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

Percentage Accurate: 82.1% → 92.9%

Time: 7.5s

Alternatives: 8

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: 82.1% 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: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+152}:\\ \;\;\;\;x - t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))
   (if (<= t_1 1e+152) (- x t_1) (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y * 2.0) * z) / (((z * 2.0) * z) - (y * t));
	double tmp;
	if (t_1 <= 1e+152) {
		tmp = x - t_1;
	} 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 = ((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t))
    if (t_1 <= 1d+152) then
        tmp = x - t_1
    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) / (((z * 2.0) * z) - (y * t));
	double tmp;
	if (t_1 <= 1e+152) {
		tmp = x - t_1;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y * 2.0) * z) / (((z * 2.0) * z) - (y * t))
	tmp = 0
	if t_1 <= 1e+152:
		tmp = x - t_1
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t)))
	tmp = 0.0
	if (t_1 <= 1e+152)
		tmp = Float64(x - t_1);
	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) / (((z * 2.0) * z) - (y * t));
	tmp = 0.0;
	if (t_1 <= 1e+152)
		tmp = x - t_1;
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+152], N[(x - t$95$1), $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))) < 1e152

   1. Initial program 94.6%

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

   if 1e152 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 2: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \frac{2}{\mathsf{fma}\left(-2, z \cdot z, t \cdot y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) 1e+152)
   (fma (* z y) (/ 2.0 (fma -2.0 (* z z) (* t y))) x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y * 2.0) * z) / (((z * 2.0) * z) - (y * t))) <= 1e+152) {
		tmp = fma((z * y), (2.0 / fma(-2.0, (z * z), (t * y))), x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))) <= 1e+152)
		tmp = fma(Float64(z * y), Float64(2.0 / fma(-2.0, Float64(z * z), Float64(t * y))), x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+152], N[(N[(z * y), $MachinePrecision] * N[(2.0 / N[(-2.0 * N[(z * z), $MachinePrecision] + N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $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))) < 1e152

   1. Initial program 94.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 94.5%

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

   if 1e152 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 3: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-30} \lor \neg \left(z \leq 5.5 \cdot 10^{+27}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.4e-30) (not (<= z 5.5e+27)))
   (- x (/ y z))
   (fma (/ z t) 2.0 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.4e-30) || !(z <= 5.5e+27)) {
		tmp = x - (y / z);
	} else {
		tmp = fma((z / t), 2.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.4e-30) || !(z <= 5.5e+27))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = fma(Float64(z / t), 2.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.4e-30], N[Not[LessEqual[z, 5.5e+27]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999985e-30 or 5.49999999999999966e27 < z

   1. Initial program 71.0%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   if -2.39999999999999985e-30 < z < 5.49999999999999966e27

   1. Initial program 89.5%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-30} \lor \neg \left(z \leq 5.5 \cdot 10^{+27}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-30} \lor \neg \left(z \leq 5.5 \cdot 10^{+27}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.4e-30) (not (<= z 5.5e+27)))
   (- x (/ y z))
   (fma (/ 2.0 t) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.4e-30) || !(z <= 5.5e+27)) {
		tmp = x - (y / z);
	} else {
		tmp = fma((2.0 / t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.4e-30) || !(z <= 5.5e+27))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = fma(Float64(2.0 / t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.4e-30], N[Not[LessEqual[z, 5.5e+27]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999985e-30 or 5.49999999999999966e27 < z

   1. Initial program 71.0%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   if -2.39999999999999985e-30 < z < 5.49999999999999966e27

   1. Initial program 89.5%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   7. 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. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 87.4

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

   8. Applied rewrites 87.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-30} \lor \neg \left(z \leq 5.5 \cdot 10^{+27}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+190}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.6e+190) (- x (/ y z)) (* (/ z t) 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+190) {
		tmp = x - (y / z);
	} else {
		tmp = (z / t) * 2.0;
	}
	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 <= 2.6d+190) then
        tmp = x - (y / z)
    else
        tmp = (z / t) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+190) {
		tmp = x - (y / z);
	} else {
		tmp = (z / t) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.6e+190:
		tmp = x - (y / z)
	else:
		tmp = (z / t) * 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.6e+190)
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(Float64(z / t) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.6e+190)
		tmp = x - (y / z);
	else
		tmp = (z / t) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.6e+190], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.60000000000000011e190

   1. Initial program 81.0%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   if 2.60000000000000011e190 < y

   1. Initial program 75.2%

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

   3. Taylor expanded in y around 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.1%

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

Alternative 6: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+190}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.6e+190) (- x (/ y z)) (* (/ 2.0 t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+190) {
		tmp = x - (y / z);
	} else {
		tmp = (2.0 / t) * 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 (y <= 2.6d+190) then
        tmp = x - (y / z)
    else
        tmp = (2.0d0 / t) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+190) {
		tmp = x - (y / z);
	} else {
		tmp = (2.0 / t) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.6e+190:
		tmp = x - (y / z)
	else:
		tmp = (2.0 / t) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.6e+190)
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(Float64(2.0 / t) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.6e+190)
		tmp = x - (y / z);
	else
		tmp = (2.0 / t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.6e+190], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.60000000000000011e190

   1. Initial program 81.0%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   if 2.60000000000000011e190 < y

   1. Initial program 75.2%

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

   3. Taylor expanded in y around 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.1%

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

   10. Applied rewrites 40.0%

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

Alternative 7: 63.1% 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 80.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 62.8

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

5. Applied rewrites 62.8%

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

Alternative 8: 14.8% 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(Float64(-y) / 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 80.6%

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

3. Taylor expanded in 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. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

   11. distribute-neg-in N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

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

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

   15. associate-*r* N/A

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

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

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-fma.f64 N/A

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

5. Applied rewrites 18.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 13.9%

   \[\leadsto \frac{-y}{\color{blue}{z}} \]
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 2024318 
(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)))))