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

Percentage Accurate: 81.2% → 94.4%

Time: 10.2s

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: 81.2% 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: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot 2, \frac{y}{\mathsf{fma}\left(y, t, -2 \cdot \left(z \cdot z\right)\right)}, x\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+134}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-169}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* z 2.0) (/ y (fma y t (* -2.0 (* z z)))) x)))
   (if (<= z -1.5e+134)
     (- x (/ y z))
     (if (<= z -1.2e-88) t_1 (if (<= z 1e-169) (- x (/ (* z -2.0) t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z * 2.0), (y / fma(y, t, (-2.0 * (z * z)))), x);
	double tmp;
	if (z <= -1.5e+134) {
		tmp = x - (y / z);
	} else if (z <= -1.2e-88) {
		tmp = t_1;
	} else if (z <= 1e-169) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z * 2.0), Float64(y / fma(y, t, Float64(-2.0 * Float64(z * z)))), x)
	tmp = 0.0
	if (z <= -1.5e+134)
		tmp = Float64(x - Float64(y / z));
	elseif (z <= -1.2e-88)
		tmp = t_1;
	elseif (z <= 1e-169)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * 2.0), $MachinePrecision] * N[(y / N[(y * t + N[(-2.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.5e+134], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.2e-88], t$95$1, If[LessEqual[z, 1e-169], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999998e134

   1. Initial program 55.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 94.8

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

   5. Simplified 94.8%

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

   if -1.49999999999999998e134 < z < -1.2e-88 or 1.00000000000000002e-169 < z

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 94.8%

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

   if -1.2e-88 < z < 1.00000000000000002e-169

   1. Initial program 83.8%

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

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

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

      4. lower-*.f64 98.9

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

   5. Simplified 98.9%

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

Alternative 2: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-87}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 2, \frac{y \cdot -0.5}{z \cdot z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.35e-87)
   (- x (/ y z))
   (if (<= z 3.6e-7)
     (- x (/ (* z -2.0) t))
     (fma (* z 2.0) (/ (* y -0.5) (* z z)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.35e-87) {
		tmp = x - (y / z);
	} else if (z <= 3.6e-7) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = fma((z * 2.0), ((y * -0.5) / (z * z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.35e-87)
		tmp = Float64(x - Float64(y / z));
	elseif (z <= 3.6e-7)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	else
		tmp = fma(Float64(z * 2.0), Float64(Float64(y * -0.5) / Float64(z * z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.35e-87], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e-7], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(z * 2.0), $MachinePrecision] * N[(N[(y * -0.5), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.35e-87

   1. Initial program 73.2%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 84.5

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

   5. Simplified 84.5%

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

   if -2.35e-87 < z < 3.59999999999999994e-7

   1. Initial program 85.3%

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

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

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

      4. lower-*.f64 95.8

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

   5. Simplified 95.8%

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

   if 3.59999999999999994e-7 < z

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 91.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(z \cdot 2, \color{blue}{\frac{-1}{2} \cdot \frac{y}{{z}^{2}}}, x\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.9

         \[\leadsto \mathsf{fma}\left(z \cdot 2, \frac{-0.5 \cdot y}{\color{blue}{z \cdot z}}, x\right) \]

   8. Simplified 86.9%

      \[\leadsto \mathsf{fma}\left(z \cdot 2, \color{blue}{\frac{-0.5 \cdot y}{z \cdot z}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-87}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 2, \frac{y \cdot -0.5}{z \cdot z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -2.35e-87) t_1 (if (<= z 3.6e-7) (- 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 <= -2.35e-87) {
		tmp = t_1;
	} else if (z <= 3.6e-7) {
		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 <= (-2.35d-87)) then
        tmp = t_1
    else if (z <= 3.6d-7) 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 <= -2.35e-87) {
		tmp = t_1;
	} else if (z <= 3.6e-7) {
		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 <= -2.35e-87:
		tmp = t_1
	elif z <= 3.6e-7:
		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 <= -2.35e-87)
		tmp = t_1;
	elseif (z <= 3.6e-7)
		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 <= -2.35e-87)
		tmp = t_1;
	elseif (z <= 3.6e-7)
		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, -2.35e-87], t$95$1, If[LessEqual[z, 3.6e-7], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.35e-87 or 3.59999999999999994e-7 < z

   1. Initial program 72.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 85.5

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

   5. Simplified 85.5%

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

   if -2.35e-87 < z < 3.59999999999999994e-7

   1. Initial program 85.3%

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

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

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

      4. lower-*.f64 95.8

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

   5. Simplified 95.8%

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

Alternative 4: 88.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -2.35e-87) t_1 (if (<= z 3.6e-7) (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 <= -2.35e-87) {
		tmp = t_1;
	} else if (z <= 3.6e-7) {
		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 <= -2.35e-87)
		tmp = t_1;
	elseif (z <= 3.6e-7)
		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, -2.35e-87], t$95$1, If[LessEqual[z, 3.6e-7], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.35e-87 or 3.59999999999999994e-7 < z

   1. Initial program 72.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 85.5

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

   5. Simplified 85.5%

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

   if -2.35e-87 < z < 3.59999999999999994e-7

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 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 95.7

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

   5. Simplified 95.7%

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

Alternative 5: 77.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{-116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-42}:\\ \;\;\;\;\frac{x \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -5.4e-116) t_1 (if (<= z 1.28e-42) (/ (* x t) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -5.4e-116) {
		tmp = t_1;
	} else if (z <= 1.28e-42) {
		tmp = (x * t) / 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 <= (-5.4d-116)) then
        tmp = t_1
    else if (z <= 1.28d-42) then
        tmp = (x * t) / 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 <= -5.4e-116) {
		tmp = t_1;
	} else if (z <= 1.28e-42) {
		tmp = (x * t) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -5.4e-116:
		tmp = t_1
	elif z <= 1.28e-42:
		tmp = (x * t) / 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 <= -5.4e-116)
		tmp = t_1;
	elseif (z <= 1.28e-42)
		tmp = Float64(Float64(x * t) / 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 <= -5.4e-116)
		tmp = t_1;
	elseif (z <= 1.28e-42)
		tmp = (x * t) / 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, -5.4e-116], t$95$1, If[LessEqual[z, 1.28e-42], N[(N[(x * t), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e-116 or 1.27999999999999994e-42 < z

   1. Initial program 74.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 84.2

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

   5. Simplified 84.2%

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

   if -5.4e-116 < z < 1.27999999999999994e-42

   1. Initial program 83.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   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 95.3

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

   5. Simplified 95.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 90.4

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

   8. Simplified 90.4%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-*.f64 87.2

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

   11. Simplified 87.2%

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

   12. Taylor expanded in z around 0

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

       1. lower-*.f64 68.6

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

   14. Simplified 68.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-116}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-42}:\\ \;\;\;\;\frac{x \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. 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 78.3%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
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 61.8

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

5. Simplified 61.8%

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

Alternative 7: 15.4% 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 78.3%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
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 61.8

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

5. Simplified 61.8%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{y}{\color{blue}{-1 \cdot z}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{-1 \cdot z}} \]

   5. mul-1-neg N/A

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

   6. lower-neg.f64 14.6

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

8. Simplified 14.6%

   \[\leadsto \color{blue}{\frac{y}{-z}} \]
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 2024215 
(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)))))