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

Percentage Accurate: 81.8% → 98.3%

Time: 5.8s

Alternatives: 5

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 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 81.4%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;x + \left(z \cdot -2\right) \cdot \frac{y}{2 \cdot \left(z \cdot z\right) - y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.5e+142)
   (+ x (* (* z -2.0) (/ y (- (* 2.0 (* z z)) (* y t)))))
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.5e+142) {
		tmp = x + ((z * -2.0) * (y / ((2.0 * (z * z)) - (y * t))));
	} else {
		tmp = x - (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) :: tmp
    if (z <= 2.5d+142) then
        tmp = x + ((z * (-2.0d0)) * (y / ((2.0d0 * (z * z)) - (y * t))))
    else
        tmp = x - (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.5e+142) {
		tmp = x + ((z * -2.0) * (y / ((2.0 * (z * z)) - (y * t))));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 2.5e+142:
		tmp = x + ((z * -2.0) * (y / ((2.0 * (z * z)) - (y * t))))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.5e+142)
		tmp = Float64(x + Float64(Float64(z * -2.0) * Float64(y / Float64(Float64(2.0 * Float64(z * z)) - Float64(y * t)))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2.5e+142)
		tmp = x + ((z * -2.0) * (y / ((2.0 * (z * z)) - (y * t))));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2.5e+142], N[(x + N[(N[(z * -2.0), $MachinePrecision] * N[(y / N[(N[(2.0 * N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5000000000000001e142

   1. Initial program 87.6%

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

      1. sub-neg 87.6%

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

      2. associate-*l* 87.6%

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

      3. *-commutative 87.6%

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

      4. associate-*l/ 96.0%

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

      5. distribute-rgt-neg-in 96.0%

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

      6. *-commutative 96.0%

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

      7. associate-*l* 96.0%

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

      8. distribute-rgt-neg-in 96.0%

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

      9. metadata-eval 96.0%

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

   3. Simplified 96.0%

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

   if 2.5000000000000001e142 < z

   1. Initial program 47.9%

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

      1. sub-neg 47.9%

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

      2. associate-*l* 47.9%

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

      3. *-commutative 47.9%

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

      4. associate-*l/ 70.9%

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

      5. distribute-rgt-neg-in 70.9%

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

      6. *-commutative 70.9%

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

      7. associate-*l* 70.9%

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

      8. distribute-rgt-neg-in 70.9%

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

      9. metadata-eval 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in y around 0 95.3%

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

      1. mul-1-neg 95.3%

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

      2. sub-neg 95.3%

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

   6. Simplified 95.3%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;x + \left(z \cdot -2\right) \cdot \frac{y}{2 \cdot \left(z \cdot z\right) - y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

Alternative 3: 88.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-84} \lor \neg \left(z \leq 2.2 \cdot 10^{+37}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e-84) (not (<= z 2.2e+37)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e-84) || !(z <= 2.2e+37)) {
		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 <= (-1.25d-84)) .or. (.not. (z <= 2.2d+37))) 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 <= -1.25e-84) || !(z <= 2.2e+37)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e-84) or not (z <= 2.2e+37):
		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 <= -1.25e-84) || !(z <= 2.2e+37))
		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 <= -1.25e-84) || ~((z <= 2.2e+37)))
		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, -1.25e-84], N[Not[LessEqual[z, 2.2e+37]], $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 < -1.25e-84 or 2.2000000000000001e37 < z

   1. Initial program 69.0%

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

      1. sub-neg 69.0%

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

      2. associate-*l* 69.0%

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

      3. *-commutative 69.0%

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

      4. associate-*l/ 87.7%

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

      5. distribute-rgt-neg-in 87.7%

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

      6. *-commutative 87.7%

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

      7. associate-*l* 87.7%

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

      8. distribute-rgt-neg-in 87.7%

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

      9. metadata-eval 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in y around 0 88.8%

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

      1. mul-1-neg 88.8%

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

      2. sub-neg 88.8%

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

   6. Simplified 88.8%

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

   if -1.25e-84 < z < 2.2000000000000001e37

   1. Initial program 96.5%

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

      1. sub-neg 96.5%

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

      2. associate-*l* 96.5%

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

      3. *-commutative 96.5%

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

      4. associate-*l/ 97.4%

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

      5. distribute-rgt-neg-in 97.4%

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

      6. *-commutative 97.4%

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

      7. associate-*l* 97.4%

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

      8. distribute-rgt-neg-in 97.4%

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

      9. metadata-eval 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in y around inf 93.3%

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

      1. metadata-eval 93.3%

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

      2. cancel-sign-sub-inv 93.3%

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

      3. associate-*r/ 93.3%

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

      4. *-commutative 93.3%

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

   6. Simplified 93.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-84} \lor \neg \left(z \leq 2.2 \cdot 10^{+37}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]

Alternative 4: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-84} \lor \neg \left(z \leq 6.8 \cdot 10^{+37}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e-84) (not (<= z 6.8e+37))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e-84) || !(z <= 6.8e+37)) {
		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 <= (-1.25d-84)) .or. (.not. (z <= 6.8d+37))) 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 <= -1.25e-84) || !(z <= 6.8e+37)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e-84) or not (z <= 6.8e+37):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.25e-84) || !(z <= 6.8e+37))
		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 <= -1.25e-84) || ~((z <= 6.8e+37)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.25e-84], N[Not[LessEqual[z, 6.8e+37]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e-84 or 6.80000000000000011e37 < z

   1. Initial program 69.0%

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

      1. sub-neg 69.0%

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

      2. associate-*l* 69.0%

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

      3. *-commutative 69.0%

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

      4. associate-*l/ 87.7%

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

      5. distribute-rgt-neg-in 87.7%

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

      6. *-commutative 87.7%

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

      7. associate-*l* 87.7%

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

      8. distribute-rgt-neg-in 87.7%

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

      9. metadata-eval 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in y around 0 88.8%

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

      1. mul-1-neg 88.8%

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

      2. sub-neg 88.8%

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

   6. Simplified 88.8%

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

   if -1.25e-84 < z < 6.80000000000000011e37

   1. Initial program 96.5%

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

      1. sub-neg 96.5%

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

      2. associate-*l* 96.5%

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

      3. *-commutative 96.5%

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

      4. associate-*l/ 97.4%

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

      5. distribute-rgt-neg-in 97.4%

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

      6. *-commutative 97.4%

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

      7. associate-*l* 97.4%

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

      8. distribute-rgt-neg-in 97.4%

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

      9. metadata-eval 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in x around inf 80.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-84} \lor \neg \left(z \leq 6.8 \cdot 10^{+37}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 75.5% 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 81.4%

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

   1. sub-neg 81.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l/ 92.1%

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

   5. distribute-rgt-neg-in 92.1%

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

   6. *-commutative 92.1%

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

   7. associate-*l* 92.1%

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

   8. distribute-rgt-neg-in 92.1%

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

   9. metadata-eval 92.1%

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

3. Simplified 92.1%

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

4. Taylor expanded in x around inf 74.1%

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

5. Final simplification 74.1%

   \[\leadsto x \]

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 2023308 
(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)))))