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

Percentage Accurate: 82.0% → 98.2%

Time: 6.7s

Alternatives: 6

Speedup: 1.5×

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: 98.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. sub-neg 81.7%

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

   2. +-commutative 81.7%

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

   3. distribute-neg-frac 81.7%

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

   4. distribute-rgt-neg-out 81.7%

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

   5. remove-double-neg 81.7%

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

   6. distribute-rgt-neg-in 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. distribute-lft-neg-out 81.7%

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

   9. associate-/l* 90.5%

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

   10. associate-*l/ 90.5%

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

   11. fma-def 90.5%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(z \cdot 2\right) - y \cdot t\\ \mathbf{if}\;\frac{z \cdot \left(y \cdot 2\right)}{t_1} \leq 2 \cdot 10^{+154}:\\ \;\;\;\;x - \frac{y \cdot 2}{\frac{t_1}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z (* z 2.0)) (* y t))))
   (if (<= (/ (* z (* y 2.0)) t_1) 2e+154)
     (- x (/ (* y 2.0) (/ t_1 z)))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * (z * 2.0)) - (y * t);
	double tmp;
	if (((z * (y * 2.0)) / t_1) <= 2e+154) {
		tmp = x - ((y * 2.0) / (t_1 / z));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (z * (z * 2.0d0)) - (y * t)
    if (((z * (y * 2.0d0)) / t_1) <= 2d+154) then
        tmp = x - ((y * 2.0d0) / (t_1 / z))
    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 t_1 = (z * (z * 2.0)) - (y * t);
	double tmp;
	if (((z * (y * 2.0)) / t_1) <= 2e+154) {
		tmp = x - ((y * 2.0) / (t_1 / z));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * (z * 2.0)) - (y * t)
	tmp = 0
	if ((z * (y * 2.0)) / t_1) <= 2e+154:
		tmp = x - ((y * 2.0) / (t_1 / z))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * Float64(z * 2.0)) - Float64(y * t))
	tmp = 0.0
	if (Float64(Float64(z * Float64(y * 2.0)) / t_1) <= 2e+154)
		tmp = Float64(x - Float64(Float64(y * 2.0) / Float64(t_1 / z)));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * (z * 2.0)) - (y * t);
	tmp = 0.0;
	if (((z * (y * 2.0)) / t_1) <= 2e+154)
		tmp = x - ((y * 2.0) / (t_1 / z));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(z * N[(y * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 2e+154], N[(x - N[(N[(y * 2.0), $MachinePrecision] / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $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))) < 2.00000000000000007e154

   1. Initial program 95.9%

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

      1. associate-/l* 98.1%

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

      2. associate-*l* 98.1%

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

   3. Simplified 98.1%

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

   if 2.00000000000000007e154 < (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))

   1. Initial program 2.6%

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

      1. sub-neg 2.6%

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

      2. associate-/l* 48.1%

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

      3. distribute-neg-frac 48.1%

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

      4. distribute-lft-neg-out 48.1%

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

      5. associate-/r/ 48.1%

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

      6. distribute-lft-neg-out 48.1%

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

      7. distribute-rgt-neg-in 48.1%

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

      8. metadata-eval 48.1%

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

      9. *-commutative 48.1%

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

      10. associate-*l* 48.1%

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

      11. fma-neg 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in y around 0 82.7%

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

      1. mul-1-neg 82.7%

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

      2. sub-neg 82.7%

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

   6. Simplified 82.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(y \cdot 2\right)}{z \cdot \left(z \cdot 2\right) - y \cdot t} \leq 2 \cdot 10^{+154}:\\ \;\;\;\;x - \frac{y \cdot 2}{\frac{z \cdot \left(z \cdot 2\right) - y \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

Alternative 3: 89.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+39} \lor \neg \left(z \leq 0.0035\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.5e+39) (not (<= z 0.0035)))
   (- x (/ y z))
   (+ x (* z (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e+39) || !(z <= 0.0035)) {
		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 <= (-2.5d+39)) .or. (.not. (z <= 0.0035d0))) 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 <= -2.5e+39) || !(z <= 0.0035)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.5e+39) or not (z <= 0.0035):
		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 <= -2.5e+39) || !(z <= 0.0035))
		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 <= -2.5e+39) || ~((z <= 0.0035)))
		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, -2.5e+39], N[Not[LessEqual[z, 0.0035]], $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 < -2.50000000000000008e39 or 0.00350000000000000007 < z

   1. Initial program 69.7%

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

      1. sub-neg 69.7%

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

      2. associate-/l* 88.0%

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

      3. distribute-neg-frac 88.0%

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

      4. distribute-lft-neg-out 88.0%

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

      5. associate-/r/ 87.6%

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

      6. distribute-lft-neg-out 87.6%

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

      7. distribute-rgt-neg-in 87.6%

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

      8. metadata-eval 87.6%

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

      9. *-commutative 87.6%

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

      10. associate-*l* 87.6%

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

      11. fma-neg 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in y around 0 92.7%

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

      1. mul-1-neg 92.7%

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

      2. sub-neg 92.7%

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

   6. Simplified 92.7%

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

   if -2.50000000000000008e39 < z < 0.00350000000000000007

   1. Initial program 90.1%

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

      1. sub-neg 90.1%

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

      2. associate-/l* 92.2%

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

      3. distribute-neg-frac 92.2%

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

      4. distribute-lft-neg-out 92.2%

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

      5. associate-/r/ 93.4%

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

      6. distribute-lft-neg-out 93.4%

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

      7. distribute-rgt-neg-in 93.4%

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

      8. metadata-eval 93.4%

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

      9. *-commutative 93.4%

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

      10. associate-*l* 93.4%

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

      11. fma-neg 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in y around inf 86.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+39} \lor \neg \left(z \leq 0.0035\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]

Alternative 4: 89.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+39} \lor \neg \left(z \leq 0.0031\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.6e+39) (not (<= z 0.0031)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.6e+39) || !(z <= 0.0031)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-3.6d+39)) .or. (.not. (z <= 0.0031d0))) then
        tmp = x - (y / z)
    else
        tmp = x - ((z * (-2.0d0)) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.6e+39) || !(z <= 0.0031)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.6e+39) or not (z <= 0.0031):
		tmp = x - (y / z)
	else:
		tmp = x - ((z * -2.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.6e+39) || !(z <= 0.0031))
		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 <= -3.6e+39) || ~((z <= 0.0031)))
		tmp = x - (y / z);
	else
		tmp = x - ((z * -2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.6e+39], N[Not[LessEqual[z, 0.0031]], $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 < -3.59999999999999984e39 or 0.00309999999999999989 < z

   1. Initial program 69.7%

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

      1. sub-neg 69.7%

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

      2. associate-/l* 88.0%

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

      3. distribute-neg-frac 88.0%

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

      4. distribute-lft-neg-out 88.0%

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

      5. associate-/r/ 87.6%

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

      6. distribute-lft-neg-out 87.6%

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

      7. distribute-rgt-neg-in 87.6%

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

      8. metadata-eval 87.6%

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

      9. *-commutative 87.6%

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

      10. associate-*l* 87.6%

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

      11. fma-neg 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in y around 0 92.7%

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

      1. mul-1-neg 92.7%

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

      2. sub-neg 92.7%

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

   6. Simplified 92.7%

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

   if -3.59999999999999984e39 < z < 0.00309999999999999989

   1. Initial program 90.1%

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

      1. associate-/l* 92.2%

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

      2. associate-*l* 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in y around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. *-commutative 86.0%

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

   6. Simplified 86.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+39} \lor \neg \left(z \leq 0.0031\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]

Alternative 5: 81.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+40} \lor \neg \left(z \leq 3.2 \cdot 10^{+136}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.3e+40) (not (<= z 3.2e+136))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.3e+40) || !(z <= 3.2e+136)) {
		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 <= (-4.3d+40)) .or. (.not. (z <= 3.2d+136))) 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 <= -4.3e+40) || !(z <= 3.2e+136)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.3e+40) or not (z <= 3.2e+136):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.3e+40) || !(z <= 3.2e+136))
		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 <= -4.3e+40) || ~((z <= 3.2e+136)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.3e+40], N[Not[LessEqual[z, 3.2e+136]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -4.3000000000000002e40 or 3.19999999999999988e136 < z

   1. Initial program 65.5%

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

      1. sub-neg 65.5%

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

      2. associate-/l* 85.5%

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

      3. distribute-neg-frac 85.5%

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

      4. distribute-lft-neg-out 85.5%

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

      5. associate-/r/ 85.1%

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

      6. distribute-lft-neg-out 85.1%

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

      7. distribute-rgt-neg-in 85.1%

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

      8. metadata-eval 85.1%

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

      9. *-commutative 85.1%

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

      10. associate-*l* 85.1%

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

      11. fma-neg 85.1%

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

   3. Simplified 85.1%

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

   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%

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

   if -4.3000000000000002e40 < z < 3.19999999999999988e136

   1. Initial program 89.3%

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

      1. sub-neg 89.3%

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

      2. associate-/l* 92.8%

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

      3. distribute-neg-frac 92.8%

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

      4. distribute-lft-neg-out 92.8%

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

      5. associate-/r/ 93.8%

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

      6. distribute-lft-neg-out 93.8%

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

      7. distribute-rgt-neg-in 93.8%

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

      8. metadata-eval 93.8%

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

      9. *-commutative 93.8%

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

      10. associate-*l* 93.8%

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

      11. fma-neg 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in x around inf 80.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+40} \lor \neg \left(z \leq 3.2 \cdot 10^{+136}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 74.9% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. sub-neg 81.7%

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

   2. associate-/l* 90.5%

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

   3. distribute-neg-frac 90.5%

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

   4. distribute-lft-neg-out 90.5%

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

   5. associate-/r/ 91.0%

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

   6. distribute-lft-neg-out 91.0%

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

   7. distribute-rgt-neg-in 91.0%

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

   8. metadata-eval 91.0%

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

   9. *-commutative 91.0%

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

   10. associate-*l* 91.0%

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

   11. fma-neg 91.0%

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

3. Simplified 91.0%

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

4. Taylor expanded in x around inf 80.5%

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

5. Final simplification 80.5%

   \[\leadsto x \]

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 2023297 
(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)))))