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

Percentage Accurate: 81.7% → 98.3%

Time: 10.0s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.1× 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 83.2%

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

   1. sub-neg 83.2%

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

   2. associate-/l* 89.5%

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

   3. distribute-neg-frac 89.5%

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

   4. distribute-rgt-neg-in 89.5%

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

   5. metadata-eval 89.5%

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

   6. div-sub 89.5%

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

   7. *-commutative 89.5%

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

   8. associate-/l* 95.6%

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

   9. associate-/r/ 95.6%

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

   10. *-inverses 95.6%

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

   11. *-lft-identity 95.6%

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

   12. *-commutative 95.6%

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

   13. *-commutative 95.6%

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

   14. associate-/l* 97.1%

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

   15. associate-/r/ 98.5%

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

3. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-26} \lor \neg \left(z \leq 2.3 \cdot 10^{-34}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.3e-26) (not (<= z 2.3e-34)))
   (- x (/ y z))
   (+ x (* z (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.3e-26) || !(z <= 2.3e-34)) {
		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 <= (-4.3d-26)) .or. (.not. (z <= 2.3d-34))) 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 <= -4.3e-26) || !(z <= 2.3e-34)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.3e-26) or not (z <= 2.3e-34):
		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 <= -4.3e-26) || !(z <= 2.3e-34))
		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 <= -4.3e-26) || ~((z <= 2.3e-34)))
		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, -4.3e-26], N[Not[LessEqual[z, 2.3e-34]], $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 < -4.29999999999999988e-26 or 2.30000000000000011e-34 < z

   1. Initial program 77.3%

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

      1. sub-neg 77.3%

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

      2. associate-*l* 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-/l* 87.1%

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

      5. associate-/r/ 86.4%

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

      6. distribute-rgt-neg-in 86.4%

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

      7. *-commutative 86.4%

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

      8. associate-*l* 86.4%

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

      9. fma-neg 86.4%

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

      10. distribute-rgt-neg-in 86.4%

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

      11. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in y around 0 87.8%

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

      1. mul-1-neg 87.8%

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

      2. sub-neg 87.8%

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

   7. Simplified 87.8%

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

   if -4.29999999999999988e-26 < z < 2.30000000000000011e-34

   1. Initial program 90.3%

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

      1. sub-neg 90.3%

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

      2. associate-*l* 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-/l* 91.4%

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

      5. associate-/r/ 92.9%

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

      6. distribute-rgt-neg-in 92.9%

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

      7. *-commutative 92.9%

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

      8. associate-*l* 92.9%

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

      9. fma-neg 92.9%

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

      10. distribute-rgt-neg-in 92.9%

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

      11. metadata-eval 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in y around inf 95.2%

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

      1. associate-*l/ 95.3%

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

      2. neg-mul-1 95.3%

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

      3. *-commutative 95.3%

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

      4. distribute-lft-neg-in 95.3%

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

      5. metadata-eval 95.3%

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

      6. associate-*r/ 95.3%

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

      7. clear-num 95.2%

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

      8. un-div-inv 95.2%

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

   7. Applied egg-rr 95.2%

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

      1. associate-/r/ 95.2%

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

   9. Simplified 95.2%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-26} \lor \neg \left(z \leq 2.3 \cdot 10^{-34}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-26} \lor \neg \left(z \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-2 \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.5e-26) (not (<= z 2e-34)))
   (- x (/ y z))
   (- x (/ (* -2.0 z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.5e-26) || !(z <= 2e-34)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((-2.0 * z) / 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 <= (-4.5d-26)) .or. (.not. (z <= 2d-34))) then
        tmp = x - (y / z)
    else
        tmp = x - (((-2.0d0) * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.5e-26) || !(z <= 2e-34)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((-2.0 * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.5e-26) or not (z <= 2e-34):
		tmp = x - (y / z)
	else:
		tmp = x - ((-2.0 * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.5e-26) || !(z <= 2e-34))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(-2.0 * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.5e-26) || ~((z <= 2e-34)))
		tmp = x - (y / z);
	else
		tmp = x - ((-2.0 * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.5e-26], N[Not[LessEqual[z, 2e-34]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(-2.0 * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4999999999999999e-26 or 1.99999999999999986e-34 < z

   1. Initial program 77.3%

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

      1. sub-neg 77.3%

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

      2. associate-*l* 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-/l* 87.1%

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

      5. associate-/r/ 86.4%

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

      6. distribute-rgt-neg-in 86.4%

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

      7. *-commutative 86.4%

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

      8. associate-*l* 86.4%

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

      9. fma-neg 86.4%

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

      10. distribute-rgt-neg-in 86.4%

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

      11. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in y around 0 87.8%

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

      1. mul-1-neg 87.8%

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

      2. sub-neg 87.8%

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

   7. Simplified 87.8%

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

   if -4.4999999999999999e-26 < z < 1.99999999999999986e-34

   1. Initial program 90.3%

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

   3. Taylor expanded in y around inf 95.3%

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

      1. associate-*r/ 95.3%

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

      2. *-commutative 95.3%

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

   5. Simplified 95.3%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-26} \lor \neg \left(z \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-2 \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-50} \lor \neg \left(z \leq 1.3 \cdot 10^{+177}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.55e-50) (not (<= z 1.3e+177))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.55e-50) || !(z <= 1.3e+177)) {
		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 <= (-1.55d-50)) .or. (.not. (z <= 1.3d+177))) 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 <= -1.55e-50) || !(z <= 1.3e+177)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.55e-50) or not (z <= 1.3e+177):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.55e-50) || !(z <= 1.3e+177))
		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 <= -1.55e-50) || ~((z <= 1.3e+177)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.55e-50], N[Not[LessEqual[z, 1.3e+177]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.5500000000000001e-50 or 1.2999999999999999e177 < z

   1. Initial program 70.4%

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

      1. sub-neg 70.4%

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

      2. associate-*l* 69.4%

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

      3. *-commutative 69.4%

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

      4. associate-/l* 82.8%

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

      5. associate-/r/ 81.9%

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

      6. distribute-rgt-neg-in 81.9%

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

      7. *-commutative 81.9%

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

      8. associate-*l* 81.9%

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

      9. fma-neg 81.9%

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

      10. distribute-rgt-neg-in 81.9%

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

      11. metadata-eval 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in y around 0 89.0%

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

      1. mul-1-neg 89.0%

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

      2. sub-neg 89.0%

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

   7. Simplified 89.0%

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

   if -1.5500000000000001e-50 < z < 1.2999999999999999e177

   1. Initial program 92.1%

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

      1. sub-neg 92.1%

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

      2. associate-*l* 92.1%

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

      3. *-commutative 92.1%

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

      4. associate-/l* 93.4%

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

      5. associate-/r/ 94.5%

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

      6. distribute-rgt-neg-in 94.5%

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

      7. *-commutative 94.5%

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

      8. associate-*l* 94.5%

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

      9. fma-neg 94.5%

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

      10. distribute-rgt-neg-in 94.5%

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

      11. metadata-eval 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around inf 76.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-50} \lor \neg \left(z \leq 1.3 \cdot 10^{+177}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-260}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.7e-260) x (if (<= x 9e-241) (* 2.0 (/ z t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.7e-260) {
		tmp = x;
	} else if (x <= 9e-241) {
		tmp = 2.0 * (z / 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 <= (-5.7d-260)) then
        tmp = x
    else if (x <= 9d-241) then
        tmp = 2.0d0 * (z / 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 <= -5.7e-260) {
		tmp = x;
	} else if (x <= 9e-241) {
		tmp = 2.0 * (z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.7e-260:
		tmp = x
	elif x <= 9e-241:
		tmp = 2.0 * (z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.7e-260)
		tmp = x;
	elseif (x <= 9e-241)
		tmp = Float64(2.0 * Float64(z / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -5.7e-260)
		tmp = x;
	elseif (x <= 9e-241)
		tmp = 2.0 * (z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.7e-260], x, If[LessEqual[x, 9e-241], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6999999999999998e-260 or 8.9999999999999997e-241 < x

   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* 83.8%

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

      3. *-commutative 83.8%

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

      4. associate-/l* 90.7%

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

      5. associate-/r/ 91.0%

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

      6. distribute-rgt-neg-in 91.0%

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

      7. *-commutative 91.0%

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

      8. associate-*l* 91.0%

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

      9. fma-neg 91.0%

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

      10. distribute-rgt-neg-in 91.0%

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

      11. metadata-eval 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in x around inf 82.2%

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

   if -5.6999999999999998e-260 < x < 8.9999999999999997e-241

   1. Initial program 74.6%

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

   3. Taylor expanded in y around inf 52.7%

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

      1. associate-*r/ 52.7%

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

      2. *-commutative 52.7%

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

   5. Simplified 52.7%

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

   6. Taylor expanded in x around 0 52.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-260}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-276}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-227}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.5e-276) x (if (<= x 2.05e-227) (- (/ y z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.5e-276) {
		tmp = x;
	} else if (x <= 2.05e-227) {
		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 <= (-1.5d-276)) then
        tmp = x
    else if (x <= 2.05d-227) 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 <= -1.5e-276) {
		tmp = x;
	} else if (x <= 2.05e-227) {
		tmp = -(y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.5e-276:
		tmp = x
	elif x <= 2.05e-227:
		tmp = -(y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.5e-276)
		tmp = x;
	elseif (x <= 2.05e-227)
		tmp = Float64(-Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.5e-276)
		tmp = x;
	elseif (x <= 2.05e-227)
		tmp = -(y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.5e-276], x, If[LessEqual[x, 2.05e-227], (-N[(y / z), $MachinePrecision]), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999994e-276 or 2.05000000000000005e-227 < x

   1. Initial program 84.6%

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

      1. sub-neg 84.6%

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

      2. associate-*l* 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-/l* 90.4%

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

      5. associate-/r/ 90.7%

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

      6. distribute-rgt-neg-in 90.7%

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

      7. *-commutative 90.7%

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

      8. associate-*l* 90.7%

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

      9. fma-neg 90.7%

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

      10. distribute-rgt-neg-in 90.7%

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

      11. metadata-eval 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in x around inf 82.0%

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

   if -1.49999999999999994e-276 < x < 2.05000000000000005e-227

   1. Initial program 70.1%

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

      1. sub-neg 70.1%

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

      2. associate-/l* 77.0%

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

      3. distribute-neg-frac 77.0%

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

      4. distribute-rgt-neg-in 77.0%

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

      5. metadata-eval 77.0%

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

      6. div-sub 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-/l* 84.7%

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

      9. associate-/r/ 84.7%

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

      10. *-inverses 84.7%

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

      11. *-lft-identity 84.7%

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

      12. *-commutative 84.7%

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

      13. *-commutative 84.7%

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

      14. associate-/l* 88.1%

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

      15. associate-/r/ 92.4%

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

   3. Simplified 92.4%

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

      1. +-commutative 92.4%

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

      2. div-inv 92.2%

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

      3. fma-define 92.2%

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

      4. *-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in z around inf 53.4%

      \[\leadsto \mathsf{fma}\left(y \cdot -2, \color{blue}{\frac{0.5}{z}}, x\right) \]

   8. Taylor expanded in y around inf 45.9%

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

      1. associate-*r/ 45.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot y}{z}} \]

      2. neg-mul-1 45.9%

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

   10. Simplified 45.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-276}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-227}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 83.2%

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

   1. sub-neg 83.2%

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

   2. associate-*l* 82.8%

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

   3. *-commutative 82.8%

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

   4. associate-/l* 89.1%

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

   5. associate-/r/ 89.3%

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

   6. distribute-rgt-neg-in 89.3%

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

   7. *-commutative 89.3%

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

   8. associate-*l* 89.3%

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

   9. fma-neg 89.3%

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

   10. distribute-rgt-neg-in 89.3%

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

   11. metadata-eval 89.3%

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

3. Simplified 89.3%

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

5. Taylor expanded in x around inf 75.2%

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

6. Final simplification 75.2%

   \[\leadsto x \]
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 2024034 
(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)))))