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

Percentage Accurate: 81.0% → 93.9%

Time: 4.5s

Alternatives: 6

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: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{\left(2 \cdot y\right) \cdot z}{\left(z \cdot 2\right) \cdot z - t \cdot y} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(-2, z \cdot z, t \cdot y\right)}, 2 \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x (/ (* (* 2.0 y) z) (- (* (* z 2.0) z) (* t y)))) 2e+293)
   (fma (/ z (fma -2.0 (* z z) (* t y))) (* 2.0 y) x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - (((2.0 * y) * z) / (((z * 2.0) * z) - (t * y)))) <= 2e+293) {
		tmp = fma((z / fma(-2.0, (z * z), (t * y))), (2.0 * y), x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - Float64(Float64(Float64(2.0 * y) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(t * y)))) <= 2e+293)
		tmp = fma(Float64(z / fma(-2.0, Float64(z * z), Float64(t * y))), Float64(2.0 * y), x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - N[(N[(N[(2.0 * y), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+293], N[(N[(z / N[(-2.0 * N[(z * z), $MachinePrecision] + N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * y), $MachinePrecision] + x), $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)))) < 1.9999999999999998e293

   1. Initial program 94.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 96.4%

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

   if 1.9999999999999998e293 < (-.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 10.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 92.6

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

   5. Applied rewrites 92.6%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(2 \cdot y\right) \cdot z}{\left(z \cdot 2\right) \cdot z - t \cdot y} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(-2, z \cdot z, t \cdot y\right)}, 2 \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.15e-17) t_1 (if (<= z 8.4e-45) (fma (/ z t) 2.0 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.15e-17) {
		tmp = t_1;
	} else if (z <= 8.4e-45) {
		tmp = fma((z / t), 2.0, 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 <= -1.15e-17)
		tmp = t_1;
	elseif (z <= 8.4e-45)
		tmp = fma(Float64(z / t), 2.0, 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, -1.15e-17], t$95$1, If[LessEqual[z, 8.4e-45], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000004e-17 or 8.3999999999999998e-45 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -1.15000000000000004e-17 < z < 8.3999999999999998e-45

   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. 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 93.2

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

   5. Applied rewrites 93.2%

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

Alternative 3: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.15e-17) t_1 (if (<= z 8.4e-45) (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 <= -1.15e-17) {
		tmp = t_1;
	} else if (z <= 8.4e-45) {
		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 <= -1.15e-17)
		tmp = t_1;
	elseif (z <= 8.4e-45)
		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, -1.15e-17], t$95$1, If[LessEqual[z, 8.4e-45], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000004e-17 or 8.3999999999999998e-45 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -1.15000000000000004e-17 < z < 8.3999999999999998e-45

   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. 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 93.2

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

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, 2, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.4%

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

Alternative 4: 80.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-97}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.12e+72) t_1 (if (<= z 4.8e-97) (* 1.0 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.12e+72) {
		tmp = t_1;
	} else if (z <= 4.8e-97) {
		tmp = 1.0 * x;
	} 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 <= (-1.12d+72)) then
        tmp = t_1
    else if (z <= 4.8d-97) then
        tmp = 1.0d0 * x
    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 <= -1.12e+72) {
		tmp = t_1;
	} else if (z <= 4.8e-97) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1.12e+72:
		tmp = t_1
	elif z <= 4.8e-97:
		tmp = 1.0 * 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 <= -1.12e+72)
		tmp = t_1;
	elseif (z <= 4.8e-97)
		tmp = Float64(1.0 * x);
	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 <= -1.12e+72)
		tmp = t_1;
	elseif (z <= 4.8e-97)
		tmp = 1.0 * x;
	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, -1.12e+72], t$95$1, If[LessEqual[z, 4.8e-97], N[(1.0 * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12000000000000001e72 or 4.8e-97 < z

   1. Initial program 70.5%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.12000000000000001e72 < z < 4.8e-97

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 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 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 78.5%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-243}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;x \leq 10^{-193}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -8.8e-243) (* 1.0 x) (if (<= x 1e-193) (/ (- y) z) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8.8e-243) {
		tmp = 1.0 * x;
	} else if (x <= 1e-193) {
		tmp = -y / z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-8.8d-243)) then
        tmp = 1.0d0 * x
    else if (x <= 1d-193) then
        tmp = -y / z
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8.8e-243) {
		tmp = 1.0 * x;
	} else if (x <= 1e-193) {
		tmp = -y / z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -8.8e-243:
		tmp = 1.0 * x
	elif x <= 1e-193:
		tmp = -y / z
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -8.8e-243)
		tmp = Float64(1.0 * x);
	elseif (x <= 1e-193)
		tmp = Float64(Float64(-y) / z);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -8.8e-243)
		tmp = 1.0 * x;
	elseif (x <= 1e-193)
		tmp = -y / z;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -8.8e-243], N[(1.0 * x), $MachinePrecision], If[LessEqual[x, 1e-193], N[((-y) / z), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.7999999999999996e-243 or 1e-193 < x

   1. Initial program 83.5%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 84.4%

       \[\leadsto 1 \cdot x \]

   if -8.7999999999999996e-243 < x < 1e-193

   1. Initial program 65.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{y \cdot z}{2 \cdot {z}^{2} - t \cdot y}} \]
   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 54.5%

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

Alternative 6: 73.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Python

def code(x, y, z, t):
	return 1.0 * x

Julia

function code(x, y, z, t)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 81.2%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
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 65.2

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

5. Applied rewrites 65.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 64.7%

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

9. Taylor expanded in x around inf

   \[\leadsto 1 \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 75.0%

    \[\leadsto 1 \cdot x \]
12. 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 2024308 
(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)))))