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

Percentage Accurate: 82.2% → 99.9%

Time: 11.7s

Alternatives: 8

Speedup: 1.3×

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: 82.2% 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: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{\frac{\frac{t}{z}}{-2} + \frac{z}{y}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (/ -1.0 (+ (/ (/ t z) -2.0) (/ z y)))))

C

double code(double x, double y, double z, double t) {
	return x + (-1.0 / (((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 + ((-1.0d0) / (((t / z) / (-2.0d0)) + (z / y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (-1.0 / (((t / z) / -2.0) + (z / y)));
}

Python

def code(x, y, z, t):
	return x + (-1.0 / (((t / z) / -2.0) + (z / y)))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(-1.0 / Float64(Float64(Float64(t / z) / -2.0) + Float64(z / y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (-1.0 / (((t / z) / -2.0) + (z / y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(-1.0 / N[(N[(N[(t / z), $MachinePrecision] / -2.0), $MachinePrecision] + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

3. Simplified 88.4%

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

   1. clear-num 88.0%

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

   2. un-div-inv 88.0%

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

   3. *-commutative 88.0%

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

   4. associate-*l* 88.0%

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

   5. pow2 88.0%

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

6. Applied egg-rr 88.0%

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

7. Taylor expanded in y around 0 93.9%

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

8. Taylor expanded in x around 0 93.9%

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

   1. cancel-sign-sub-inv 93.9%

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

   2. metadata-eval 93.9%

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

   3. associate-*r/ 93.9%

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

   4. *-commutative 93.9%

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

   5. +-commutative 93.9%

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

   6. *-commutative 93.9%

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

   7. mul-1-neg 93.9%

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

   8. associate-*r/ 95.8%

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

   9. unsub-neg 95.8%

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

   10. *-commutative 95.8%

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

10. Simplified 95.8%

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

    1. clear-num 95.8%

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

    2. inv-pow 95.8%

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

    3. associate-*l/ 93.8%

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

    4. associate-/l* 97.8%

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

12. Applied egg-rr 97.8%

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

    1. unpow-1 97.8%

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

    2. div-sub 97.8%

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

    3. *-commutative 97.8%

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

    4. *-commutative 97.8%

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

    5. times-frac 97.8%

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

    6. metadata-eval 97.8%

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

    7. neg-mul-1 97.8%

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

    8. distribute-neg-frac2 97.8%

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

    9. times-frac 99.9%

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

    10. *-inverses 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

    \[\leadsto x + \frac{-1}{\frac{\frac{t}{z}}{-2} + \frac{z}{y}} \]
16. Add Preprocessing

Alternative 2: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7600 \lor \neg \left(z \leq 1.12 \cdot 10^{-10}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7600.0) (not (<= z 1.12e-10)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7600.0) || !(z <= 1.12e-10)) {
		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 <= (-7600.0d0)) .or. (.not. (z <= 1.12d-10))) 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 <= -7600.0) || !(z <= 1.12e-10)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7600.0) or not (z <= 1.12e-10):
		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 <= -7600.0) || !(z <= 1.12e-10))
		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 <= -7600.0) || ~((z <= 1.12e-10)))
		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, -7600.0], N[Not[LessEqual[z, 1.12e-10]], $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 < -7600 or 1.12e-10 < z

   1. Initial program 75.1%

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

      1. sub-neg 75.1%

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

      2. +-commutative 75.1%

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

      3. associate-/l* 85.1%

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

      4. distribute-rgt-neg-in 85.1%

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

      5. fma-define 85.1%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in y around 0 87.9%

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

      1. mul-1-neg 87.9%

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

      2. sub-neg 87.9%

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

   7. Simplified 87.9%

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

   if -7600 < z < 1.12e-10

   1. Initial program 88.4%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in y around inf 90.3%

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

      1. associate-*r/ 90.3%

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

      2. *-commutative 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7600 \lor \neg \left(z \leq 1.12 \cdot 10^{-10}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -780000 \lor \neg \left(z \leq 1.16 \cdot 10^{-10}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{2}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -780000.0) (not (<= z 1.16e-10)))
   (- x (/ y z))
   (+ x (/ 2.0 (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -780000.0) || !(z <= 1.16e-10)) {
		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 <= (-780000.0d0)) .or. (.not. (z <= 1.16d-10))) 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 <= -780000.0) || !(z <= 1.16e-10)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (2.0 / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -780000.0) or not (z <= 1.16e-10):
		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 <= -780000.0) || !(z <= 1.16e-10))
		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 <= -780000.0) || ~((z <= 1.16e-10)))
		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, -780000.0], N[Not[LessEqual[z, 1.16e-10]], $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 < -7.8e5 or 1.16e-10 < z

   1. Initial program 75.1%

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

      1. sub-neg 75.1%

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

      2. +-commutative 75.1%

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

      3. associate-/l* 85.1%

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

      4. distribute-rgt-neg-in 85.1%

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

      5. fma-define 85.1%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in y around 0 87.9%

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

      1. mul-1-neg 87.9%

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

      2. sub-neg 87.9%

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

   7. Simplified 87.9%

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

   if -7.8e5 < z < 1.16e-10

   1. Initial program 88.4%

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

   3. Simplified 90.9%

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

      1. clear-num 90.2%

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

      2. un-div-inv 90.2%

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

      3. *-commutative 90.2%

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

      4. associate-*l* 90.2%

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

      5. pow2 90.2%

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

   6. Applied egg-rr 90.2%

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

   7. Taylor expanded in y around 0 93.8%

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

   8. Taylor expanded in x around 0 93.8%

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

      1. cancel-sign-sub-inv 93.8%

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

      2. metadata-eval 93.8%

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

      3. associate-*r/ 93.8%

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

      4. *-commutative 93.8%

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

      5. +-commutative 93.8%

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

      6. *-commutative 93.8%

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

      7. mul-1-neg 93.8%

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

      8. associate-*r/ 93.4%

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

      9. unsub-neg 93.4%

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

      10. *-commutative 93.4%

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

   10. Simplified 93.4%

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

   11. Taylor expanded in y around inf 90.3%

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

       1. cancel-sign-sub-inv 90.3%

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

       2. metadata-eval 90.3%

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

       3. +-commutative 90.3%

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

       4. associate-*r/ 90.3%

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

       5. associate-*l/ 90.2%

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

       6. associate-/r/ 90.2%

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

   13. Simplified 90.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -780000 \lor \neg \left(z \leq 1.16 \cdot 10^{-10}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{2}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+30} \lor \neg \left(z \leq 3.8 \cdot 10^{-116}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.25e+30) (not (<= z 3.8e-116))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.25e+30) || !(z <= 3.8e-116)) {
		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 <= (-2.25d+30)) .or. (.not. (z <= 3.8d-116))) 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 <= -2.25e+30) || !(z <= 3.8e-116)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.25e+30) or not (z <= 3.8e-116):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.25e+30) || !(z <= 3.8e-116))
		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 <= -2.25e+30) || ~((z <= 3.8e-116)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.25e+30], N[Not[LessEqual[z, 3.8e-116]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.24999999999999997e30 or 3.8000000000000001e-116 < z

   1. Initial program 79.9%

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

      1. sub-neg 79.9%

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

      2. +-commutative 79.9%

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

      3. associate-/l* 88.2%

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

      4. distribute-rgt-neg-in 88.2%

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

      5. fma-define 88.2%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in y around 0 83.2%

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

      1. mul-1-neg 83.2%

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

      2. sub-neg 83.2%

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

   7. Simplified 83.2%

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

   if -2.24999999999999997e30 < z < 3.8000000000000001e-116

   1. Initial program 85.7%

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

      1. sub-neg 85.7%

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

      2. +-commutative 85.7%

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

      3. associate-/l* 88.6%

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

      4. distribute-rgt-neg-in 88.6%

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

      5. fma-define 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in y around 0 74.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+30} \lor \neg \left(z \leq 3.8 \cdot 10^{-116}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-244}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-181}:\\ \;\;\;\;z \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.7e-244) x (if (<= x 6.8e-181) (* z (/ 2.0 t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e-244) {
		tmp = x;
	} else if (x <= 6.8e-181) {
		tmp = z * (2.0 / t);
	} 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 (x <= (-1.7d-244)) then
        tmp = x
    else if (x <= 6.8d-181) then
        tmp = z * (2.0d0 / t)
    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 (x <= -1.7e-244) {
		tmp = x;
	} else if (x <= 6.8e-181) {
		tmp = z * (2.0 / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.7e-244:
		tmp = x
	elif x <= 6.8e-181:
		tmp = z * (2.0 / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.7e-244)
		tmp = x;
	elseif (x <= 6.8e-181)
		tmp = Float64(z * Float64(2.0 / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.7e-244)
		tmp = x;
	elseif (x <= 6.8e-181)
		tmp = z * (2.0 / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.7e-244], x, If[LessEqual[x, 6.8e-181], N[(z * N[(2.0 / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000004e-244 or 6.8000000000000001e-181 < x

   1. Initial program 85.3%

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

      1. sub-neg 85.3%

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

      2. +-commutative 85.3%

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

      3. associate-/l* 91.2%

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

      4. distribute-rgt-neg-in 91.2%

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

      5. fma-define 91.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in y around 0 82.9%

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

   if -1.70000000000000004e-244 < x < 6.8000000000000001e-181

   1. Initial program 70.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in y around inf 63.7%

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

      1. associate-*r/ 63.7%

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

      2. *-commutative 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in x around 0 56.2%

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

      1. associate-*r/ 56.2%

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

      2. *-commutative 56.2%

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

   10. Simplified 56.2%

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

       1. associate-/l* 56.0%

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

   12. Applied egg-rr 56.0%

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

Alternative 6: 96.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Simplified 88.4%

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

   1. clear-num 88.0%

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

   2. un-div-inv 88.0%

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

   3. *-commutative 88.0%

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

   4. associate-*l* 88.0%

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

   5. pow2 88.0%

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

6. Applied egg-rr 88.0%

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

7. Taylor expanded in y around 0 93.9%

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

8. Taylor expanded in x around 0 93.9%

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

   1. cancel-sign-sub-inv 93.9%

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

   2. metadata-eval 93.9%

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

   3. associate-*r/ 93.9%

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

   4. *-commutative 93.9%

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

   5. +-commutative 93.9%

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

   6. *-commutative 93.9%

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

   7. mul-1-neg 93.9%

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

   8. associate-*r/ 95.8%

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

   9. unsub-neg 95.8%

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

   10. *-commutative 95.8%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 7: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-243}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.8e-243) x (if (<= x 4.8e-184) (/ y (- z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.8e-243) {
		tmp = x;
	} else if (x <= 4.8e-184) {
		tmp = 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 (x <= (-4.8d-243)) then
        tmp = x
    else if (x <= 4.8d-184) then
        tmp = 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 (x <= -4.8e-243) {
		tmp = x;
	} else if (x <= 4.8e-184) {
		tmp = y / -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.8e-243:
		tmp = x
	elif x <= 4.8e-184:
		tmp = y / -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.8e-243)
		tmp = x;
	elseif (x <= 4.8e-184)
		tmp = Float64(y / Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.8e-243)
		tmp = x;
	elseif (x <= 4.8e-184)
		tmp = y / -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.8e-243], x, If[LessEqual[x, 4.8e-184], N[(y / (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000002e-243 or 4.80000000000000049e-184 < x

   1. Initial program 85.3%

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

      1. sub-neg 85.3%

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

      2. +-commutative 85.3%

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

      3. associate-/l* 91.2%

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

      4. distribute-rgt-neg-in 91.2%

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

      5. fma-define 91.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in y around 0 82.9%

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

   if -4.8000000000000002e-243 < x < 4.80000000000000049e-184

   1. Initial program 70.5%

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

      1. sub-neg 70.5%

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

      2. +-commutative 70.5%

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

      3. associate-/l* 75.5%

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

      4. distribute-rgt-neg-in 75.5%

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

      5. fma-define 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in y around 0 39.4%

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

      1. mul-1-neg 39.4%

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

      2. sub-neg 39.4%

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

   7. Simplified 39.4%

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

   8. Taylor expanded in x around 0 37.2%

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

      1. neg-mul-1 37.2%

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

      2. distribute-neg-frac2 37.2%

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

   10. Simplified 37.2%

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

Alternative 8: 75.4% 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 82.7%

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

   1. sub-neg 82.7%

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

   2. +-commutative 82.7%

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

   3. associate-/l* 88.4%

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

   4. distribute-rgt-neg-in 88.4%

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

   5. fma-define 88.4%

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

3. Simplified 89.2%

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

5. Taylor expanded in y around 0 70.0%

   \[\leadsto \color{blue}{x} \]
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 2024158 
(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)))))