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

Percentage Accurate: 81.8% → 95.1%

Time: 9.0s

Alternatives: 4

Speedup: 1.4×

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: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(z + z, \frac{y}{\mathsf{fma}\left(-2, z \cdot z, t \cdot y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) 2e+208)
   (fma (+ z z) (/ y (fma -2.0 (* z z) (* t y))) 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+208) {
		tmp = fma((z + z), (y / fma(-2.0, (z * z), (t * y))), 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(Float64(z * 2.0) * z) - Float64(y * t))) <= 2e+208)
		tmp = fma(Float64(z + z), Float64(y / fma(-2.0, Float64(z * z), Float64(t * y))), 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[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+208], N[(N[(z + z), $MachinePrecision] * N[(y / N[(-2.0 * N[(z * z), $MachinePrecision] + N[(t * y), $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))) < 2e208

   1. Initial program 94.8%

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

   4. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 95.1

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

   6. Applied rewrites 95.1%

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

   if 2e208 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 2.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

Alternative 2: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+21} \lor \neg \left(z \leq 5.1 \cdot 10^{-17}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.8e+21) (not (<= z 5.1e-17)))
   (- x (/ y z))
   (fma (/ z t) 2.0 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.8e+21) || !(z <= 5.1e-17)) {
		tmp = x - (y / z);
	} else {
		tmp = fma((z / t), 2.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.8e+21) || !(z <= 5.1e-17))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = fma(Float64(z / t), 2.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.8e+21], N[Not[LessEqual[z, 5.1e-17]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8e21 or 5.1000000000000003e-17 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -8.8e21 < z < 5.1000000000000003e-17

   1. Initial program 87.6%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+21} \lor \neg \left(z \leq 5.1 \cdot 10^{-17}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+21} \lor \neg \left(z \leq 5.1 \cdot 10^{-17}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.8e+21) (not (<= z 5.1e-17)))
   (- x (/ y z))
   (fma z (/ 2.0 t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.8e+21) || !(z <= 5.1e-17)) {
		tmp = x - (y / z);
	} else {
		tmp = fma(z, (2.0 / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.8e+21) || !(z <= 5.1e-17))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = fma(z, Float64(2.0 / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.8e+21], N[Not[LessEqual[z, 5.1e-17]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8e21 or 5.1000000000000003e-17 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -8.8e21 < z < 5.1000000000000003e-17

   1. Initial program 87.6%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 90.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+21} \lor \neg \left(z \leq 5.1 \cdot 10^{-17}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (y / 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 - (y / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 65.2

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

5. Applied rewrites 65.2%

   \[\leadsto x - \color{blue}{\frac{y}{z}} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× 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 2024337 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (/ 1 (- (/ z y) (/ (/ t 2) z)))))

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