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

Percentage Accurate: 82.2% → 93.9%

Time: 10.2s

Alternatives: 7

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: 82.2% 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: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 4 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(z \cdot z, -2, y \cdot t\right)}, y \cdot 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t)))) 4e+276)
   (fma (/ z (fma (* z z) -2.0 (* y t))) (* y 2.0) x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - (((y * 2.0) * z) / ((z * (2.0 * z)) - (y * t)))) <= 4e+276) {
		tmp = fma((z / fma((z * z), -2.0, (y * t))), (y * 2.0), 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(z * Float64(2.0 * z)) - Float64(y * t)))) <= 4e+276)
		tmp = fma(Float64(z / fma(Float64(z * z), -2.0, Float64(y * t))), Float64(y * 2.0), 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[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+276], N[(N[(z / N[(N[(z * z), $MachinePrecision] * -2.0 + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 2.0), $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)))) < 4.0000000000000002e276

   1. Initial program 94.7%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.7%

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

   if 4.0000000000000002e276 < (-.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 4.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 4 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(z \cdot z, -2, y \cdot t\right)}, y \cdot 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 4 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t)))) 4e+276)
   (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) / ((z * (2.0 * z)) - (y * t)))) <= 4e+276) {
		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(z * Float64(2.0 * z)) - Float64(y * t)))) <= 4e+276)
		tmp = fma(Float64(y * 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[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+276], N[(N[(y * z), $MachinePrecision] * N[(2.0 / 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 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 4.0000000000000002e276

   1. Initial program 94.7%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 94.7%

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

   if 4.0000000000000002e276 < (-.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 4.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 4 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 4 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{2}{\mathsf{fma}\left(z \cdot z, -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t)))) 4e+276)
   (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) / ((z * (2.0 * z)) - (y * t)))) <= 4e+276) {
		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(z * Float64(2.0 * z)) - Float64(y * t)))) <= 4e+276)
		tmp = fma(Float64(y * z), Float64(2.0 / fma(Float64(z * 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[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+276], N[(N[(y * z), $MachinePrecision] * N[(2.0 / N[(N[(z * z), $MachinePrecision] * -2.0 + 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 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 4.0000000000000002e276

   1. Initial program 94.7%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 94.7%

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

   if 4.0000000000000002e276 < (-.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 4.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 4 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{2}{\mathsf{fma}\left(z \cdot z, -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1150000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+44}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1150000000.0)
     t_1
     (if (<= z 1.65e+44) (- x (/ (* z -2.0) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1150000000.0) {
		tmp = t_1;
	} else if (z <= 1.65e+44) {
		tmp = x - ((z * -2.0) / 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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (y / z)
    if (z <= (-1150000000.0d0)) then
        tmp = t_1
    else if (z <= 1.65d+44) then
        tmp = x - ((z * (-2.0d0)) / t)
    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 tmp;
	if (z <= -1150000000.0) {
		tmp = t_1;
	} else if (z <= 1.65e+44) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1150000000.0:
		tmp = t_1
	elif z <= 1.65e+44:
		tmp = x - ((z * -2.0) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -1150000000.0)
		tmp = t_1;
	elseif (z <= 1.65e+44)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -1150000000.0)
		tmp = t_1;
	elseif (z <= 1.65e+44)
		tmp = x - ((z * -2.0) / t);
	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]}, If[LessEqual[z, -1150000000.0], t$95$1, If[LessEqual[z, 1.65e+44], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e9 or 1.65000000000000007e44 < z

   1. Initial program 66.2%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 91.7

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

   5. Applied rewrites 91.7%

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

   if -1.15e9 < z < 1.65000000000000007e44

   1. Initial program 87.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1150000000:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+44}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1150000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1150000000.0) t_1 (if (<= z 1.65e+44) (fma z (/ 2.0 t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1150000000.0) {
		tmp = t_1;
	} else if (z <= 1.65e+44) {
		tmp = fma(z, (2.0 / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -1150000000.0)
		tmp = t_1;
	elseif (z <= 1.65e+44)
		tmp = fma(z, Float64(2.0 / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1150000000.0], t$95$1, If[LessEqual[z, 1.65e+44], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e9 or 1.65000000000000007e44 < z

   1. Initial program 66.2%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 91.7

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

   5. Applied rewrites 91.7%

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

   if -1.15e9 < z < 1.65000000000000007e44

   1. Initial program 87.4%

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

   3. Taylor expanded in y around inf

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

      1. cancel-sign-sub-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. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 6: 63.4% 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 77.5%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
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 61.6

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

5. Applied rewrites 61.6%

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

Alternative 7: 14.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

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

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

   3. associate-*r/ N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-in N/A

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

   15. unpow2 N/A

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

   16. associate-*r* N/A

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

   17. distribute-lft-neg-out N/A

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

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

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

   19. metadata-eval N/A

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

   20. associate-*r* N/A

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

   21. unpow2 N/A

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

5. Applied rewrites 20.7%

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

6. Taylor expanded in y around 0

   \[\leadsto -1 \cdot \color{blue}{\frac{y}{z}} \]
7. Step-by-step derivation

8. Applied rewrites 16.4%

   \[\leadsto \frac{y}{\color{blue}{-z}} \]
9. 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 2024233 
(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)))))