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

Percentage Accurate: 81.7% → 98.3%

Time: 8.2s

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: 81.7% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(y \cdot 2\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(y \cdot 2\right) \cdot \frac{1}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{y \cdot 2}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot 2\right), \color{blue}{\left(\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(\frac{\color{blue}{\left(z \cdot 2\right) \cdot z - y \cdot t}}{z}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\left(\left(z \cdot 2\right) \cdot z - y \cdot t\right), \color{blue}{z}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(z \cdot 2\right) \cdot z\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(2 \cdot z\right) \cdot z\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(2 \cdot \left(z \cdot z\right)\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(z \cdot z\right)\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(z, z\right)\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   12. *-lowering-*.f64 88.0%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(y, t\right)\right), z\right)\right)\right) \]

4. Applied egg-rr 88.0%

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

5. Taylor expanded in y around inf

   \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \color{blue}{\left(y \cdot \left(-1 \cdot \frac{t}{z} + 2 \cdot \frac{z}{y}\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(2 \cdot \frac{z}{y} + \color{blue}{-1 \cdot \frac{t}{z}}\right)\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(\frac{2 \cdot z}{y} + \color{blue}{-1} \cdot \frac{t}{z}\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(\frac{\left(\mathsf{neg}\left(-2\right)\right) \cdot z}{y} + -1 \cdot \frac{t}{z}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(\frac{\mathsf{neg}\left(-2 \cdot z\right)}{y} + -1 \cdot \frac{t}{z}\right)\right)\right)\right) \]

   5. distribute-frac-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(\left(\mathsf{neg}\left(\frac{-2 \cdot z}{y}\right)\right) + \color{blue}{-1} \cdot \frac{t}{z}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(\left(\mathsf{neg}\left(-2 \cdot \frac{z}{y}\right)\right) + -1 \cdot \frac{t}{z}\right)\right)\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(\left(\mathsf{neg}\left(-2 \cdot \frac{z}{y}\right)\right) + \left(\mathsf{neg}\left(\frac{t}{z}\right)\right)\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot \frac{z}{y} + \frac{t}{z}\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot \frac{z}{y} + \frac{t}{z}\right)\right)\right)}\right)\right)\right) \]

   10. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(-2 \cdot \frac{z}{y}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{z}\right)\right)}\right)\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{-2 \cdot z}{y}\right)\right) + \left(\mathsf{neg}\left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right)\right) \]

   12. distribute-frac-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \left(\frac{\mathsf{neg}\left(-2 \cdot z\right)}{y} + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(-2\right)\right) \cdot z}{y} + \left(\mathsf{neg}\left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \left(\frac{2 \cdot z}{y} + \left(\mathsf{neg}\left(\frac{t}{z}\right)\right)\right)\right)\right)\right) \]

   15. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \left(2 \cdot \frac{z}{y} + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)\right)\right)\right) \]

   16. unsub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \left(2 \cdot \frac{z}{y} - \color{blue}{\frac{t}{z}}\right)\right)\right)\right) \]

   17. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(2 \cdot \frac{z}{y}\right), \color{blue}{\left(\frac{t}{z}\right)}\right)\right)\right)\right) \]

   18. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{2 \cdot z}{y}\right), \left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right) \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot z\right), y\right), \left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(z \cdot 2\right), y\right), \left(\frac{t}{z}\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, 2\right), y\right), \left(\frac{t}{z}\right)\right)\right)\right)\right) \]

   22. /-lowering-/.f64 97.7%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, 2\right), y\right), \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right)\right)\right) \]

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 2: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -8500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -8500000000.0)
     t_1
     (if (<= z 1.8e-6) (- 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 <= -8500000000.0) {
		tmp = t_1;
	} else if (z <= 1.8e-6) {
		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 <= (-8500000000.0d0)) then
        tmp = t_1
    else if (z <= 1.8d-6) 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 <= -8500000000.0) {
		tmp = t_1;
	} else if (z <= 1.8e-6) {
		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 <= -8500000000.0:
		tmp = t_1
	elif z <= 1.8e-6:
		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 <= -8500000000.0)
		tmp = t_1;
	elseif (z <= 1.8e-6)
		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 <= -8500000000.0)
		tmp = t_1;
	elseif (z <= 1.8e-6)
		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, -8500000000.0], t$95$1, If[LessEqual[z, 1.8e-6], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e9 or 1.79999999999999992e-6 < z

   1. Initial program 71.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\frac{y}{z}\right)\right) \]

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      4. /-lowering-/.f64 93.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 93.4%

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

   if -8.5e9 < z < 1.79999999999999992e-6

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(-2 \cdot \frac{z}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{-2 \cdot z}{\color{blue}{t}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(-2 \cdot z\right), \color{blue}{t}\right)\right) \]

      4. *-lowering-*.f64 94.3%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, z\right), t\right)\right) \]

   5. Simplified 94.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8500000000:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1700000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1700000000000.0) t_1 (if (<= z 7e-40) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1700000000000.0) {
		tmp = t_1;
	} else if (z <= 7e-40) {
		tmp = x;
	} 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 <= (-1700000000000.0d0)) then
        tmp = t_1
    else if (z <= 7d-40) then
        tmp = x
    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 <= -1700000000000.0) {
		tmp = t_1;
	} else if (z <= 7e-40) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1700000000000.0:
		tmp = t_1
	elif z <= 7e-40:
		tmp = 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 <= -1700000000000.0)
		tmp = t_1;
	elseif (z <= 7e-40)
		tmp = x;
	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 <= -1700000000000.0)
		tmp = t_1;
	elseif (z <= 7e-40)
		tmp = x;
	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, -1700000000000.0], t$95$1, If[LessEqual[z, 7e-40], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e12 or 7.0000000000000003e-40 < z

   1. Initial program 73.2%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\frac{y}{z}\right)\right) \]

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      4. /-lowering-/.f64 91.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 91.7%

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

   if -1.7e12 < z < 7.0000000000000003e-40

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 78.1%

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

Alternative 4: 95.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y * 2.0) / (((y * t) / z) - (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 + ((y * 2.0d0) / (((y * t) / z) - (2.0d0 * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(y \cdot 2\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(y \cdot 2\right) \cdot \frac{1}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{y \cdot 2}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot 2\right), \color{blue}{\left(\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(\frac{\color{blue}{\left(z \cdot 2\right) \cdot z - y \cdot t}}{z}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\left(\left(z \cdot 2\right) \cdot z - y \cdot t\right), \color{blue}{z}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(z \cdot 2\right) \cdot z\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(2 \cdot z\right) \cdot z\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(2 \cdot \left(z \cdot z\right)\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(z \cdot z\right)\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(z, z\right)\right), \left(y \cdot t\right)\right), z\right)\right)\right) \]

   12. *-lowering-*.f64 88.0%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(y, t\right)\right), z\right)\right)\right) \]

4. Applied egg-rr 88.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \color{blue}{\left(-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z\right)}\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(2 \cdot z + \color{blue}{-1 \cdot \frac{t \cdot y}{z}}\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(2 \cdot z + \left(\mathsf{neg}\left(\frac{t \cdot y}{z}\right)\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \left(2 \cdot z - \color{blue}{\frac{t \cdot y}{z}}\right)\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\left(2 \cdot z\right), \color{blue}{\left(\frac{t \cdot y}{z}\right)}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\left(z \cdot 2\right), \left(\frac{\color{blue}{t \cdot y}}{z}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, 2\right), \left(\frac{\color{blue}{t \cdot y}}{z}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, 2\right), \mathsf{/.f64}\left(\left(t \cdot y\right), \color{blue}{z}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, 2\right), \mathsf{/.f64}\left(\left(y \cdot t\right), z\right)\right)\right)\right) \]

   9. *-lowering-*.f64 95.2%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, 2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, t\right), z\right)\right)\right)\right) \]

7. Simplified 95.2%

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

8. Final simplification 95.2%

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

Alternative 5: 74.9% 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 80.6%

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

3. Taylor expanded in x around inf

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

5. Simplified 75.2%

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

Developer Target 1: 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 2024150 
(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)))))