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

Percentage Accurate: 81.5% → 96.2%

Time: 8.8s

Alternatives: 5

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: 81.5% 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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 91.9%

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

5. Taylor expanded in t around 0

   \[\leadsto x - \frac{2 \cdot y}{\color{blue}{-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 97.8

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

7. Applied rewrites 97.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

9. Applied rewrites 97.8%

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

10. Final simplification 97.8%

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

Alternative 2: 88.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 14500:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.6e+85) t_1 (if (<= z 14500.0) (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+85) {
		tmp = t_1;
	} else if (z <= 14500.0) {
		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+85)
		tmp = t_1;
	elseif (z <= 14500.0)
		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+85], t$95$1, If[LessEqual[z, 14500.0], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000009e85 or 14500 < z

   1. Initial program 70.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.60000000000000009e85 < z < 14500

   1. Initial program 91.5%

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

   3. Taylor expanded in 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 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.8%

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

Alternative 3: 88.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 14500:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.6e+85) t_1 (if (<= z 14500.0) (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 <= -1.6e+85) {
		tmp = t_1;
	} else if (z <= 14500.0) {
		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 <= -1.6e+85)
		tmp = t_1;
	elseif (z <= 14500.0)
		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, -1.6e+85], t$95$1, If[LessEqual[z, 14500.0], N[(N[(2.0 / t), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000009e85 or 14500 < z

   1. Initial program 70.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.60000000000000009e85 < z < 14500

   1. Initial program 91.5%

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

   3. Taylor expanded in 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 88.7

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

   5. Applied rewrites 88.7%

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

Alternative 4: 62.4% 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 82.1%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
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 64.0

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

5. Applied rewrites 64.0%

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

Alternative 5: 14.6% 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 82.1%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
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 64.0

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

5. Applied rewrites 64.0%

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

6. Taylor expanded in x around 0

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

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

   7. mul-1-neg N/A

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

   8. distribute-neg-in N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

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

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

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

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. unpow2 N/A

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

   16. mul-1-neg N/A

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

   17. remove-double-neg N/A

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

8. Applied rewrites 16.5%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 13.2%

    \[\leadsto \frac{-y}{\color{blue}{z}} \]
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 2024236 
(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)))))