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

Percentage Accurate: 81.9% → 98.3%

Time: 7.5s

Alternatives: 7

Speedup: 1.5×

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: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - ((y * 2.0) / ((2.0 * z) - (y / (z / 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) / ((2.0d0 * z) - (y / (z / t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

   1. remove-double-neg 85.1%

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

   2. neg-mul-1 85.1%

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

   3. *-commutative 85.1%

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

   4. *-commutative 85.1%

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

   5. neg-mul-1 85.1%

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

   6. remove-double-neg 85.1%

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

   7. associate-/l* 90.3%

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

   8. associate-*l* 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in z around 0 95.5%

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

   1. +-commutative 95.5%

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

   2. mul-1-neg 95.5%

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

   3. associate-/l* 96.2%

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

   4. unsub-neg 96.2%

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

   5. *-commutative 96.2%

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

   6. associate-/r/ 98.1%

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

   7. *-commutative 98.1%

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

6. Simplified 98.1%

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

   1. clear-num 98.1%

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

   2. un-div-inv 98.1%

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

8. Applied egg-rr 98.1%

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

9. Final simplification 98.1%

   \[\leadsto x - \frac{y \cdot 2}{2 \cdot z - \frac{y}{\frac{z}{t}}} \]

Alternative 2: 96.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (y * (2.0 / ((2.0 * z) - (t * (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 * (2.0d0 / ((2.0d0 * z) - (t * (y / z)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

   1. remove-double-neg 85.1%

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

   2. neg-mul-1 85.1%

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

   3. *-commutative 85.1%

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

   4. *-commutative 85.1%

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

   5. neg-mul-1 85.1%

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

   6. remove-double-neg 85.1%

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

   7. associate-/l* 90.3%

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

   8. associate-*l* 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in z around 0 95.5%

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

   1. +-commutative 95.5%

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

   2. mul-1-neg 95.5%

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

   3. associate-/l* 96.2%

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

   4. unsub-neg 96.2%

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

   5. *-commutative 96.2%

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

   6. associate-/r/ 98.1%

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

   7. *-commutative 98.1%

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

6. Simplified 98.1%

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

   1. expm1-log1p-u 91.6%

      \[\leadsto x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}}\right)\right)} \]

   2. expm1-udef 79.2%

      \[\leadsto x - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y \cdot 2}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right)} \]

   3. *-commutative 79.2%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{2 \cdot y}}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right) \]

   4. *-un-lft-identity 79.2%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\frac{2 \cdot y}{\color{blue}{1 \cdot \left(z \cdot 2 - y \cdot \frac{t}{z}\right)}}\right)} - 1\right) \]

   5. times-frac 79.2%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{1} \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}}\right)} - 1\right) \]

   6. metadata-eval 79.2%

      \[\leadsto x - \left(e^{\mathsf{log1p}\left(\color{blue}{2} \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right) \]

8. Applied egg-rr 79.2%

   \[\leadsto x - \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 91.6%

      \[\leadsto x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{y}{z \cdot 2 - y \cdot \frac{t}{z}}\right)\right)} \]

   2. expm1-log1p 98.1%

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

   3. *-commutative 98.1%

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

   4. associate-*l/ 98.1%

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

   5. associate-*r/ 98.1%

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

   6. *-commutative 98.1%

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

   7. associate-*l/ 95.4%

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

   8. associate-*r/ 96.2%

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

10. Simplified 96.2%

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

11. Final simplification 96.2%

    \[\leadsto x - y \cdot \frac{2}{2 \cdot z - t \cdot \frac{y}{z}} \]

Alternative 3: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - ((y * 2.0) / ((2.0 * z) - (y * (t / 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 * 2.0d0) / ((2.0d0 * z) - (y * (t / z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

   1. remove-double-neg 85.1%

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

   2. neg-mul-1 85.1%

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

   3. *-commutative 85.1%

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

   4. *-commutative 85.1%

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

   5. neg-mul-1 85.1%

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

   6. remove-double-neg 85.1%

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

   7. associate-/l* 90.3%

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

   8. associate-*l* 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in z around 0 95.5%

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

   1. +-commutative 95.5%

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

   2. mul-1-neg 95.5%

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

   3. associate-/l* 96.2%

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

   4. unsub-neg 96.2%

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

   5. *-commutative 96.2%

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

   6. associate-/r/ 98.1%

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

   7. *-commutative 98.1%

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

6. Simplified 98.1%

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

7. Final simplification 98.1%

   \[\leadsto x - \frac{y \cdot 2}{2 \cdot z - y \cdot \frac{t}{z}} \]

Alternative 4: 89.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+19} \lor \neg \left(z \leq 3.7 \cdot 10^{-5}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.2e+19) (not (<= z 3.7e-5)))
   (- x (/ y z))
   (+ x (* z (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.2e+19) || !(z <= 3.7e-5)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-8.2d+19)) .or. (.not. (z <= 3.7d-5))) then
        tmp = x - (y / z)
    else
        tmp = x + (z * (2.0d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.2e+19) || !(z <= 3.7e-5)) {
		tmp = x - (y / z);
	} else {
		tmp = x + (z * (2.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.2e+19) or not (z <= 3.7e-5):
		tmp = x - (y / z)
	else:
		tmp = x + (z * (2.0 / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.2e+19) || !(z <= 3.7e-5))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x + Float64(z * Float64(2.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -8.2e+19) || ~((z <= 3.7e-5)))
		tmp = x - (y / z);
	else
		tmp = x + (z * (2.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.2e+19], N[Not[LessEqual[z, 3.7e-5]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e19 or 3.69999999999999981e-5 < z

   1. Initial program 77.2%

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

      1. remove-double-neg 77.2%

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

      2. neg-mul-1 77.2%

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

      3. *-commutative 77.2%

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

      4. *-commutative 77.2%

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

      5. neg-mul-1 77.2%

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

      6. remove-double-neg 77.2%

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

      7. associate-/l* 87.8%

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

      8. associate-*l* 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in y around 0 92.5%

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

   if -8.2e19 < z < 3.69999999999999981e-5

   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. Step-by-step derivation

      1. sub-neg 92.6%

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

      2. associate-/l* 92.7%

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

      3. distribute-neg-frac 92.7%

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

      4. distribute-lft-neg-out 92.7%

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

      5. associate-/r/ 94.8%

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

      6. distribute-lft-neg-out 94.8%

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

      7. distribute-rgt-neg-in 94.8%

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

      8. metadata-eval 94.8%

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

      9. *-commutative 94.8%

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

      10. associate-*l* 94.8%

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

      11. fma-neg 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around inf 89.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+19} \lor \neg \left(z \leq 3.7 \cdot 10^{-5}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]

Alternative 5: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+29} \lor \neg \left(z \leq 0.058\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{t}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.5e+29) (not (<= z 0.058)))
   (- x (/ y z))
   (- x (/ z (/ t -2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e+29) || !(z <= 0.058)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (z / (t / -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-5.5d+29)) .or. (.not. (z <= 0.058d0))) then
        tmp = x - (y / z)
    else
        tmp = x - (z / (t / (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e+29) || !(z <= 0.058)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (z / (t / -2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.5e+29) or not (z <= 0.058):
		tmp = x - (y / z)
	else:
		tmp = x - (z / (t / -2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.5e+29) || !(z <= 0.058))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(z / Float64(t / -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.5e+29) || ~((z <= 0.058)))
		tmp = x - (y / z);
	else
		tmp = x - (z / (t / -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.5e+29], N[Not[LessEqual[z, 0.058]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e29 or 0.0580000000000000029 < z

   1. Initial program 76.8%

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

      1. remove-double-neg 76.8%

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

      2. neg-mul-1 76.8%

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

      3. *-commutative 76.8%

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

      4. *-commutative 76.8%

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

      5. neg-mul-1 76.8%

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

      6. remove-double-neg 76.8%

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

      7. associate-/l* 87.6%

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

      8. associate-*l* 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in y around 0 93.2%

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

   if -5.5e29 < z < 0.0580000000000000029

   1. Initial program 92.7%

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

      1. remove-double-neg 92.7%

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

      2. neg-mul-1 92.7%

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

      3. *-commutative 92.7%

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

      4. *-commutative 92.7%

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

      5. neg-mul-1 92.7%

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

      6. remove-double-neg 92.7%

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

      7. associate-/l* 92.8%

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

      8. associate-*l* 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in y around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. associate-/l* 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+29} \lor \neg \left(z \leq 0.058\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{t}{-2}}\\ \end{array} \]

Alternative 6: 78.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z 1.15e+21) x (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.15e+21) {
		tmp = x;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.15d+21) then
        tmp = x
    else
        tmp = x - (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.15e+21) {
		tmp = x;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.15e+21:
		tmp = x
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.15e+21)
		tmp = x;
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.15e+21)
		tmp = x;
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.15e+21], x, N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.15e21

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. associate-/l* 92.8%

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

      3. distribute-neg-frac 92.8%

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

      4. distribute-lft-neg-out 92.8%

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

      5. associate-/r/ 94.2%

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

      6. distribute-lft-neg-out 94.2%

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

      7. distribute-rgt-neg-in 94.2%

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

      8. metadata-eval 94.2%

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

      9. *-commutative 94.2%

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

      10. associate-*l* 94.2%

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

      11. fma-neg 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in y around inf 73.9%

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

   5. Taylor expanded in x around inf 78.1%

      \[\leadsto \color{blue}{x} \]

   if 1.15e21 < z

   1. Initial program 73.4%

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

      1. remove-double-neg 73.4%

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

      2. neg-mul-1 73.4%

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

      3. *-commutative 73.4%

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

      4. *-commutative 73.4%

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

      5. neg-mul-1 73.4%

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

      6. remove-double-neg 73.4%

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

      7. associate-/l* 83.6%

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

      8. associate-*l* 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in y around 0 97.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

Alternative 7: 75.7% accurate, 17.0× 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 85.1%

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

   1. sub-neg 85.1%

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

   2. associate-/l* 90.3%

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

   3. distribute-neg-frac 90.3%

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

   4. distribute-lft-neg-out 90.3%

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

   5. associate-/r/ 91.4%

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

   6. distribute-lft-neg-out 91.4%

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

   7. distribute-rgt-neg-in 91.4%

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

   8. metadata-eval 91.4%

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

   9. *-commutative 91.4%

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

   10. associate-*l* 91.4%

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

   11. fma-neg 91.4%

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

3. Simplified 91.4%

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

4. Taylor expanded in y around inf 61.0%

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

5. Taylor expanded in x around inf 77.7%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 77.7%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.3× 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 2023316 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))