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

Percentage Accurate: 81.0% → 94.8%

Time: 9.7s

Alternatives: 6

Speedup: 1.6×

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: 94.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t))) INFINITY)
   (fma (/ y (fma z (* z -2.0) (* y t))) (* 2.0 z) 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))) <= ((double) INFINITY)) {
		tmp = fma((y / fma(z, (z * -2.0), (y * t))), (2.0 * z), 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(z * Float64(2.0 * z)) - Float64(y * t))) <= Inf)
		tmp = fma(Float64(y / fma(z, Float64(z * -2.0), Float64(y * t))), Float64(2.0 * z), 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[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y / N[(z * N[(z * -2.0), $MachinePrecision] + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * z), $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))) < +inf.0

   1. Initial program 94.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-frac2 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 97.7%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-lowering-*.f64 97.7

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

   6. Applied egg-rr 97.7%

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

   if +inf.0 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 82.0

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

   5. Simplified 82.0%

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

4. Final simplification 95.8%

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

Alternative 2: 94.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t))) INFINITY)
   (fma (* 2.0 z) (/ y (fma y t (* -2.0 (* z z)))) 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))) <= ((double) INFINITY)) {
		tmp = fma((2.0 * z), (y / fma(y, t, (-2.0 * (z * z)))), 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(z * Float64(2.0 * z)) - Float64(y * t))) <= Inf)
		tmp = fma(Float64(2.0 * z), Float64(y / fma(y, t, Float64(-2.0 * Float64(z * z)))), 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[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(2.0 * z), $MachinePrecision] * N[(y / N[(y * t + N[(-2.0 * N[(z * z), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 94.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-frac2 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 97.7%

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

   if +inf.0 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 0.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 82.0

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

   5. Simplified 82.0%

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

4. Final simplification 95.8%

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

Alternative 3: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -3.2e-18) t_1 (if (<= z 1.75e-65) (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 <= -3.2e-18) {
		tmp = t_1;
	} else if (z <= 1.75e-65) {
		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 <= -3.2e-18)
		tmp = t_1;
	elseif (z <= 1.75e-65)
		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, -3.2e-18], t$95$1, If[LessEqual[z, 1.75e-65], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999999e-18 or 1.75000000000000002e-65 < z

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 91.8

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

   5. Simplified 91.8%

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

   if -3.1999999999999999e-18 < z < 1.75000000000000002e-65

   1. Initial program 88.2%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. 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. accelerator-lowering-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. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 96.0

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

   7. Applied egg-rr 96.0%

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

Alternative 4: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.1e-17) t_1 (if (<= z 1.4e-65) (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 <= -1.1e-17) {
		tmp = t_1;
	} else if (z <= 1.4e-65) {
		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 <= -1.1e-17)
		tmp = t_1;
	elseif (z <= 1.4e-65)
		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, -1.1e-17], t$95$1, If[LessEqual[z, 1.4e-65], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e-17 or 1.4e-65 < z

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 91.8

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

   5. Simplified 91.8%

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

   if -1.1e-17 < z < 1.4e-65

   1. Initial program 88.2%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. 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. accelerator-lowering-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. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 5: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.08e-17) t_1 (if (<= z 4.8e-65) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.08e-17) {
		tmp = t_1;
	} else if (z <= 4.8e-65) {
		tmp = 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 <= (-1.08d-17)) then
        tmp = t_1
    else if (z <= 4.8d-65) then
        tmp = 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 <= -1.08e-17) {
		tmp = t_1;
	} else if (z <= 4.8e-65) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1.08e-17:
		tmp = t_1
	elif z <= 4.8e-65:
		tmp = 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 <= -1.08e-17)
		tmp = t_1;
	elseif (z <= 4.8e-65)
		tmp = 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 <= -1.08e-17)
		tmp = t_1;
	elseif (z <= 4.8e-65)
		tmp = 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, -1.08e-17], t$95$1, If[LessEqual[z, 4.8e-65], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.07999999999999995e-17 or 4.8000000000000003e-65 < z

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 91.8

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

   5. Simplified 91.8%

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

   if -1.07999999999999995e-17 < z < 4.8000000000000003e-65

   1. Initial program 88.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 80.3%

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

Alternative 6: 75.4% accurate, 43.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 77.9%

   \[\leadsto \color{blue}{x} \]
6. 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 2024199 
(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)))))