0 → 3.7

Time: 8.5s

Precision: binary64

Cost: 1484

Math

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

↓

\[\begin{array}{l} t_1 := x - z \cdot \frac{y \cdot 2}{z \cdot \left(2 \cdot z\right) - y \cdot t}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+172}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-103}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (* z (/ (* y 2.0) (- (* z (* 2.0 z)) (* y t)))))))
   (if (<= z -1.8e+172)
     (- x (/ y z))
     (if (<= z -1e-151) t_1 (if (<= z 3.6e-103) (- x (* -2.0 (/ z t))) t_1)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = x - (z * ((y * 2.0) / ((z * (2.0 * z)) - (y * t))));
	double tmp;
	if (z <= -1.8e+172) {
		tmp = x - (y / z);
	} else if (z <= -1e-151) {
		tmp = t_1;
	} else if (z <= 3.6e-103) {
		tmp = x - (-2.0 * (z / 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
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

↓

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 - (z * ((y * 2.0d0) / ((z * (2.0d0 * z)) - (y * t))))
    if (z <= (-1.8d+172)) then
        tmp = x - (y / z)
    else if (z <= (-1d-151)) then
        tmp = t_1
    else if (z <= 3.6d-103) then
        tmp = x - ((-2.0d0) * (z / t))
    else
        tmp = t_1
    end if
    code = tmp
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)));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (z * ((y * 2.0) / ((z * (2.0 * z)) - (y * t))));
	double tmp;
	if (z <= -1.8e+172) {
		tmp = x - (y / z);
	} else if (z <= -1e-151) {
		tmp = t_1;
	} else if (z <= 3.6e-103) {
		tmp = x - (-2.0 * (z / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = x - (z * ((y * 2.0) / ((z * (2.0 * z)) - (y * t))))
	tmp = 0
	if z <= -1.8e+172:
		tmp = x - (y / z)
	elif z <= -1e-151:
		tmp = t_1
	elif z <= 3.6e-103:
		tmp = x - (-2.0 * (z / t))
	else:
		tmp = t_1
	return tmp

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

↓

function code(x, y, z, t)
	t_1 = Float64(x - Float64(z * Float64(Float64(y * 2.0) / Float64(Float64(z * Float64(2.0 * z)) - Float64(y * t)))))
	tmp = 0.0
	if (z <= -1.8e+172)
		tmp = Float64(x - Float64(y / z));
	elseif (z <= -1e-151)
		tmp = t_1;
	elseif (z <= 3.6e-103)
		tmp = Float64(x - Float64(-2.0 * Float64(z / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x - (z * ((y * 2.0) / ((z * (2.0 * z)) - (y * t))));
	tmp = 0.0;
	if (z <= -1.8e+172)
		tmp = x - (y / z);
	elseif (z <= -1e-151)
		tmp = t_1;
	elseif (z <= 3.6e-103)
		tmp = x - (-2.0 * (z / t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(z * N[(N[(y * 2.0), $MachinePrecision] / N[(N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e+172], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1e-151], t$95$1, If[LessEqual[z, 3.6e-103], N[(x - N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	12.0
Target	0.1
Herbie	3.7

\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

Derivation

1. Split input into 3 regimes

2. if z < -1.79999999999999987e172

   1. Initial program 27.1

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

   2. Simplified 14.4

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

      [Start] 27.1	\[ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
      rational.json-simplify-50 [=>] 27.1	\[ x - \color{blue}{\frac{-\left(y \cdot 2\right) \cdot z}{y \cdot t - \left(z \cdot 2\right) \cdot z}} \]
      rational.json-simplify-10 [=>] 27.1	\[ x - \frac{\color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{-1}}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      rational.json-simplify-47 [=>] 27.1	\[ x - \color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)}} \]
      rational.json-simplify-2 [=>] 27.1	\[ x - \frac{\color{blue}{z \cdot \left(y \cdot 2\right)}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-43 [=>] 27.1	\[ x - \frac{\color{blue}{y \cdot \left(2 \cdot z\right)}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-2 [<=] 27.1	\[ x - \frac{y \cdot \color{blue}{\left(z \cdot 2\right)}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-2 [=>] 27.1	\[ x - \frac{\color{blue}{\left(z \cdot 2\right) \cdot y}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-49 [=>] 14.4	\[ x - \color{blue}{y \cdot \frac{z \cdot 2}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)}} \]
      rational.json-simplify-46 [=>] 14.4	\[ x - y \cdot \color{blue}{\frac{\frac{z \cdot 2}{-1}}{y \cdot t - \left(z \cdot 2\right) \cdot z}} \]
      rational.json-simplify-2 [=>] 14.4	\[ x - y \cdot \frac{\frac{\color{blue}{2 \cdot z}}{-1}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      rational.json-simplify-49 [=>] 14.4	\[ x - y \cdot \frac{\color{blue}{z \cdot \frac{2}{-1}}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      metadata-eval [=>] 14.4	\[ x - y \cdot \frac{z \cdot \color{blue}{-2}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      rational.json-simplify-2 [=>] 14.4	\[ x - y \cdot \frac{z \cdot -2}{y \cdot t - \color{blue}{z \cdot \left(z \cdot 2\right)}} \]
      rational.json-simplify-43 [<=] 14.4	\[ x - y \cdot \frac{z \cdot -2}{y \cdot t - \color{blue}{2 \cdot \left(z \cdot z\right)}} \]

   3. Taylor expanded in y around 0 1.3

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

   if -1.79999999999999987e172 < z < -9.9999999999999994e-152 or 3.5999999999999998e-103 < z

   1. Initial program 10.9

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

   2. Simplified 4.9

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

      [Start] 10.9	\[ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
      rational.json-simplify-49 [=>] 4.9	\[ x - \color{blue}{z \cdot \frac{y \cdot 2}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]
      rational.json-simplify-50 [=>] 4.9	\[ x - z \cdot \color{blue}{\frac{-y \cdot 2}{y \cdot t - \left(z \cdot 2\right) \cdot z}} \]
      rational.json-simplify-5 [<=] 4.9	\[ x - z \cdot \frac{-y \cdot 2}{\color{blue}{\left(y \cdot t - \left(z \cdot 2\right) \cdot z\right) - 0}} \]
      rational.json-simplify-50 [<=] 4.9	\[ x - z \cdot \color{blue}{\frac{y \cdot 2}{0 - \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)}} \]
      rational.json-simplify-45 [=>] 4.9	\[ x - z \cdot \frac{y \cdot 2}{\color{blue}{\left(z \cdot 2\right) \cdot z - \left(y \cdot t - 0\right)}} \]
      rational.json-simplify-2 [=>] 4.9	\[ x - z \cdot \frac{y \cdot 2}{\color{blue}{z \cdot \left(z \cdot 2\right)} - \left(y \cdot t - 0\right)} \]
      rational.json-simplify-2 [=>] 4.9	\[ x - z \cdot \frac{y \cdot 2}{z \cdot \color{blue}{\left(2 \cdot z\right)} - \left(y \cdot t - 0\right)} \]
      rational.json-simplify-5 [=>] 4.9	\[ x - z \cdot \frac{y \cdot 2}{z \cdot \left(2 \cdot z\right) - \color{blue}{y \cdot t}} \]

   if -9.9999999999999994e-152 < z < 3.5999999999999998e-103

   1. Initial program 8.3

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

   2. Simplified 7.5

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

      [Start] 8.3	\[ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
      rational.json-simplify-50 [=>] 8.3	\[ x - \color{blue}{\frac{-\left(y \cdot 2\right) \cdot z}{y \cdot t - \left(z \cdot 2\right) \cdot z}} \]
      rational.json-simplify-10 [=>] 8.3	\[ x - \frac{\color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{-1}}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      rational.json-simplify-47 [=>] 8.3	\[ x - \color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)}} \]
      rational.json-simplify-2 [=>] 8.3	\[ x - \frac{\color{blue}{z \cdot \left(y \cdot 2\right)}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-43 [=>] 8.2	\[ x - \frac{\color{blue}{y \cdot \left(2 \cdot z\right)}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-2 [<=] 8.2	\[ x - \frac{y \cdot \color{blue}{\left(z \cdot 2\right)}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-2 [=>] 8.2	\[ x - \frac{\color{blue}{\left(z \cdot 2\right) \cdot y}}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)} \]
      rational.json-simplify-49 [=>] 7.5	\[ x - \color{blue}{y \cdot \frac{z \cdot 2}{-1 \cdot \left(y \cdot t - \left(z \cdot 2\right) \cdot z\right)}} \]
      rational.json-simplify-46 [=>] 7.5	\[ x - y \cdot \color{blue}{\frac{\frac{z \cdot 2}{-1}}{y \cdot t - \left(z \cdot 2\right) \cdot z}} \]
      rational.json-simplify-2 [=>] 7.5	\[ x - y \cdot \frac{\frac{\color{blue}{2 \cdot z}}{-1}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      rational.json-simplify-49 [=>] 7.5	\[ x - y \cdot \frac{\color{blue}{z \cdot \frac{2}{-1}}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      metadata-eval [=>] 7.5	\[ x - y \cdot \frac{z \cdot \color{blue}{-2}}{y \cdot t - \left(z \cdot 2\right) \cdot z} \]
      rational.json-simplify-2 [=>] 7.5	\[ x - y \cdot \frac{z \cdot -2}{y \cdot t - \color{blue}{z \cdot \left(z \cdot 2\right)}} \]
      rational.json-simplify-43 [<=] 7.5	\[ x - y \cdot \frac{z \cdot -2}{y \cdot t - \color{blue}{2 \cdot \left(z \cdot z\right)}} \]

   3. Taylor expanded in y around inf 2.3

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+172}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-151}:\\ \;\;\;\;x - z \cdot \frac{y \cdot 2}{z \cdot \left(2 \cdot z\right) - y \cdot t}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-103}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y \cdot 2}{z \cdot \left(2 \cdot z\right) - y \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.7
Cost	1484

\[\begin{array}{l} t_1 := x - y \cdot \frac{z \cdot -2}{y \cdot t - 2 \cdot \left(z \cdot z\right)}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+172}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-103}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	7.3
Cost	712

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-69}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.9
Cost	584

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023075 
(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)))))