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

Percentage Accurate: 82.5% → 94.3%

Time: 8.3s

Alternatives: 7

Speedup: 1.4×

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: 82.5% 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: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;x + \frac{y \cdot 2}{\frac{\mathsf{fma}\left(z \cdot z, -2, y \cdot t\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x (/ (* (* y 2.0) z) (- (* y t) (* z (* 2.0 z))))) 5e+280)
   (+ x (/ (* y 2.0) (/ (fma (* z z) -2.0 (* y t)) z)))
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + (((y * 2.0) * z) / ((y * t) - (z * (2.0 * z))))) <= 5e+280) {
		tmp = x + ((y * 2.0) / (fma((z * z), -2.0, (y * t)) / z));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(y * t) - Float64(z * Float64(2.0 * z))))) <= 5e+280)
		tmp = Float64(x + Float64(Float64(y * 2.0) / Float64(fma(Float64(z * z), -2.0, Float64(y * t)) / z)));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+280], N[(x + N[(N[(y * 2.0), $MachinePrecision] / N[(N[(N[(z * z), $MachinePrecision] * -2.0 + N[(y * t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 5.0000000000000002e280

   1. Initial program 92.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 94.5%

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

   if 5.0000000000000002e280 < (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))))

   1. Initial program 10.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;x + \frac{y \cdot 2}{\frac{\mathsf{fma}\left(z \cdot z, -2, y \cdot t\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot 2\right) \cdot z\\ t_2 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;\frac{t\_1}{t\_2 - y \cdot t} \leq 10^{+168}:\\ \;\;\;\;x + \frac{t\_1}{y \cdot t - t\_2}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* y 2.0) z)) (t_2 (* z (* 2.0 z))))
   (if (<= (/ t_1 (- t_2 (* y t))) 1e+168)
     (+ x (/ t_1 (- (* y t) t_2)))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * 2.0) * z;
	double t_2 = z * (2.0 * z);
	double tmp;
	if ((t_1 / (t_2 - (y * t))) <= 1e+168) {
		tmp = x + (t_1 / ((y * t) - t_2));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (y * 2.0d0) * z
    t_2 = z * (2.0d0 * z)
    if ((t_1 / (t_2 - (y * t))) <= 1d+168) then
        tmp = x + (t_1 / ((y * t) - t_2))
    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 = (y * 2.0) * z;
	double t_2 = z * (2.0 * z);
	double tmp;
	if ((t_1 / (t_2 - (y * t))) <= 1e+168) {
		tmp = x + (t_1 / ((y * t) - t_2));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * 2.0) * z
	t_2 = z * (2.0 * z)
	tmp = 0
	if (t_1 / (t_2 - (y * t))) <= 1e+168:
		tmp = x + (t_1 / ((y * t) - t_2))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * 2.0) * z)
	t_2 = Float64(z * Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t_1 / Float64(t_2 - Float64(y * t))) <= 1e+168)
		tmp = Float64(x + Float64(t_1 / Float64(Float64(y * t) - t_2)));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * 2.0) * z;
	t_2 = z * (2.0 * z);
	tmp = 0.0;
	if ((t_1 / (t_2 - (y * t))) <= 1e+168)
		tmp = x + (t_1 / ((y * t) - t_2));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(t$95$2 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+168], N[(x + N[(t$95$1 / N[(N[(y * t), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))) < 9.9999999999999993e167

   1. Initial program 93.6%

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

   if 9.9999999999999993e167 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 5.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 10^{+168}:\\ \;\;\;\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -7.8e-8) t_1 (if (<= z 9.5e-45) (- x (/ (* z -2.0) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -7.8e-8) {
		tmp = t_1;
	} else if (z <= 9.5e-45) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = t_1;
	}
	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 = x - (y / z)
    if (z <= (-7.8d-8)) then
        tmp = t_1
    else if (z <= 9.5d-45) then
        tmp = x - ((z * (-2.0d0)) / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -7.8e-8) {
		tmp = t_1;
	} else if (z <= 9.5e-45) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -7.8e-8:
		tmp = t_1
	elif z <= 9.5e-45:
		tmp = x - ((z * -2.0) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -7.8e-8)
		tmp = t_1;
	elseif (z <= 9.5e-45)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -7.8e-8)
		tmp = t_1;
	elseif (z <= 9.5e-45)
		tmp = x - ((z * -2.0) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e-8], t$95$1, If[LessEqual[z, 9.5e-45], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999997e-8 or 9.5000000000000002e-45 < z

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -7.7999999999999997e-8 < z < 9.5000000000000002e-45

   1. Initial program 90.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -7.8e-8) t_1 (if (<= z 1.05e-53) (fma z (/ 2.0 t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -7.8e-8) {
		tmp = t_1;
	} else if (z <= 1.05e-53) {
		tmp = fma(z, (2.0 / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -7.8e-8)
		tmp = t_1;
	elseif (z <= 1.05e-53)
		tmp = fma(z, Float64(2.0 / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e-8], t$95$1, If[LessEqual[z, 1.05e-53], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999997e-8 or 1.04999999999999989e-53 < z

   1. Initial program 69.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -7.7999999999999997e-8 < z < 1.04999999999999989e-53

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 91.1

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

   5. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{2}{t}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+268}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7.5e+268) (- x (/ y z)) (* 2.0 (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.5e+268) {
		tmp = x - (y / z);
	} else {
		tmp = 2.0 * (z / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 7.5d+268) then
        tmp = x - (y / z)
    else
        tmp = 2.0d0 * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.5e+268) {
		tmp = x - (y / z);
	} else {
		tmp = 2.0 * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 7.5e+268:
		tmp = x - (y / z)
	else:
		tmp = 2.0 * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7.5e+268)
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(2.0 * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 7.5e+268)
		tmp = x - (y / z);
	else
		tmp = 2.0 * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7.5e+268], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000004e268

   1. Initial program 80.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 7.5000000000000004e268 < y

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. associate-*r/ N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

          \[\leadsto \frac{y \cdot \left(z \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {z}^{2} + \left(\mathsf{neg}\left(t \cdot y\right)\right)\right)}\right)} \]

      13. mul-1-neg N/A

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

      14. distribute-neg-in N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. distribute-lft-neg-out N/A

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

      18. distribute-lft-neg-in N/A

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

      19. metadata-eval N/A

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

      20. associate-*r* N/A

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

      21. unpow2 N/A

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.8%

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

   10. Applied rewrites 67.8%

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

Alternative 6: 63.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 60.7

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

5. Applied rewrites 60.7%

   \[\leadsto x - \color{blue}{\frac{y}{z}} \]
6. Add Preprocessing

Alternative 7: 14.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{y}{-z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. distribute-lft-neg-in N/A

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

   3. associate-*r/ N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

       \[\leadsto \frac{y \cdot \left(z \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {z}^{2} + \left(\mathsf{neg}\left(t \cdot y\right)\right)\right)}\right)} \]

   13. mul-1-neg N/A

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

   14. distribute-neg-in N/A

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

   15. unpow2 N/A

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

   16. associate-*r* N/A

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

   17. distribute-lft-neg-out N/A

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

   18. distribute-lft-neg-in N/A

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

   19. metadata-eval N/A

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

   20. associate-*r* N/A

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

   21. unpow2 N/A

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

5. Applied rewrites 17.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 15.8%

   \[\leadsto \frac{y}{\color{blue}{-z}} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× 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 2024226 
(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)))))