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

Percentage Accurate: 81.4% → 98.5%

Time: 10.6s

Alternatives: 8

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: 81.4% 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: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Simplified 97.5%

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

5. Final simplification 97.5%

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

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \left(y \cdot 2\right)}{z \cdot \left(z \cdot 2\right) - y \cdot t}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-242}:\\ \;\;\;\;x - t_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z (* y 2.0)) (- (* z (* z 2.0)) (* y t)))))
   (if (<= t_1 -2e-242)
     (- x t_1)
     (- x (/ (* y 2.0) (- (* z 2.0) (* t (/ y z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * (y * 2.0)) / ((z * (z * 2.0)) - (y * t));
	double tmp;
	if (t_1 <= -2e-242) {
		tmp = x - t_1;
	} else {
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (y / 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) :: t_1
    real(8) :: tmp
    t_1 = (z * (y * 2.0d0)) / ((z * (z * 2.0d0)) - (y * t))
    if (t_1 <= (-2d-242)) then
        tmp = x - t_1
    else
        tmp = x - ((y * 2.0d0) / ((z * 2.0d0) - (t * (y / z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * (y * 2.0)) / ((z * (z * 2.0)) - (y * t));
	double tmp;
	if (t_1 <= -2e-242) {
		tmp = x - t_1;
	} else {
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (y / z))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * (y * 2.0)) / ((z * (z * 2.0)) - (y * t))
	tmp = 0
	if t_1 <= -2e-242:
		tmp = x - t_1
	else:
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (y / z))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * Float64(y * 2.0)) / Float64(Float64(z * Float64(z * 2.0)) - Float64(y * t)))
	tmp = 0.0
	if (t_1 <= -2e-242)
		tmp = Float64(x - t_1);
	else
		tmp = Float64(x - Float64(Float64(y * 2.0) / Float64(Float64(z * 2.0) - Float64(t * Float64(y / z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * (y * 2.0)) / ((z * (z * 2.0)) - (y * t));
	tmp = 0.0;
	if (t_1 <= -2e-242)
		tmp = x - t_1;
	else
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (y / z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * N[(y * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-242], N[(x - t$95$1), $MachinePrecision], 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. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))) < -2e-242

   1. Initial program 95.5%

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

   if -2e-242 < (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))

   1. Initial program 76.7%

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

      1. associate-/l* 82.6%

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

      2. associate-*l* 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]
   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. *-commutative 94.5%

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

      4. associate-*l/ 97.3%

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

      5. unsub-neg 97.3%

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

      6. *-commutative 97.3%

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

      7. *-commutative 97.3%

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

   7. Simplified 97.3%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(y \cdot 2\right)}{z \cdot \left(z \cdot 2\right) - y \cdot t} \leq -2 \cdot 10^{-242}:\\ \;\;\;\;x - \frac{z \cdot \left(y \cdot 2\right)}{z \cdot \left(z \cdot 2\right) - y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+53} \lor \neg \left(z \leq 1.1 \cdot 10^{+121}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.3e+53) (not (<= z 1.1e+121)))
   (- x (/ y z))
   (+ x (* z (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.3e+53) || !(z <= 1.1e+121)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.3d+53)) .or. (.not. (z <= 1.1d+121))) then
        tmp = x - (y / z)
    else
        tmp = x + (z * (2.0d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.3e+53) || !(z <= 1.1e+121)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.3e+53) or not (z <= 1.1e+121):
		tmp = x - (y / z)
	else:
		tmp = x + (z * (2.0 / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.3e+53) || !(z <= 1.1e+121))
		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 <= -2.3e+53) || ~((z <= 1.1e+121)))
		tmp = x - (y / z);
	else
		tmp = x + (z * (2.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.3e+53], N[Not[LessEqual[z, 1.1e+121]], $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 < -2.3000000000000002e53 or 1.10000000000000001e121 < z

   1. Initial program 61.4%

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

      1. sub-neg 61.4%

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

      2. associate-/l* 72.8%

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

      3. distribute-neg-frac 72.8%

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

      4. distribute-lft-neg-out 72.8%

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

      5. associate-/r/ 72.8%

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

      6. distribute-lft-neg-out 72.8%

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

      7. distribute-rgt-neg-in 72.8%

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

      8. metadata-eval 72.8%

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

      9. *-commutative 72.8%

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

      10. associate-*l* 72.8%

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

      11. fma-neg 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in y around 0 96.8%

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

      1. mul-1-neg 96.8%

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

      2. sub-neg 96.8%

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

   7. Simplified 96.8%

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

   if -2.3000000000000002e53 < z < 1.10000000000000001e121

   1. Initial program 89.4%

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

      1. sub-neg 89.4%

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

      2. associate-/l* 90.7%

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

      3. distribute-neg-frac 90.7%

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

      4. distribute-lft-neg-out 90.7%

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

      5. associate-/r/ 92.8%

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

      6. distribute-lft-neg-out 92.8%

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

      7. distribute-rgt-neg-in 92.8%

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

      8. metadata-eval 92.8%

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

      9. *-commutative 92.8%

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

      10. associate-*l* 92.8%

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

      11. fma-neg 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in y around inf 86.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+53} \lor \neg \left(z \leq 1.1 \cdot 10^{+121}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+52} \lor \neg \left(z \leq 1.1 \cdot 10^{+121}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.95e+52) (not (<= z 1.1e+121)))
   (- x (/ y z))
   (- x (* -2.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.95e+52) || !(z <= 1.1e+121)) {
		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 <= (-1.95d+52)) .or. (.not. (z <= 1.1d+121))) 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 <= -1.95e+52) || !(z <= 1.1e+121)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (-2.0 * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.95e+52) or not (z <= 1.1e+121):
		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 <= -1.95e+52) || !(z <= 1.1e+121))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(-2.0 * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.95e+52) || ~((z <= 1.1e+121)))
		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, -1.95e+52], N[Not[LessEqual[z, 1.1e+121]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95e52 or 1.10000000000000001e121 < z

   1. Initial program 61.4%

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

      1. sub-neg 61.4%

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

      2. associate-/l* 72.8%

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

      3. distribute-neg-frac 72.8%

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

      4. distribute-lft-neg-out 72.8%

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

      5. associate-/r/ 72.8%

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

      6. distribute-lft-neg-out 72.8%

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

      7. distribute-rgt-neg-in 72.8%

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

      8. metadata-eval 72.8%

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

      9. *-commutative 72.8%

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

      10. associate-*l* 72.8%

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

      11. fma-neg 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in y around 0 96.8%

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

      1. mul-1-neg 96.8%

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

      2. sub-neg 96.8%

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

   7. Simplified 96.8%

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

   if -1.95e52 < z < 1.10000000000000001e121

   1. Initial program 89.4%

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

      1. associate-/l* 90.7%

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

      2. associate-*l* 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around inf 86.7%

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

      1. *-commutative 86.7%

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

   7. Simplified 86.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+52} \lor \neg \left(z \leq 1.1 \cdot 10^{+121}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+55} \lor \neg \left(z \leq 1.1 \cdot 10^{+121}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9.6e+55) (not (<= z 1.1e+121))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.6e+55) || !(z <= 1.1e+121)) {
		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 <= (-9.6d+55)) .or. (.not. (z <= 1.1d+121))) 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 <= -9.6e+55) || !(z <= 1.1e+121)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -9.6e+55) or not (z <= 1.1e+121):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -9.6e+55) || !(z <= 1.1e+121))
		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 <= -9.6e+55) || ~((z <= 1.1e+121)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -9.6e+55], N[Not[LessEqual[z, 1.1e+121]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -9.5999999999999997e55 or 1.10000000000000001e121 < z

   1. Initial program 61.6%

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

      1. sub-neg 61.6%

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

      2. associate-/l* 72.1%

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

      3. distribute-neg-frac 72.1%

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

      4. distribute-lft-neg-out 72.1%

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

      5. associate-/r/ 72.1%

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

      6. distribute-lft-neg-out 72.1%

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

      7. distribute-rgt-neg-in 72.1%

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

      8. metadata-eval 72.1%

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

      9. *-commutative 72.1%

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

      10. associate-*l* 72.1%

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

      11. fma-neg 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. mul-1-neg 98.0%

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

      2. sub-neg 98.0%

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

   7. Simplified 98.0%

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

   if -9.5999999999999997e55 < z < 1.10000000000000001e121

   1. Initial program 89.0%

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

      1. sub-neg 89.0%

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

      2. associate-/l* 90.8%

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

      3. distribute-neg-frac 90.8%

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

      4. distribute-lft-neg-out 90.8%

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

      5. associate-/r/ 92.9%

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

      6. distribute-lft-neg-out 92.9%

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

      7. distribute-rgt-neg-in 92.9%

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

      8. metadata-eval 92.9%

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

      9. *-commutative 92.9%

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

      10. associate-*l* 92.9%

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

      11. fma-neg 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around inf 75.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+55} \lor \neg \left(z \leq 1.1 \cdot 10^{+121}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% 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 80.4%

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

   1. associate-/l* 85.0%

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

   2. associate-*l* 85.0%

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

3. Simplified 85.0%

   \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]
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. *-commutative 94.5%

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

   4. associate-*l/ 95.2%

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

   5. unsub-neg 95.2%

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

   6. *-commutative 95.2%

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

   7. *-commutative 95.2%

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

7. Simplified 95.2%

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

8. Final simplification 95.2%

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

Alternative 7: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-275}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-217}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.2e-275) x (if (<= x 1.7e-217) (/ (- y) z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.2e-275) {
		tmp = x;
	} else if (x <= 1.7e-217) {
		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 <= (-3.2d-275)) then
        tmp = x
    else if (x <= 1.7d-217) 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 <= -3.2e-275) {
		tmp = x;
	} else if (x <= 1.7e-217) {
		tmp = -y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.2e-275:
		tmp = x
	elif x <= 1.7e-217:
		tmp = -y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.2e-275)
		tmp = x;
	elseif (x <= 1.7e-217)
		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 <= -3.2e-275)
		tmp = x;
	elseif (x <= 1.7e-217)
		tmp = -y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.2e-275], x, If[LessEqual[x, 1.7e-217], N[((-y) / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2e-275 or 1.70000000000000008e-217 < x

   1. Initial program 83.3%

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

      1. sub-neg 83.3%

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

      2. associate-/l* 87.8%

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

      3. distribute-neg-frac 87.8%

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

      4. distribute-lft-neg-out 87.8%

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

      5. associate-/r/ 88.9%

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

      6. distribute-lft-neg-out 88.9%

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

      7. distribute-rgt-neg-in 88.9%

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

      8. metadata-eval 88.9%

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

      9. *-commutative 88.9%

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

      10. associate-*l* 88.9%

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

      11. fma-neg 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in x around inf 79.9%

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

   if -3.2e-275 < x < 1.70000000000000008e-217

   1. Initial program 46.5%

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

      1. associate-/l* 51.9%

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

      2. associate-*l* 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in z around 0 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. *-commutative 80.7%

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

      4. associate-*l/ 80.8%

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

      5. unsub-neg 80.8%

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

      6. *-commutative 80.8%

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

      7. *-commutative 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in x around 0 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-*r/ 71.5%

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

      3. *-commutative 71.5%

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

   10. Simplified 71.5%

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

   11. Taylor expanded in y around 0 56.4%

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

       1. mul-1-neg 56.4%

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

       2. distribute-frac-neg 56.4%

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

   13. Simplified 56.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-275}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-217}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

double code(double x, double y, double z, double t) {
	return x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

public static double code(double x, double y, double z, double t) {
	return x;
}

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 80.4%

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

   1. sub-neg 80.4%

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

   2. associate-/l* 85.0%

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

   3. distribute-neg-frac 85.0%

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

   4. distribute-lft-neg-out 85.0%

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

   5. associate-/r/ 86.4%

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

   6. distribute-lft-neg-out 86.4%

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

   7. distribute-rgt-neg-in 86.4%

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

   8. metadata-eval 86.4%

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

   9. *-commutative 86.4%

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

   10. associate-*l* 86.4%

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

   11. fma-neg 86.4%

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

3. Simplified 86.4%

   \[\leadsto \color{blue}{x + \frac{y \cdot -2}{\mathsf{fma}\left(2, z \cdot z, -y \cdot t\right)} \cdot z} \]
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 2024024 
(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)))))