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

Percentage Accurate: 81.8% → 98.3%

Time: 9.7s

Alternatives: 6

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: 81.8% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. remove-double-neg 84.3%

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

   2. neg-mul-1 84.3%

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

   3. *-commutative 84.3%

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

   4. *-commutative 84.3%

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

   5. neg-mul-1 84.3%

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

   6. remove-double-neg 84.3%

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

   7. associate-/l* 91.7%

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

   8. associate-*l* 91.7%

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

3. Simplified 91.7%

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

5. Taylor expanded in z around 0 94.5%

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

   1. +-commutative 94.5%

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

   2. mul-1-neg 94.5%

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

   3. associate-*r/ 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-/r/ 98.3%

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

   6. unsub-neg 98.3%

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

   7. *-commutative 98.3%

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

   8. associate-/r/ 97.1%

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

   9. *-commutative 97.1%

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

   10. associate-*r/ 94.5%

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

   11. *-commutative 94.5%

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

   12. associate-*r/ 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 2: 88.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+16} \lor \neg \left(z \leq 2.5 \cdot 10^{-61}\right) \land \left(z \leq 1400000000000 \lor \neg \left(z \leq 1.26 \cdot 10^{+51}\right)\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.6e+16)
         (and (not (<= z 2.5e-61))
              (or (<= z 1400000000000.0) (not (<= z 1.26e+51)))))
   (- x (/ y z))
   (+ x (* z (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.6e+16) || (!(z <= 2.5e-61) && ((z <= 1400000000000.0) || !(z <= 1.26e+51)))) {
		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 <= (-3.6d+16)) .or. (.not. (z <= 2.5d-61)) .and. (z <= 1400000000000.0d0) .or. (.not. (z <= 1.26d+51))) 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 <= -3.6e+16) || (!(z <= 2.5e-61) && ((z <= 1400000000000.0) || !(z <= 1.26e+51)))) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.6e+16) or (not (z <= 2.5e-61) and ((z <= 1400000000000.0) or not (z <= 1.26e+51))):
		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 <= -3.6e+16) || (!(z <= 2.5e-61) && ((z <= 1400000000000.0) || !(z <= 1.26e+51))))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x + Float64(z * Float64(2.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.6e+16) || (~((z <= 2.5e-61)) && ((z <= 1400000000000.0) || ~((z <= 1.26e+51)))))
		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, -3.6e+16], And[N[Not[LessEqual[z, 2.5e-61]], $MachinePrecision], Or[LessEqual[z, 1400000000000.0], N[Not[LessEqual[z, 1.26e+51]], $MachinePrecision]]]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6e16 or 2.4999999999999999e-61 < z < 1.4e12 or 1.25999999999999997e51 < z

   1. Initial program 71.9%

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

      1. sub-neg 71.9%

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

      2. associate-/l* 87.4%

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

      3. distribute-neg-frac 87.4%

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

      4. distribute-lft-neg-out 87.4%

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

      5. associate-/r/ 86.6%

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

      6. distribute-lft-neg-out 86.6%

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

      7. distribute-rgt-neg-in 86.6%

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

      8. metadata-eval 86.6%

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

      9. *-commutative 86.6%

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

      10. associate-*l* 86.6%

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

      11. fma-neg 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in y around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. sub-neg 86.8%

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

   7. Simplified 86.8%

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

   if -3.6e16 < z < 2.4999999999999999e-61 or 1.4e12 < z < 1.25999999999999997e51

   1. Initial program 94.8%

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

      1. sub-neg 94.8%

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

      2. associate-/l* 95.3%

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

      3. distribute-neg-frac 95.3%

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

      4. distribute-lft-neg-out 95.3%

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

      5. associate-/r/ 97.0%

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

      6. distribute-lft-neg-out 97.0%

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

      7. distribute-rgt-neg-in 97.0%

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

      8. metadata-eval 97.0%

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

      9. *-commutative 97.0%

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

      10. associate-*l* 97.0%

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

      11. fma-neg 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in y around inf 92.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+16} \lor \neg \left(z \leq 2.5 \cdot 10^{-61}\right) \land \left(z \leq 1400000000000 \lor \neg \left(z \leq 1.26 \cdot 10^{+51}\right)\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+14} \lor \neg \left(z \leq 2.5 \cdot 10^{-61} \lor \neg \left(z \leq 1800000000000\right) \land z \leq 1.85 \cdot 10^{+58}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.8e+14)
         (not
          (or (<= z 2.5e-61)
              (and (not (<= z 1800000000000.0)) (<= z 1.85e+58)))))
   (- x (/ y z))
   (- x (* (/ z t) -2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.8e+14) || !((z <= 2.5e-61) || (!(z <= 1800000000000.0) && (z <= 1.85e+58)))) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z / t) * -2.0);
	}
	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+14)) .or. (.not. (z <= 2.5d-61) .or. (.not. (z <= 1800000000000.0d0)) .and. (z <= 1.85d+58))) then
        tmp = x - (y / z)
    else
        tmp = x - ((z / t) * (-2.0d0))
    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+14) || !((z <= 2.5e-61) || (!(z <= 1800000000000.0) && (z <= 1.85e+58)))) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z / t) * -2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.8e+14) or not ((z <= 2.5e-61) or (not (z <= 1800000000000.0) and (z <= 1.85e+58))):
		tmp = x - (y / z)
	else:
		tmp = x - ((z / t) * -2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.8e+14) || !((z <= 2.5e-61) || (!(z <= 1800000000000.0) && (z <= 1.85e+58))))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(z / t) * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.8e+14) || ~(((z <= 2.5e-61) || (~((z <= 1800000000000.0)) && (z <= 1.85e+58)))))
		tmp = x - (y / z);
	else
		tmp = x - ((z / t) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.8e+14], N[Not[Or[LessEqual[z, 2.5e-61], And[N[Not[LessEqual[z, 1800000000000.0]], $MachinePrecision], LessEqual[z, 1.85e+58]]]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e14 or 2.4999999999999999e-61 < z < 1.8e12 or 1.8500000000000001e58 < z

   1. Initial program 71.9%

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

      1. sub-neg 71.9%

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

      2. associate-/l* 87.4%

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

      3. distribute-neg-frac 87.4%

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

      4. distribute-lft-neg-out 87.4%

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

      5. associate-/r/ 86.6%

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

      6. distribute-lft-neg-out 86.6%

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

      7. distribute-rgt-neg-in 86.6%

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

      8. metadata-eval 86.6%

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

      9. *-commutative 86.6%

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

      10. associate-*l* 86.6%

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

      11. fma-neg 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in y around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. sub-neg 86.8%

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

   7. Simplified 86.8%

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

   if -2.8e14 < z < 2.4999999999999999e-61 or 1.8e12 < z < 1.8500000000000001e58

   1. Initial program 94.8%

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

      1. remove-double-neg 94.8%

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

      2. neg-mul-1 94.8%

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

      3. *-commutative 94.8%

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

      4. *-commutative 94.8%

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

      5. neg-mul-1 94.8%

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

      6. remove-double-neg 94.8%

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

      7. associate-/l* 95.3%

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

      8. associate-*l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in y around inf 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+14} \lor \neg \left(z \leq 2.5 \cdot 10^{-61} \lor \neg \left(z \leq 1800000000000\right) \land z \leq 1.85 \cdot 10^{+58}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. remove-double-neg 84.3%

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

   2. neg-mul-1 84.3%

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

   3. *-commutative 84.3%

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

   4. *-commutative 84.3%

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

   5. neg-mul-1 84.3%

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

   6. remove-double-neg 84.3%

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

   7. associate-/l* 91.7%

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

   8. associate-*l* 91.7%

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

3. Simplified 91.7%

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

5. Taylor expanded in z around 0 94.5%

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

   1. +-commutative 94.5%

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

   2. mul-1-neg 94.5%

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

   3. associate-*r/ 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-/r/ 98.3%

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

   6. unsub-neg 98.3%

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

   7. *-commutative 98.3%

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

   8. associate-/r/ 97.1%

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

   9. *-commutative 97.1%

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

   10. associate-*r/ 94.5%

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

   11. *-commutative 94.5%

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

   12. associate-*r/ 98.2%

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

7. Simplified 98.2%

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

   1. expm1-log1p-u 88.9%

      \[\leadsto x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}}\right)\right)} \]

   2. expm1-udef 76.6%

      \[\leadsto x - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right)} \]

   3. *-commutative 76.6%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\frac{y \cdot 2}{\color{blue}{2 \cdot z} - y \cdot \frac{t}{z}}\right)} - 1\right) \]

9. Applied egg-rr 76.6%

   \[\leadsto x - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y \cdot 2}{2 \cdot z - y \cdot \frac{t}{z}}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 88.9%

       \[\leadsto x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot 2}{2 \cdot z - y \cdot \frac{t}{z}}\right)\right)} \]

    2. expm1-log1p 98.2%

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

    3. associate-*r/ 98.2%

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

    4. *-commutative 98.2%

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

    5. associate-*r/ 94.4%

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

    6. associate-/l* 98.2%

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

11. Simplified 98.2%

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

12. Final simplification 98.2%

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

Alternative 5: 80.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-141} \lor \neg \left(z \leq 1.55 \cdot 10^{-76}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.5e-141) (not (<= z 1.55e-76))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.5e-141) || !(z <= 1.55e-76)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-8.5d-141)) .or. (.not. (z <= 1.55d-76))) then
        tmp = x - (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.5e-141) || !(z <= 1.55e-76)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.5e-141) or not (z <= 1.55e-76):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.5e-141) || !(z <= 1.55e-76))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -8.5e-141) || ~((z <= 1.55e-76)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.5e-141], N[Not[LessEqual[z, 1.55e-76]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -8.50000000000000021e-141 or 1.54999999999999985e-76 < z

   1. Initial program 78.3%

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

      1. sub-neg 78.3%

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

      2. associate-/l* 90.5%

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

      3. distribute-neg-frac 90.5%

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

      4. distribute-lft-neg-out 90.5%

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

      5. associate-/r/ 89.9%

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

      6. distribute-lft-neg-out 89.9%

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

      7. distribute-rgt-neg-in 89.9%

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

      8. metadata-eval 89.9%

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

      9. *-commutative 89.9%

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

      10. associate-*l* 89.9%

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

      11. fma-neg 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around 0 78.8%

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

      1. mul-1-neg 78.8%

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

      2. sub-neg 78.8%

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

   7. Simplified 78.8%

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

   if -8.50000000000000021e-141 < z < 1.54999999999999985e-76

   1. Initial program 93.8%

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

      1. sub-neg 93.8%

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

      2. associate-/l* 93.5%

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

      3. distribute-neg-frac 93.5%

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

      4. distribute-lft-neg-out 93.5%

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

      5. associate-/r/ 95.9%

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

      6. distribute-lft-neg-out 95.9%

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

      7. distribute-rgt-neg-in 95.9%

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

      8. metadata-eval 95.9%

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

      9. *-commutative 95.9%

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

      10. associate-*l* 95.9%

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

      11. fma-neg 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in x around inf 82.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-141} \lor \neg \left(z \leq 1.55 \cdot 10^{-76}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% 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 84.3%

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

   1. sub-neg 84.3%

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

   2. associate-/l* 91.7%

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

   3. distribute-neg-frac 91.7%

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

   4. distribute-lft-neg-out 91.7%

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

   5. associate-/r/ 92.2%

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

   6. distribute-lft-neg-out 92.2%

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

   7. distribute-rgt-neg-in 92.2%

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

   8. metadata-eval 92.2%

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

   9. *-commutative 92.2%

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

   10. associate-*l* 92.2%

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

   11. fma-neg 92.2%

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

3. Simplified 92.2%

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

5. Taylor expanded in x around inf 74.7%

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

6. Final simplification 74.7%

   \[\leadsto x \]
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 2024019 
(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)))))