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

Percentage Accurate: 82.0% → 93.1%

Time: 10.1s

Alternatives: 6

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: 82.0% 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: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (* y 2.0) z) (- (* y t) (* z (* 2.0 z)))))))
   (if (<= t_1 5e+289) t_1 (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((y * 2.0) * z) / ((y * t) - (z * (2.0 * z))));
	double tmp;
	if (t_1 <= 5e+289) {
		tmp = 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 = x + (((y * 2.0d0) * z) / ((y * t) - (z * (2.0d0 * z))))
    if (t_1 <= 5d+289) then
        tmp = 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 = x + (((y * 2.0) * z) / ((y * t) - (z * (2.0 * z))));
	double tmp;
	if (t_1 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (((y * 2.0) * z) / ((y * t) - (z * (2.0 * z))))
	tmp = 0
	if t_1 <= 5e+289:
		tmp = t_1
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(y * t) - Float64(z * Float64(2.0 * z)))))
	tmp = 0.0
	if (t_1 <= 5e+289)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (((y * 2.0) * z) / ((y * t) - (z * (2.0 * z))));
	tmp = 0.0;
	if (t_1 <= 5e+289)
		tmp = t_1;
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+289], t$95$1, N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 5.00000000000000031e289

   1. Initial program 97.4%

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

   if 5.00000000000000031e289 < (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))))

   1. Initial program 6.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x (/ (* (* y 2.0) z) (- (* y t) (* z (* 2.0 z))))) 5e+289)
   (fma y (* z (/ 2.0 (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 ((x + (((y * 2.0) * z) / ((y * t) - (z * (2.0 * z))))) <= 5e+289) {
		tmp = fma(y, (z * (2.0 / 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(x + Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(y * t) - Float64(z * Float64(2.0 * z))))) <= 5e+289)
		tmp = fma(y, Float64(z * Float64(2.0 / 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[(x + N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+289], N[(y * N[(z * N[(2.0 / N[(z * N[(z * -2.0), $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 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 5.00000000000000031e289

   1. Initial program 95.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(z \cdot -2\right) \cdot \frac{-1}{\mathsf{fma}\left(z \cdot z, -2, y \cdot t\right)}, x\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 96.4

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 96.4

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

   6. Applied rewrites 96.4%

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

   if 5.00000000000000031e289 < (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))))

   1. Initial program 9.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. 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 2024223 
(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)))))