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

Percentage Accurate: 81.7% → 98.2%

Time: 7.7s

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: 81.7% 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: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+112}:\\ \;\;\;\;x - \frac{z}{\mathsf{fma}\left(-0.5, t, \frac{z \cdot z}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.7e+147)
     t_1
     (if (<= z 3.7e+112) (- x (/ z (fma -0.5 t (/ (* z z) y)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.7e+147) {
		tmp = t_1;
	} else if (z <= 3.7e+112) {
		tmp = x - (z / fma(-0.5, t, ((z * z) / y)));
	} 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 <= -1.7e+147)
		tmp = t_1;
	elseif (z <= 3.7e+112)
		tmp = Float64(x - Float64(z / fma(-0.5, t, Float64(Float64(z * z) / y))));
	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, -1.7e+147], t$95$1, If[LessEqual[z, 3.7e+112], N[(x - N[(z / N[(-0.5 * t + N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e147 or 3.70000000000000004e112 < z

   1. Initial program 59.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1.7e147 < z < 3.70000000000000004e112

   1. Initial program 84.5%

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

      1. 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. clear-num N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-/r/ N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 90.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 97.6

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(t, -0.5, \frac{\color{blue}{z \cdot z}}{y}\right)}, z, x\right) \]

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\mathsf{fma}\left(t, -0.5, \frac{z \cdot z}{y}\right)}}, z, x\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 97.8

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

   9. Applied rewrites 97.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+147}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+112}:\\ \;\;\;\;x - \frac{z}{\mathsf{fma}\left(-0.5, t, \frac{z \cdot z}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.2e-26) t_1 (if (<= z 1.28e-5) (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 <= -1.2e-26) {
		tmp = t_1;
	} else if (z <= 1.28e-5) {
		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 <= -1.2e-26)
		tmp = t_1;
	elseif (z <= 1.28e-5)
		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, -1.2e-26], t$95$1, If[LessEqual[z, 1.28e-5], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e-26 or 1.2799999999999999e-5 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 87.0

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

   5. Applied rewrites 87.0%

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

   if -1.2e-26 < z < 1.2799999999999999e-5

   1. Initial program 87.8%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.2%

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

Alternative 3: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.2e-26) t_1 (if (<= z 1.28e-5) (fma (/ 2.0 t) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.2e-26) {
		tmp = t_1;
	} else if (z <= 1.28e-5) {
		tmp = fma((2.0 / t), z, 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 <= -1.2e-26)
		tmp = t_1;
	elseif (z <= 1.28e-5)
		tmp = fma(Float64(2.0 / t), z, 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, -1.2e-26], t$95$1, If[LessEqual[z, 1.28e-5], N[(N[(2.0 / t), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e-26 or 1.2799999999999999e-5 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 87.0

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

   5. Applied rewrites 87.0%

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

   if -1.2e-26 < z < 1.2799999999999999e-5

   1. Initial program 87.8%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

Alternative 4: 63.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{z}{t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.1e-202) t_1 (if (<= z 3.5e-233) (* (/ z t) 2.0) 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-202) {
		tmp = t_1;
	} else if (z <= 3.5e-233) {
		tmp = (z / t) * 2.0;
	} 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 <= (-1.1d-202)) then
        tmp = t_1
    else if (z <= 3.5d-233) then
        tmp = (z / t) * 2.0d0
    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 <= -1.1e-202) {
		tmp = t_1;
	} else if (z <= 3.5e-233) {
		tmp = (z / t) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1.1e-202:
		tmp = t_1
	elif z <= 3.5e-233:
		tmp = (z / t) * 2.0
	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-202)
		tmp = t_1;
	elseif (z <= 3.5e-233)
		tmp = Float64(Float64(z / t) * 2.0);
	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 <= -1.1e-202)
		tmp = t_1;
	elseif (z <= 3.5e-233)
		tmp = (z / t) * 2.0;
	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, -1.1e-202], t$95$1, If[LessEqual[z, 3.5e-233], N[(N[(z / t), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000004e-202 or 3.49999999999999991e-233 < z

   1. Initial program 75.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -1.10000000000000004e-202 < z < 3.49999999999999991e-233

   1. Initial program 87.7%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 33.8%

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

Alternative 5: 63.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{2}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.1e-202) t_1 (if (<= z 3.5e-233) (* (/ 2.0 t) z) 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-202) {
		tmp = t_1;
	} else if (z <= 3.5e-233) {
		tmp = (2.0 / t) * z;
	} 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 <= (-1.1d-202)) then
        tmp = t_1
    else if (z <= 3.5d-233) then
        tmp = (2.0d0 / t) * z
    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 <= -1.1e-202) {
		tmp = t_1;
	} else if (z <= 3.5e-233) {
		tmp = (2.0 / t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1.1e-202:
		tmp = t_1
	elif z <= 3.5e-233:
		tmp = (2.0 / t) * z
	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-202)
		tmp = t_1;
	elseif (z <= 3.5e-233)
		tmp = Float64(Float64(2.0 / t) * z);
	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 <= -1.1e-202)
		tmp = t_1;
	elseif (z <= 3.5e-233)
		tmp = (2.0 / t) * z;
	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, -1.1e-202], t$95$1, If[LessEqual[z, 3.5e-233], N[(N[(2.0 / t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000004e-202 or 3.49999999999999991e-233 < z

   1. Initial program 75.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -1.10000000000000004e-202 < z < 3.49999999999999991e-233

   1. Initial program 87.7%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 33.8%

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

   10. Applied rewrites 33.6%

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

Alternative 6: 63.7% 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 77.3%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 60.3

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

5. Applied rewrites 60.3%

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

Alternative 7: 15.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{-y}{z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Taylor expanded in 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. associate-*r* N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. distribute-neg-in N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

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

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

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

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

   17. metadata-eval N/A

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

   18. associate-*r* N/A

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

5. Applied rewrites 20.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 14.7%

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

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z)))))

C

double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (1.0d0 / ((z / y) - ((t / 2.0d0) / z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Python

def code(x, y, z, t):
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(1.0 / Float64(Float64(z / y) - Float64(Float64(t / 2.0) / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (1.0 / ((z / y) - ((t / 2.0) / z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(1.0 / N[(N[(z / y), $MachinePrecision] - N[(N[(t / 2.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024257 
(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)))))