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

Percentage Accurate: 81.6% → 98.4%

Time: 11.8s

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: 81.6% 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: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Simplified 90.0%

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

   1. clear-num 89.8%

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

   2. un-div-inv 89.8%

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

   3. *-commutative 89.8%

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

   4. *-commutative 89.8%

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

   5. associate-*l* 89.8%

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

   6. pow2 89.8%

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

6. Applied egg-rr 89.8%

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

7. Taylor expanded in z around 0 95.6%

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

   1. +-commutative 95.6%

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

   2. *-commutative 95.6%

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

   3. neg-mul-1 95.6%

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

   4. unsub-neg 95.6%

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

   5. *-commutative 95.6%

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

   6. associate-/l* 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-24} \lor \neg \left(z \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-2}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.2e-24) (not (<= z 1.2e-7)))
   (- x (/ y z))
   (- x (/ -2.0 (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.2e-24) || !(z <= 1.2e-7)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (-2.0 / (t / 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) :: tmp
    if ((z <= (-5.2d-24)) .or. (.not. (z <= 1.2d-7))) then
        tmp = x - (y / z)
    else
        tmp = x - ((-2.0d0) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.2e-24) || !(z <= 1.2e-7)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (-2.0 / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.2e-24) or not (z <= 1.2e-7):
		tmp = x - (y / z)
	else:
		tmp = x - (-2.0 / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.2e-24) || !(z <= 1.2e-7))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(-2.0 / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.2e-24) || ~((z <= 1.2e-7)))
		tmp = x - (y / z);
	else
		tmp = x - (-2.0 / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.2e-24], N[Not[LessEqual[z, 1.2e-7]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(-2.0 / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e-24 or 1.19999999999999989e-7 < z

   1. Initial program 75.8%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in y around 0 86.3%

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

   if -5.2e-24 < z < 1.19999999999999989e-7

   1. Initial program 89.0%

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

   3. Simplified 92.1%

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

      1. clear-num 91.7%

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

      2. un-div-inv 91.8%

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

      3. *-commutative 91.8%

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

      4. *-commutative 91.8%

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

      5. associate-*l* 91.8%

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

      6. pow2 91.8%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in z around 0 79.1%

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

      1. associate-*r* 79.1%

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

      2. neg-mul-1 79.1%

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

      3. *-commutative 79.1%

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

   9. Simplified 79.1%

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

       1. frac-2neg 79.1%

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

       2. div-inv 79.1%

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

       3. distribute-neg-frac2 79.1%

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

       4. distribute-rgt-neg-out 79.1%

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

       5. frac-2neg 79.1%

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

       6. add-sqr-sqrt 40.1%

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

       7. times-frac 42.9%

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

       8. add-sqr-sqrt 21.3%

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

       9. sqrt-unprod 31.3%

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

       10. sqr-neg 31.3%

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

       11. sqrt-unprod 16.8%

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

       12. add-sqr-sqrt 35.2%

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

       13. times-frac 32.1%

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

       14. add-sqr-sqrt 64.6%

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

       15. clear-num 64.6%

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

       16. distribute-lft-neg-in 64.6%

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

       17. clear-num 64.6%

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

   11. Applied egg-rr 80.6%

       \[\leadsto x - \color{blue}{\left(0 - 2 \cdot \left(y \cdot \frac{\frac{z}{y}}{t}\right)\right)} \]
   12. Step-by-step derivation

       1. neg-sub0 80.6%

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

       2. associate-*r* 80.6%

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

       3. associate-*r/ 82.4%

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

       4. associate-*r/ 86.4%

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

       5. /-rgt-identity 86.4%

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

       6. associate-/r/ 86.3%

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

       7. associate-/r* 79.1%

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

       8. *-commutative 79.1%

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

       9. associate-/r* 81.1%

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

       10. associate-/r/ 81.1%

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

       11. /-rgt-identity 81.1%

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

       12. associate-*l/ 75.8%

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

       13. associate-/r/ 81.8%

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

       14. associate-/r* 84.9%

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

       15. times-frac 87.9%

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

       16. *-inverses 87.9%

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

       17. *-rgt-identity 87.9%

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

       18. distribute-neg-frac 87.9%

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

       19. metadata-eval 87.9%

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

   13. Simplified 87.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-24} \lor \neg \left(z \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-2}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-26} \lor \neg \left(z \leq 0.0022\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.8e-26) (not (<= z 0.0022)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.8e-26) || !(z <= 0.0022)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	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 <= (-2.8d-26)) .or. (.not. (z <= 0.0022d0))) then
        tmp = x - (y / z)
    else
        tmp = x - ((z * (-2.0d0)) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.8e-26) || !(z <= 0.0022)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.8e-26) or not (z <= 0.0022):
		tmp = x - (y / z)
	else:
		tmp = x - ((z * -2.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.8e-26) || !(z <= 0.0022))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.8e-26) || ~((z <= 0.0022)))
		tmp = x - (y / z);
	else
		tmp = x - ((z * -2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.8e-26], N[Not[LessEqual[z, 0.0022]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-26 or 0.00220000000000000013 < z

   1. Initial program 75.8%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in y around 0 86.3%

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

   if -2.8000000000000001e-26 < z < 0.00220000000000000013

   1. Initial program 89.0%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in y around inf 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-*l/ 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-26} \lor \neg \left(z \leq 0.0022\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.1 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-92}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.1e-19) x (if (<= t 5.2e-92) (- x (/ y z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.1e-19) {
		tmp = x;
	} else if (t <= 5.2e-92) {
		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 (t <= (-8.1d-19)) then
        tmp = x
    else if (t <= 5.2d-92) 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 (t <= -8.1e-19) {
		tmp = x;
	} else if (t <= 5.2e-92) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -8.1e-19:
		tmp = x
	elif t <= 5.2e-92:
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.1e-19)
		tmp = x;
	elseif (t <= 5.2e-92)
		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 (t <= -8.1e-19)
		tmp = x;
	elseif (t <= 5.2e-92)
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -8.1e-19], x, If[LessEqual[t, 5.2e-92], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -8.10000000000000023e-19 or 5.2e-92 < t

   1. Initial program 84.2%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in x around inf 83.6%

      \[\leadsto \color{blue}{x} \]

   if -8.10000000000000023e-19 < t < 5.2e-92

   1. Initial program 79.6%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in y around 0 75.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.1 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-92}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% 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 82.3%

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

3. Simplified 90.0%

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

5. Taylor expanded in x around inf 76.0%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 76.0%

   \[\leadsto x \]
7. 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 2024043 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

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