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

Percentage Accurate: 81.9% → 93.5%

Time: 10.9s

Alternatives: 8

Speedup: 1.0×

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: 93.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 1.5 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x (/ (* (* y 2.0) z) (- (* y t) (* z (* 2.0 z))))) 1.5e+263)
   (fma y (* z (/ 2.0 (fma z (* z -2.0) (* y t)))) x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + (((y * 2.0) * z) / ((y * t) - (z * (2.0 * z))))) <= 1.5e+263) {
		tmp = fma(y, (z * (2.0 / fma(z, (z * -2.0), (y * t)))), x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(y * t) - Float64(z * Float64(2.0 * z))))) <= 1.5e+263)
		tmp = fma(y, Float64(z * Float64(2.0 / fma(z, Float64(z * -2.0), Float64(y * t)))), x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5e+263], N[(y * N[(z * N[(2.0 / N[(z * N[(z * -2.0), $MachinePrecision] + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 1.49999999999999995e263

   1. Initial program 95.2%

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

   3. Simplified 97.3%

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

   if 1.49999999999999995e263 < (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))))

   1. Initial program 16.5%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in y around 0 82.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 1.5 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot t - z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{t\_1} \leq 1.5 \cdot 10^{+263}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y t) (* z (* 2.0 z)))))
   (if (<= (+ x (/ (* (* y 2.0) z) t_1)) 1.5e+263)
     (+ x (* (* y 2.0) (/ z t_1)))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * t) - (z * (2.0 * z));
	double tmp;
	if ((x + (((y * 2.0) * z) / t_1)) <= 1.5e+263) {
		tmp = x + ((y * 2.0) * (z / t_1));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (y * t) - (z * (2.0d0 * z))
    if ((x + (((y * 2.0d0) * z) / t_1)) <= 1.5d+263) then
        tmp = x + ((y * 2.0d0) * (z / t_1))
    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 t_1 = (y * t) - (z * (2.0 * z));
	double tmp;
	if ((x + (((y * 2.0) * z) / t_1)) <= 1.5e+263) {
		tmp = x + ((y * 2.0) * (z / t_1));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * t) - (z * (2.0 * z))
	tmp = 0
	if (x + (((y * 2.0) * z) / t_1)) <= 1.5e+263:
		tmp = x + ((y * 2.0) * (z / t_1))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * t) - Float64(z * Float64(2.0 * z)))
	tmp = 0.0
	if (Float64(x + Float64(Float64(Float64(y * 2.0) * z) / t_1)) <= 1.5e+263)
		tmp = Float64(x + Float64(Float64(y * 2.0) * Float64(z / t_1)));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * t) - (z * (2.0 * z));
	tmp = 0.0;
	if ((x + (((y * 2.0) * z) / t_1)) <= 1.5e+263)
		tmp = x + ((y * 2.0) * (z / t_1));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * t), $MachinePrecision] - N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], 1.5e+263], N[(x + N[(N[(y * 2.0), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 1.49999999999999995e263

   1. Initial program 95.2%

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

   3. Simplified 97.3%

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

   if 1.49999999999999995e263 < (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))))

   1. Initial program 16.5%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in y around 0 82.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 1.5 \cdot 10^{+263}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.25:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.25)
     t_1
     (if (<= z 9.5e-199)
       x
       (if (<= z 1.2e-139) (* 2.0 (/ z t)) (if (<= z 25.0) x t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.25) {
		tmp = t_1;
	} else if (z <= 9.5e-199) {
		tmp = x;
	} else if (z <= 1.2e-139) {
		tmp = 2.0 * (z / t);
	} else if (z <= 25.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (y / z)
    if (z <= (-1.25d0)) then
        tmp = t_1
    else if (z <= 9.5d-199) then
        tmp = x
    else if (z <= 1.2d-139) then
        tmp = 2.0d0 * (z / t)
    else if (z <= 25.0d0) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.25) {
		tmp = t_1;
	} else if (z <= 9.5e-199) {
		tmp = x;
	} else if (z <= 1.2e-139) {
		tmp = 2.0 * (z / t);
	} else if (z <= 25.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -1.25:
		tmp = t_1
	elif z <= 9.5e-199:
		tmp = x
	elif z <= 1.2e-139:
		tmp = 2.0 * (z / t)
	elif z <= 25.0:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -1.25)
		tmp = t_1;
	elseif (z <= 9.5e-199)
		tmp = x;
	elseif (z <= 1.2e-139)
		tmp = Float64(2.0 * Float64(z / t));
	elseif (z <= 25.0)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -1.25)
		tmp = t_1;
	elseif (z <= 9.5e-199)
		tmp = x;
	elseif (z <= 1.2e-139)
		tmp = 2.0 * (z / t);
	elseif (z <= 25.0)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.25], t$95$1, If[LessEqual[z, 9.5e-199], x, If[LessEqual[z, 1.2e-139], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 25.0], x, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25 or 25 < z

   1. Initial program 77.5%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in y around 0 91.4%

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

   if -1.25 < z < 9.5000000000000005e-199 or 1.20000000000000007e-139 < z < 25

   1. Initial program 90.3%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in x around inf 80.8%

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

   if 9.5000000000000005e-199 < z < 1.20000000000000007e-139

   1. Initial program 64.0%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. *-commutative 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in x around 0 75.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.07:\\ \;\;\;\;x + \frac{-1}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 10^{-198}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 3800:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.07)
   (+ x (/ -1.0 (/ z y)))
   (if (<= z 1e-198)
     x
     (if (<= z 1.2e-139) (* 2.0 (/ z t)) (if (<= z 3800.0) x (- x (/ y z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.07) {
		tmp = x + (-1.0 / (z / y));
	} else if (z <= 1e-198) {
		tmp = x;
	} else if (z <= 1.2e-139) {
		tmp = 2.0 * (z / t);
	} else if (z <= 3800.0) {
		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 <= (-0.07d0)) then
        tmp = x + ((-1.0d0) / (z / y))
    else if (z <= 1d-198) then
        tmp = x
    else if (z <= 1.2d-139) then
        tmp = 2.0d0 * (z / t)
    else if (z <= 3800.0d0) 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 <= -0.07) {
		tmp = x + (-1.0 / (z / y));
	} else if (z <= 1e-198) {
		tmp = x;
	} else if (z <= 1.2e-139) {
		tmp = 2.0 * (z / t);
	} else if (z <= 3800.0) {
		tmp = x;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.07:
		tmp = x + (-1.0 / (z / y))
	elif z <= 1e-198:
		tmp = x
	elif z <= 1.2e-139:
		tmp = 2.0 * (z / t)
	elif z <= 3800.0:
		tmp = x
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.07)
		tmp = Float64(x + Float64(-1.0 / Float64(z / y)));
	elseif (z <= 1e-198)
		tmp = x;
	elseif (z <= 1.2e-139)
		tmp = Float64(2.0 * Float64(z / t));
	elseif (z <= 3800.0)
		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 <= -0.07)
		tmp = x + (-1.0 / (z / y));
	elseif (z <= 1e-198)
		tmp = x;
	elseif (z <= 1.2e-139)
		tmp = 2.0 * (z / t);
	elseif (z <= 3800.0)
		tmp = x;
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.07], N[(x + N[(-1.0 / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-198], x, If[LessEqual[z, 1.2e-139], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3800.0], x, N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -0.070000000000000007

   1. Initial program 76.4%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in z around inf 90.1%

      \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{\frac{0.5}{z}} \]
   6. Step-by-step derivation

      1. associate-*l* 90.1%

         \[\leadsto x - \color{blue}{y \cdot \left(2 \cdot \frac{0.5}{z}\right)} \]

      2. div-inv 90.1%

         \[\leadsto x - y \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{1}{z}\right)}\right) \]

      3. associate-*r* 90.1%

         \[\leadsto x - y \cdot \color{blue}{\left(\left(2 \cdot 0.5\right) \cdot \frac{1}{z}\right)} \]

      4. metadata-eval 90.1%

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

      5. *-un-lft-identity 90.1%

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

      6. div-inv 90.2%

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

      7. clear-num 90.3%

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

   7. Applied egg-rr 90.3%

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

   if -0.070000000000000007 < z < 9.9999999999999991e-199 or 1.20000000000000007e-139 < z < 3800

   1. Initial program 90.3%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in x around inf 80.8%

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

   if 9.9999999999999991e-199 < z < 1.20000000000000007e-139

   1. Initial program 64.0%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. *-commutative 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in x around 0 75.8%

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

   if 3800 < z

   1. Initial program 78.3%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in y around 0 92.3%

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.07:\\ \;\;\;\;x + \frac{-1}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 10^{-198}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 3800:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{-z}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-173}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-291}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-202}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-176}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (- z))))
   (if (<= x -7.8e-173)
     x
     (if (<= x 4e-291)
       t_1
       (if (<= x 7e-202) (* 2.0 (/ z t)) (if (<= x 2.3e-176) t_1 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / -z;
	double tmp;
	if (x <= -7.8e-173) {
		tmp = x;
	} else if (x <= 4e-291) {
		tmp = t_1;
	} else if (x <= 7e-202) {
		tmp = 2.0 * (z / t);
	} else if (x <= 2.3e-176) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	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 = y / -z
    if (x <= (-7.8d-173)) then
        tmp = x
    else if (x <= 4d-291) then
        tmp = t_1
    else if (x <= 7d-202) then
        tmp = 2.0d0 * (z / t)
    else if (x <= 2.3d-176) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / -z;
	double tmp;
	if (x <= -7.8e-173) {
		tmp = x;
	} else if (x <= 4e-291) {
		tmp = t_1;
	} else if (x <= 7e-202) {
		tmp = 2.0 * (z / t);
	} else if (x <= 2.3e-176) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / -z
	tmp = 0
	if x <= -7.8e-173:
		tmp = x
	elif x <= 4e-291:
		tmp = t_1
	elif x <= 7e-202:
		tmp = 2.0 * (z / t)
	elif x <= 2.3e-176:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(-z))
	tmp = 0.0
	if (x <= -7.8e-173)
		tmp = x;
	elseif (x <= 4e-291)
		tmp = t_1;
	elseif (x <= 7e-202)
		tmp = Float64(2.0 * Float64(z / t));
	elseif (x <= 2.3e-176)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / -z;
	tmp = 0.0;
	if (x <= -7.8e-173)
		tmp = x;
	elseif (x <= 4e-291)
		tmp = t_1;
	elseif (x <= 7e-202)
		tmp = 2.0 * (z / t);
	elseif (x <= 2.3e-176)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / (-z)), $MachinePrecision]}, If[LessEqual[x, -7.8e-173], x, If[LessEqual[x, 4e-291], t$95$1, If[LessEqual[x, 7e-202], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-176], t$95$1, x]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999974e-173 or 2.3000000000000001e-176 < x

   1. Initial program 88.4%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in x around inf 88.6%

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

   if -7.79999999999999974e-173 < x < 3.99999999999999985e-291 or 6.9999999999999998e-202 < x < 2.3000000000000001e-176

   1. Initial program 71.2%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in y around 0 61.4%

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

   6. Taylor expanded in x around 0 53.5%

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

      1. associate-*r/ 53.5%

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

      2. neg-mul-1 53.5%

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

   8. Simplified 53.5%

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

   if 3.99999999999999985e-291 < x < 6.9999999999999998e-202

   1. Initial program 64.9%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in y around inf 53.3%

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

      1. associate-*r/ 53.3%

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

      2. *-commutative 53.3%

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

   7. Simplified 53.3%

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

   8. Taylor expanded in x around 0 49.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-173}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-202}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-176}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -45:\\ \;\;\;\;x + \frac{-1}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 420:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -45.0)
   (+ x (/ -1.0 (/ z y)))
   (if (<= z 420.0) (- x (/ (* z -2.0) t)) (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -45.0) {
		tmp = x + (-1.0 / (z / y));
	} else if (z <= 420.0) {
		tmp = x - ((z * -2.0) / t);
	} 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 <= (-45.0d0)) then
        tmp = x + ((-1.0d0) / (z / y))
    else if (z <= 420.0d0) then
        tmp = x - ((z * (-2.0d0)) / t)
    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 <= -45.0) {
		tmp = x + (-1.0 / (z / y));
	} else if (z <= 420.0) {
		tmp = x - ((z * -2.0) / t);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -45.0:
		tmp = x + (-1.0 / (z / y))
	elif z <= 420.0:
		tmp = x - ((z * -2.0) / t)
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -45.0)
		tmp = Float64(x + Float64(-1.0 / Float64(z / y)));
	elseif (z <= 420.0)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	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 <= -45.0)
		tmp = x + (-1.0 / (z / y));
	elseif (z <= 420.0)
		tmp = x - ((z * -2.0) / t);
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -45.0], N[(x + N[(-1.0 / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 420.0], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -45

   1. Initial program 75.9%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in z around inf 91.9%

      \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{\frac{0.5}{z}} \]
   6. Step-by-step derivation

      1. associate-*l* 91.9%

         \[\leadsto x - \color{blue}{y \cdot \left(2 \cdot \frac{0.5}{z}\right)} \]

      2. div-inv 91.9%

         \[\leadsto x - y \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{1}{z}\right)}\right) \]

      3. associate-*r* 91.9%

         \[\leadsto x - y \cdot \color{blue}{\left(\left(2 \cdot 0.5\right) \cdot \frac{1}{z}\right)} \]

      4. metadata-eval 91.9%

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

      5. *-un-lft-identity 91.9%

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

      6. div-inv 92.0%

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

      7. clear-num 92.0%

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

   7. Applied egg-rr 92.0%

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

   if -45 < z < 420

   1. Initial program 88.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in y around inf 88.5%

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

      1. associate-*r/ 88.5%

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

      2. *-commutative 88.5%

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

   7. Simplified 88.5%

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

   if 420 < z

   1. Initial program 78.3%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in y around 0 92.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -45:\\ \;\;\;\;x + \frac{-1}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 420:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-171}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.8e-171) x (if (<= x 5.2e-177) (/ y (- z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.8e-171) {
		tmp = x;
	} else if (x <= 5.2e-177) {
		tmp = y / -z;
	} else {
		tmp = x;
	}
	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 (x <= (-2.8d-171)) then
        tmp = x
    else if (x <= 5.2d-177) then
        tmp = y / -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.8e-171) {
		tmp = x;
	} else if (x <= 5.2e-177) {
		tmp = y / -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.8e-171:
		tmp = x
	elif x <= 5.2e-177:
		tmp = y / -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.8e-171)
		tmp = x;
	elseif (x <= 5.2e-177)
		tmp = Float64(y / Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.8e-171)
		tmp = x;
	elseif (x <= 5.2e-177)
		tmp = y / -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.8e-171], x, If[LessEqual[x, 5.2e-177], N[(y / (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000023e-171 or 5.2000000000000002e-177 < x

   1. Initial program 88.4%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in x around inf 88.6%

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

   if -2.80000000000000023e-171 < x < 5.2000000000000002e-177

   1. Initial program 68.6%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in y around 0 55.7%

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

   6. Taylor expanded in x around 0 43.4%

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

      1. associate-*r/ 43.4%

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

      2. neg-mul-1 43.4%

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

   8. Simplified 43.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-171}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% 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 83.8%

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

3. Simplified 89.3%

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

5. Taylor expanded in x around inf 73.6%

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

6. Final simplification 73.6%

   \[\leadsto x \]
7. Add Preprocessing

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 2024067 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

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