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

Percentage Accurate: 82.1% → 95.9%

Time: 10.3s

Alternatives: 5

Speedup: 1.1×

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: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))

C

double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Python

def code(x, y, z, t):
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot 2}{t \cdot \frac{y}{z} - z \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (/ (* y 2.0) (- (* t (/ y z)) (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * 2.0) / ((t * (y / z)) - (z * 2.0)));
}

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) / ((t * (y / z)) - (z * 2.0d0)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y * 2.0) / ((t * (y / z)) - (z * 2.0)));
}

Python

def code(x, y, z, t):
	return x + ((y * 2.0) / ((t * (y / z)) - (z * 2.0)))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * 2.0) / Float64(Float64(t * Float64(y / z)) - Float64(z * 2.0))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y * 2.0) / ((t * (y / z)) - (z * 2.0)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y * 2.0), $MachinePrecision] / N[(N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision] - N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

3. Simplified 92.3%

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

   1. clear-num 92.3%

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

   2. un-div-inv 92.4%

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

   3. *-commutative 92.4%

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

   4. *-commutative 92.4%

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

   5. associate-*l* 92.4%

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

   6. pow2 92.4%

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

6. Applied egg-rr 92.4%

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

7. Taylor expanded in y around 0 96.9%

   \[\leadsto x - \frac{2 \cdot y}{\color{blue}{-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z}} \]
8. Step-by-step derivation

   1. +-commutative 96.9%

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

   2. mul-1-neg 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-*r/ 98.8%

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

   5. unsub-neg 98.8%

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

   6. *-commutative 98.8%

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

   7. associate-*r/ 96.9%

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

   8. *-commutative 96.9%

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

   9. associate-/l* 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

    \[\leadsto x + \frac{y \cdot 2}{t \cdot \frac{y}{z} - z \cdot 2} \]
11. Add Preprocessing

Alternative 2: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ t_2 := x - \frac{z \cdot -2}{t}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-77}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))) (t_2 (- x (/ (* z -2.0) t))))
   (if (<= z -1e+18)
     t_1
     (if (<= z 5.3e-77)
       t_2
       (if (<= z 2.8e+31) x (if (<= z 1.05e+60) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double t_2 = x - ((z * -2.0) / t);
	double tmp;
	if (z <= -1e+18) {
		tmp = t_1;
	} else if (z <= 5.3e-77) {
		tmp = t_2;
	} else if (z <= 2.8e+31) {
		tmp = x;
	} else if (z <= 1.05e+60) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = x - (y / z)
    t_2 = x - ((z * (-2.0d0)) / t)
    if (z <= (-1d+18)) then
        tmp = t_1
    else if (z <= 5.3d-77) then
        tmp = t_2
    else if (z <= 2.8d+31) then
        tmp = x
    else if (z <= 1.05d+60) then
        tmp = t_2
    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 t_2 = x - ((z * -2.0) / t);
	double tmp;
	if (z <= -1e+18) {
		tmp = t_1;
	} else if (z <= 5.3e-77) {
		tmp = t_2;
	} else if (z <= 2.8e+31) {
		tmp = x;
	} else if (z <= 1.05e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	t_2 = x - ((z * -2.0) / t)
	tmp = 0
	if z <= -1e+18:
		tmp = t_1
	elif z <= 5.3e-77:
		tmp = t_2
	elif z <= 2.8e+31:
		tmp = x
	elif z <= 1.05e+60:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	t_2 = Float64(x - Float64(Float64(z * -2.0) / t))
	tmp = 0.0
	if (z <= -1e+18)
		tmp = t_1;
	elseif (z <= 5.3e-77)
		tmp = t_2;
	elseif (z <= 2.8e+31)
		tmp = x;
	elseif (z <= 1.05e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	t_2 = x - ((z * -2.0) / t);
	tmp = 0.0;
	if (z <= -1e+18)
		tmp = t_1;
	elseif (z <= 5.3e-77)
		tmp = t_2;
	elseif (z <= 2.8e+31)
		tmp = x;
	elseif (z <= 1.05e+60)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+18], t$95$1, If[LessEqual[z, 5.3e-77], t$95$2, If[LessEqual[z, 2.8e+31], x, If[LessEqual[z, 1.05e+60], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e18 or 1.0500000000000001e60 < z

   1. Initial program 74.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in y around 0 93.1%

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

   if -1e18 < z < 5.30000000000000015e-77 or 2.80000000000000017e31 < z < 1.0500000000000001e60

   1. Initial program 92.9%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in y around inf 93.4%

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

      1. associate-*r/ 93.4%

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

      2. *-commutative 93.4%

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

   7. Simplified 93.4%

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

   if 5.30000000000000015e-77 < z < 2.80000000000000017e31

   1. Initial program 96.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 87.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+21} \lor \neg \left(z \leq 4.3 \cdot 10^{+60}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.5e+21) (not (<= z 4.3e+60))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.5e+21) || !(z <= 4.3e+60)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	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) :: tmp
    if ((z <= (-8.5d+21)) .or. (.not. (z <= 4.3d+60))) then
        tmp = x - (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.5e+21) || !(z <= 4.3e+60)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.5e+21) or not (z <= 4.3e+60):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.5e+21) || !(z <= 4.3e+60))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -8.5e+21) || ~((z <= 4.3e+60)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.5e+21], N[Not[LessEqual[z, 4.3e+60]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e21 or 4.29999999999999971e60 < z

   1. Initial program 73.6%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 93.9%

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

   if -8.5e21 < z < 4.29999999999999971e60

   1. Initial program 93.5%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in x around inf 79.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+21} \lor \neg \left(z \leq 4.3 \cdot 10^{+60}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x + y \cdot \frac{2}{\frac{y \cdot t}{z} - z \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* y (/ 2.0 (- (/ (* y t) z) (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	return x + (y * (2.0 / (((y * t) / z) - (z * 2.0))));
}

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 / (((y * t) / z) - (z * 2.0d0))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (y * (2.0 / (((y * t) / z) - (z * 2.0))));
}

Python

def code(x, y, z, t):
	return x + (y * (2.0 / (((y * t) / z) - (z * 2.0))))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(y * Float64(2.0 / Float64(Float64(Float64(y * t) / z) - Float64(z * 2.0)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (y * (2.0 / (((y * t) / z) - (z * 2.0))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(y * N[(2.0 / N[(N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision] - N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

3. Simplified 92.3%

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

   1. clear-num 92.3%

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

   2. un-div-inv 92.4%

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

   3. *-commutative 92.4%

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

   4. *-commutative 92.4%

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

   5. associate-*l* 92.4%

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

   6. pow2 92.4%

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

6. Applied egg-rr 92.4%

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

7. Taylor expanded in y around 0 96.9%

   \[\leadsto x - \frac{2 \cdot y}{\color{blue}{-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z}} \]
8. Step-by-step derivation

   1. +-commutative 96.9%

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

   2. mul-1-neg 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-*r/ 98.8%

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

   5. unsub-neg 98.8%

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

   6. *-commutative 98.8%

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

   7. associate-*r/ 96.9%

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

   8. *-commutative 96.9%

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

   9. associate-/l* 97.3%

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

9. Simplified 97.3%

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

    1. associate-/l* 97.3%

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

    2. *-commutative 97.3%

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

    3. fma-neg 97.3%

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

    4. *-commutative 97.3%

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

    5. distribute-rgt-neg-in 97.3%

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

11. Applied egg-rr 97.3%

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

    1. associate-*r/ 97.3%

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

    2. *-commutative 97.3%

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

    3. associate-*r/ 97.2%

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

    4. fma-define 97.2%

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

    5. *-commutative 97.2%

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

    6. cancel-sign-sub-inv 97.2%

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

    7. *-commutative 97.2%

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

    8. associate-*r/ 96.9%

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

    9. *-commutative 96.9%

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

13. Simplified 96.9%

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

14. Final simplification 96.9%

    \[\leadsto x + y \cdot \frac{2}{\frac{y \cdot t}{z} - z \cdot 2} \]
15. Add Preprocessing

Alternative 5: 74.6% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x;
}

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
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 85.3%

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

3. Simplified 92.3%

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

5. Taylor expanded in x around inf 80.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 99.9% accurate, 1.3× 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 2024096 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

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