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

Percentage Accurate: 82.0% → 98.4%

Time: 11.4s

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: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 88.9%

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

   1. clear-num 88.9%

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

   2. un-div-inv 88.9%

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

   3. associate-*r* 88.9%

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

   4. *-commutative 88.9%

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

   5. associate-*l* 88.9%

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

   6. pow2 88.9%

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

6. Applied egg-rr 88.9%

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

7. Taylor expanded in y around inf 97.2%

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

   1. +-commutative 97.2%

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

   2. mul-1-neg 97.2%

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

   3. unsub-neg 97.2%

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

   4. associate-*r/ 97.2%

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

   5. *-commutative 97.2%

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

9. Simplified 97.2%

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

10. Final simplification 97.2%

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

Alternative 2: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-139} \lor \neg \left(z \leq 1.55 \cdot 10^{-200}\right):\\ \;\;\;\;x + \frac{y \cdot 2}{\frac{y \cdot t}{z} - 2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.6e-139) (not (<= z 1.55e-200)))
   (+ x (/ (* y 2.0) (- (/ (* y t) z) (* 2.0 z))))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.6e-139) || !(z <= 1.55e-200)) {
		tmp = x + ((y * 2.0) / (((y * t) / z) - (2.0 * 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 <= (-1.6d-139)) .or. (.not. (z <= 1.55d-200))) then
        tmp = x + ((y * 2.0d0) / (((y * t) / z) - (2.0d0 * 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 <= -1.6e-139) || !(z <= 1.55e-200)) {
		tmp = x + ((y * 2.0) / (((y * t) / z) - (2.0 * z)));
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.6e-139) or not (z <= 1.55e-200):
		tmp = x + ((y * 2.0) / (((y * t) / z) - (2.0 * z)))
	else:
		tmp = x - ((z * -2.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.6e-139) || !(z <= 1.55e-200))
		tmp = Float64(x + Float64(Float64(y * 2.0) / Float64(Float64(Float64(y * t) / z) - Float64(2.0 * z))));
	else
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.6e-139) || ~((z <= 1.55e-200)))
		tmp = x + ((y * 2.0) / (((y * t) / z) - (2.0 * z)));
	else
		tmp = x - ((z * -2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.6e-139], N[Not[LessEqual[z, 1.55e-200]], $MachinePrecision]], N[(x + N[(N[(y * 2.0), $MachinePrecision] / N[(N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision] - N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6e-139 or 1.5499999999999999e-200 < z

   1. Initial program 82.5%

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

   3. Simplified 91.3%

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

      1. clear-num 91.3%

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

      2. un-div-inv 91.3%

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

      3. associate-*r* 91.3%

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

      4. *-commutative 91.3%

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

      5. associate-*l* 91.3%

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

      6. pow2 91.3%

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

   6. Applied egg-rr 91.3%

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

   7. Taylor expanded in y around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-*r/ 97.9%

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

      5. unsub-neg 97.9%

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

      6. *-commutative 97.9%

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

      7. associate-*r/ 96.4%

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

   9. Simplified 96.4%

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

   if -1.6e-139 < z < 1.5499999999999999e-200

   1. Initial program 80.8%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in y around inf 98.5%

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

      1. metadata-eval 98.5%

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

      2. cancel-sign-sub-inv 98.5%

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

      3. associate-*r/ 98.5%

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

      4. *-commutative 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-139} \lor \neg \left(z \leq 1.55 \cdot 10^{-200}\right):\\ \;\;\;\;x + \frac{y \cdot 2}{\frac{y \cdot t}{z} - 2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -21000 \lor \neg \left(z \leq 3.8 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -21000.0) (not (<= z 3.8e+20)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -21000.0) || !(z <= 3.8e+20)) {
		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 <= (-21000.0d0)) .or. (.not. (z <= 3.8d+20))) 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 <= -21000.0) || !(z <= 3.8e+20)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -21000.0) or not (z <= 3.8e+20):
		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 <= -21000.0) || !(z <= 3.8e+20))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -21000.0) || ~((z <= 3.8e+20)))
		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, -21000.0], N[Not[LessEqual[z, 3.8e+20]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -21000 or 3.8e20 < z

   1. Initial program 74.5%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in y around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. sub-neg 89.5%

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

   7. Simplified 89.5%

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

   if -21000 < z < 3.8e20

   1. Initial program 87.3%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around inf 90.0%

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

      1. metadata-eval 90.0%

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

      2. cancel-sign-sub-inv 90.0%

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

      3. associate-*r/ 90.0%

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

      4. *-commutative 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -21000 \lor \neg \left(z \leq 3.8 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -11500 \lor \neg \left(z \leq 3.8 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -11500.0) (not (<= z 3.8e+20)))
   (- x (/ y z))
   (- x (* z (/ -2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -11500.0) || !(z <= 3.8e+20)) {
		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 <= (-11500.0d0)) .or. (.not. (z <= 3.8d+20))) 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 <= -11500.0) || !(z <= 3.8e+20)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (z * (-2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -11500.0) or not (z <= 3.8e+20):
		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 <= -11500.0) || !(z <= 3.8e+20))
		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 <= -11500.0) || ~((z <= 3.8e+20)))
		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, -11500.0], N[Not[LessEqual[z, 3.8e+20]], $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 < -11500 or 3.8e20 < z

   1. Initial program 74.5%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in y around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. sub-neg 89.5%

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

   7. Simplified 89.5%

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

   if -11500 < z < 3.8e20

   1. Initial program 87.3%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around inf 90.0%

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

      1. metadata-eval 90.0%

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

      2. cancel-sign-sub-inv 90.0%

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

      3. associate-*r/ 90.0%

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

      4. *-commutative 90.0%

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

   7. Simplified 90.0%

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

      1. associate-/l* 90.0%

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

      2. *-commutative 90.0%

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

   9. Applied egg-rr 90.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -11500 \lor \neg \left(z \leq 3.8 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+15} \lor \neg \left(z \leq 40000000\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+15) (not (<= z 40000000.0))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+15) || !(z <= 40000000.0)) {
		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 <= (-1.3d+15)) .or. (.not. (z <= 40000000.0d0))) 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 <= -1.3e+15) || !(z <= 40000000.0)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.3e+15) or not (z <= 40000000.0):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+15) || !(z <= 40000000.0))
		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 <= -1.3e+15) || ~((z <= 40000000.0)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e+15], N[Not[LessEqual[z, 40000000.0]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e15 or 4e7 < z

   1. Initial program 73.8%

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

   3. Simplified 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)} \]
   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 -1.3e15 < z < 4e7

   1. Initial program 87.9%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 72.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+15} \lor \neg \left(z \leq 40000000\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-175}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-247}:\\ \;\;\;\;z \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.7e-175) x (if (<= x 1.7e-247) (* z (/ 2.0 t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.7e-175) {
		tmp = x;
	} else if (x <= 1.7e-247) {
		tmp = z * (2.0 / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.7d-175)) then
        tmp = x
    else if (x <= 1.7d-247) then
        tmp = z * (2.0d0 / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.7e-175) {
		tmp = x;
	} else if (x <= 1.7e-247) {
		tmp = z * (2.0 / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.7e-175:
		tmp = x
	elif x <= 1.7e-247:
		tmp = z * (2.0 / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.7e-175)
		tmp = x;
	elseif (x <= 1.7e-247)
		tmp = Float64(z * Float64(2.0 / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.7e-175)
		tmp = x;
	elseif (x <= 1.7e-247)
		tmp = z * (2.0 / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.7e-175], x, If[LessEqual[x, 1.7e-247], N[(z * N[(2.0 / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999999e-175 or 1.7000000000000001e-247 < x

   1. Initial program 85.0%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around 0 77.4%

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

   if -2.69999999999999999e-175 < x < 1.7000000000000001e-247

   1. Initial program 59.9%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in y around inf 64.6%

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

      1. metadata-eval 64.6%

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

      2. cancel-sign-sub-inv 64.6%

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

      3. associate-*r/ 64.6%

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

      4. *-commutative 64.6%

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

   7. Simplified 64.6%

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

      1. clear-num 64.4%

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

      2. inv-pow 64.4%

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

      3. *-un-lft-identity 64.4%

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

      4. *-commutative 64.4%

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

      5. times-frac 64.4%

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

      6. metadata-eval 64.4%

         \[\leadsto x - {\left(\color{blue}{-0.5} \cdot \frac{t}{z}\right)}^{-1} \]

   9. Applied egg-rr 64.4%

      \[\leadsto x - \color{blue}{{\left(-0.5 \cdot \frac{t}{z}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 64.4%

          \[\leadsto x - \color{blue}{\frac{1}{-0.5 \cdot \frac{t}{z}}} \]

       2. associate-*r/ 64.4%

          \[\leadsto x - \frac{1}{\color{blue}{\frac{-0.5 \cdot t}{z}}} \]

   11. Simplified 64.4%

       \[\leadsto x - \color{blue}{\frac{1}{\frac{-0.5 \cdot t}{z}}} \]

   12. Taylor expanded in x around 0 49.6%

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

       1. associate-*r/ 49.6%

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

       2. associate-*l/ 49.4%

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

       3. *-commutative 49.4%

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

   14. Simplified 49.4%

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

Alternative 7: 75.1% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 88.6%

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

5. Taylor expanded in y around 0 70.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (1.0d0 / ((z / y) - ((t / 2.0d0) / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024123 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (/ 1 (- (/ z y) (/ (/ t 2) z)))))

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