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

Percentage Accurate: 82.0% → 93.0%

Time: 8.0s

Alternatives: 5

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: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (* z (* 2.0 y)) (- (* (* z 2.0) z) (* t y))))))
   (if (<= t_1 1e+306) t_1 (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((z * (2.0 * y)) / (((z * 2.0) * z) - (t * y)));
	double tmp;
	if (t_1 <= 1e+306) {
		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 - ((z * (2.0d0 * y)) / (((z * 2.0d0) * z) - (t * y)))
    if (t_1 <= 1d+306) 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 - ((z * (2.0 * y)) / (((z * 2.0) * z) - (t * y)));
	double tmp;
	if (t_1 <= 1e+306) {
		tmp = t_1;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - ((z * (2.0 * y)) / (((z * 2.0) * z) - (t * y)))
	tmp = 0
	if t_1 <= 1e+306:
		tmp = t_1
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(z * Float64(2.0 * y)) / Float64(Float64(Float64(z * 2.0) * z) - Float64(t * y))))
	tmp = 0.0
	if (t_1 <= 1e+306)
		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 - ((z * (2.0 * y)) / (((z * 2.0) * z) - (t * y)));
	tmp = 0.0;
	if (t_1 <= 1e+306)
		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[(z * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+306], 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)))) < 1.00000000000000002e306

   1. Initial program 97.7%

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

   if 1.00000000000000002e306 < (-.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 0.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y} \leq 10^{+306}:\\ \;\;\;\;x - \frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y} \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(-2, z \cdot z, t \cdot y\right)}, y + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x (/ (* z (* 2.0 y)) (- (* (* z 2.0) z) (* t y)))) 1e+306)
   (fma (/ z (fma -2.0 (* z z) (* t y))) (+ y y) x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - ((z * (2.0 * y)) / (((z * 2.0) * z) - (t * y)))) <= 1e+306) {
		tmp = fma((z / fma(-2.0, (z * z), (t * y))), (y + y), x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - Float64(Float64(z * Float64(2.0 * y)) / Float64(Float64(Float64(z * 2.0) * z) - Float64(t * y)))) <= 1e+306)
		tmp = fma(Float64(z / fma(-2.0, Float64(z * z), Float64(t * y))), Float64(y + y), x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - N[(N[(z * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(z / N[(-2.0 * N[(z * z), $MachinePrecision] + N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + y), $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)))) < 1.00000000000000002e306

   1. Initial program 97.7%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 97.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if 1.00000000000000002e306 < (-.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 0.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y} \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(-2, z \cdot z, t \cdot y\right)}, y + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -20000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -20000.0) t_1 (if (<= z 4e-86) (fma (/ z t) 2.0 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -20000.0) {
		tmp = t_1;
	} else if (z <= 4e-86) {
		tmp = fma((z / t), 2.0, 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 <= -20000.0)
		tmp = t_1;
	elseif (z <= 4e-86)
		tmp = fma(Float64(z / t), 2.0, 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, -20000.0], t$95$1, If[LessEqual[z, 4e-86], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4 or 4.00000000000000034e-86 < z

   1. Initial program 66.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

   if -2e4 < z < 4.00000000000000034e-86

   1. Initial program 94.6%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. 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 91.8

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

   5. Applied rewrites 91.8%

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

Alternative 4: 62.1% 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 79.0%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 66.8

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

5. Applied rewrites 66.8%

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

Alternative 5: 15.0% 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(Float64(-y) / 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 79.0%

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

3. Taylor expanded in x around 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. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

   11. distribute-neg-in N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

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

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

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

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

   16. metadata-eval N/A

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

   17. associate-*r* N/A

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

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

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

   19. mul-1-neg N/A

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

   20. remove-double-neg N/A

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

5. Applied rewrites 15.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 12.6%

   \[\leadsto \frac{-y}{\color{blue}{z}} \]
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 2024298 
(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)))))