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

Percentage Accurate: 82.6% → 93.8%

Time: 8.0s

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: 82.6% 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: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+177}:\\ \;\;\;\;x - t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z (* 2.0 y)) (- (* (* z 2.0) z) (* t y)))))
   (if (<= t_1 5e+177) (- x t_1) (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * (2.0 * y)) / (((z * 2.0) * z) - (t * y));
	double tmp;
	if (t_1 <= 5e+177) {
		tmp = x - 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 = (z * (2.0d0 * y)) / (((z * 2.0d0) * z) - (t * y))
    if (t_1 <= 5d+177) then
        tmp = x - 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 = (z * (2.0 * y)) / (((z * 2.0) * z) - (t * y));
	double tmp;
	if (t_1 <= 5e+177) {
		tmp = x - t_1;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * (2.0 * y)) / (((z * 2.0) * z) - (t * y))
	tmp = 0
	if t_1 <= 5e+177:
		tmp = x - t_1
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * Float64(2.0 * y)) / Float64(Float64(Float64(z * 2.0) * z) - Float64(t * y)))
	tmp = 0.0
	if (t_1 <= 5e+177)
		tmp = Float64(x - t_1);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * (2.0 * y)) / (((z * 2.0) * z) - (t * y));
	tmp = 0.0;
	if (t_1 <= 5e+177)
		tmp = x - t_1;
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+177], N[(x - t$95$1), $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))) < 5.0000000000000003e177

   1. Initial program 99.0%

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

   if 5.0000000000000003e177 < (/.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 t 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%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y} \leq 5 \cdot 10^{+177}:\\ \;\;\;\;x - \frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot 2\right) \cdot z\\ \mathbf{if}\;\frac{z \cdot \left(2 \cdot y\right)}{t\_1 - t \cdot y} \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-2 \cdot y}{\mathsf{fma}\left(-t, y, t\_1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z 2.0) z)))
   (if (<= (/ (* z (* 2.0 y)) (- t_1 (* t y))) 5e+177)
     (fma z (/ (* -2.0 y) (fma (- t) y t_1)) x)
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * 2.0) * z;
	double tmp;
	if (((z * (2.0 * y)) / (t_1 - (t * y))) <= 5e+177) {
		tmp = fma(z, ((-2.0 * y) / fma(-t, y, t_1)), x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * 2.0) * z)
	tmp = 0.0
	if (Float64(Float64(z * Float64(2.0 * y)) / Float64(t_1 - Float64(t * y))) <= 5e+177)
		tmp = fma(z, Float64(Float64(-2.0 * y) / fma(Float64(-t), y, t_1)), x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(N[(z * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+177], N[(z * N[(N[(-2.0 * y), $MachinePrecision] / N[((-t) * y + t$95$1), $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))) < 5.0000000000000003e177

   1. Initial program 99.0%

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 97.9%

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

   if 5.0000000000000003e177 < (/.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 t 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%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(2 \cdot y\right)}{\left(z \cdot 2\right) \cdot z - t \cdot y} \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-2 \cdot y}{\mathsf{fma}\left(-t, y, \left(z \cdot 2\right) \cdot z\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -4.8e-36) t_1 (if (<= z 2.6e-21) (fma (/ 2.0 t) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -4.8e-36) {
		tmp = t_1;
	} else if (z <= 2.6e-21) {
		tmp = fma((2.0 / t), z, 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 <= -4.8e-36)
		tmp = t_1;
	elseif (z <= 2.6e-21)
		tmp = fma(Float64(2.0 / t), z, 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, -4.8e-36], t$95$1, If[LessEqual[z, 2.6e-21], N[(N[(2.0 / t), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e-36 or 2.60000000000000017e-21 < z

   1. Initial program 73.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if -4.8e-36 < z < 2.60000000000000017e-21

   1. Initial program 95.7%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

Alternative 4: 77.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -2.4e-57) t_1 (if (<= z 8.4e-70) (/ (* x t) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -2.4e-57) {
		tmp = t_1;
	} else if (z <= 8.4e-70) {
		tmp = (x * t) / 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 <= (-2.4d-57)) then
        tmp = t_1
    else if (z <= 8.4d-70) then
        tmp = (x * t) / 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 <= -2.4e-57) {
		tmp = t_1;
	} else if (z <= 8.4e-70) {
		tmp = (x * t) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -2.4e-57:
		tmp = t_1
	elif z <= 8.4e-70:
		tmp = (x * t) / 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 <= -2.4e-57)
		tmp = t_1;
	elseif (z <= 8.4e-70)
		tmp = Float64(Float64(x * t) / 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 <= -2.4e-57)
		tmp = t_1;
	elseif (z <= 8.4e-70)
		tmp = (x * t) / 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, -2.4e-57], t$95$1, If[LessEqual[z, 8.4e-70], N[(N[(x * t), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000006e-57 or 8.4000000000000004e-70 < z

   1. Initial program 76.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -2.40000000000000006e-57 < z < 8.4000000000000004e-70

   1. Initial program 94.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{t \cdot x}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-57}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% 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 81.6%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 68.8

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

5. Applied rewrites 68.8%

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

Alternative 6: 14.9% 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 81.6%

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

3. Taylor expanded in t around inf

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 59.8

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

5. Applied rewrites 59.8%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 11.3%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. associate-/l* N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

11. Applied rewrites 18.2%

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

12. Taylor expanded in t around 0

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

14. Applied rewrites 18.5%

    \[\leadsto \frac{-y}{\color{blue}{z}} \]
15. 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 2024244 
(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)))))