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

Percentage Accurate: 81.9% → 94.8%

Time: 10.7s

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: 81.9% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot 2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -7.2e+117)
     t_1
     (if (<= z 9.6e+139)
       (fma (/ (* z 2.0) (fma z (* z -2.0) (* y t))) y x)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -7.2e+117) {
		tmp = t_1;
	} else if (z <= 9.6e+139) {
		tmp = fma(((z * 2.0) / fma(z, (z * -2.0), (y * t))), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -7.2e+117)
		tmp = t_1;
	elseif (z <= 9.6e+139)
		tmp = fma(Float64(Float64(z * 2.0) / fma(z, Float64(z * -2.0), Float64(y * t))), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+117], t$95$1, If[LessEqual[z, 9.6e+139], N[(N[(N[(z * 2.0), $MachinePrecision] / N[(z * N[(z * -2.0), $MachinePrecision] + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.20000000000000025e117 or 9.60000000000000032e139 < z

   1. Initial program 56.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   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 96.3

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

   5. Simplified 96.3%

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

   if -7.20000000000000025e117 < z < 9.60000000000000032e139

   1. Initial program 90.5%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing
   3. 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. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. associate-*l* N/A

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

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

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

   4. Applied egg-rr 95.3%

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. metadata-eval N/A

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

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

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

      12. +-commutative N/A

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

      13. associate-*l* N/A

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

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

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

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

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

      16. *-lowering-*.f64 95.3

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

   6. Applied egg-rr 95.3%

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

Alternative 2: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.22 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(y, t, -2 \cdot \left(z \cdot z\right)\right)}, y \cdot 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -6.2e+117)
     t_1
     (if (<= z 2.22e+139)
       (fma (/ z (fma y t (* -2.0 (* z z)))) (* y 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 <= -6.2e+117) {
		tmp = t_1;
	} else if (z <= 2.22e+139) {
		tmp = fma((z / fma(y, t, (-2.0 * (z * z)))), (y * 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 <= -6.2e+117)
		tmp = t_1;
	elseif (z <= 2.22e+139)
		tmp = fma(Float64(z / fma(y, t, Float64(-2.0 * Float64(z * z)))), Float64(y * 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, -6.2e+117], t$95$1, If[LessEqual[z, 2.22e+139], N[(N[(z / N[(y * t + N[(-2.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 2.0), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1999999999999995e117 or 2.22000000000000001e139 < z

   1. Initial program 56.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   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 96.3

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

   5. Simplified 96.3%

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

   if -6.1999999999999995e117 < z < 2.22000000000000001e139

   1. Initial program 90.5%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing
   3. 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. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-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 egg-rr 95.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+117}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.22 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(y, t, -2 \cdot \left(z \cdot z\right)\right)}, y \cdot 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10200000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.6e+25)
     t_1
     (if (<= z 10200000000000.0) (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 <= -1.6e+25) {
		tmp = t_1;
	} else if (z <= 10200000000000.0) {
		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 <= -1.6e+25)
		tmp = t_1;
	elseif (z <= 10200000000000.0)
		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, -1.6e+25], t$95$1, If[LessEqual[z, 10200000000000.0], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6e25 or 1.02e13 < z

   1. Initial program 66.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 89.8

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

   5. Simplified 89.8%

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

   if -1.6e25 < z < 1.02e13

   1. Initial program 93.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. 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 85.1

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

   5. Simplified 85.1%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. /-lowering-/.f64 85.2

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

   7. Applied egg-rr 85.2%

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

Alternative 4: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.85e+28) t_1 (if (<= z 5.8e+24) (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 <= -1.85e+28) {
		tmp = t_1;
	} else if (z <= 5.8e+24) {
		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 <= -1.85e+28)
		tmp = t_1;
	elseif (z <= 5.8e+24)
		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, -1.85e+28], t$95$1, If[LessEqual[z, 5.8e+24], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.85e28 or 5.79999999999999958e24 < z

   1. Initial program 66.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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 89.8

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

   5. Simplified 89.8%

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

   if -1.85e28 < z < 5.79999999999999958e24

   1. Initial program 93.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   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. 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 85.1

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

   5. Simplified 85.1%

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

Alternative 5: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -210000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -210000.0) t_1 (if (<= z 1.7e-61) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -210000.0) {
		tmp = t_1;
	} else if (z <= 1.7e-61) {
		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 <= (-210000.0d0)) then
        tmp = t_1
    else if (z <= 1.7d-61) 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 <= -210000.0) {
		tmp = t_1;
	} else if (z <= 1.7e-61) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -210000.0:
		tmp = t_1
	elif z <= 1.7e-61:
		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 <= -210000.0)
		tmp = t_1;
	elseif (z <= 1.7e-61)
		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 <= -210000.0)
		tmp = t_1;
	elseif (z <= 1.7e-61)
		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, -210000.0], t$95$1, If[LessEqual[z, 1.7e-61], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e5 or 1.6999999999999999e-61 < z

   1. Initial program 71.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   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 84.8

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

   5. Simplified 84.8%

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

   if -2.1e5 < z < 1.6999999999999999e-61

   1. Initial program 92.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.6%

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

Alternative 6: 75.1% 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 79.0%

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

3. Taylor expanded in x around inf

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

5. Simplified 69.8%

   \[\leadsto \color{blue}{x} \]
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 2024195 
(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)))))