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

Percentage Accurate: 82.3% → 97.3%

Time: 9.4s

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: 82.3% 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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(z \cdot 2\right)}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-162}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \left(t \cdot -0.5\right) \cdot \frac{\frac{y}{z}}{\frac{z}{y}}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (* (* y 2.0) (/ z (- (* y t) (* z (* z 2.0))))))))
   (if (<= z -1e+154)
     (- x (/ y z))
     (if (<= z -6e-166)
       t_1
       (if (<= z 1.1e-162)
         (- x (/ (* z -2.0) t))
         (if (<= z 7.2e+141)
           t_1
           (- x (/ (- y (* (* t -0.5) (/ (/ y z) (/ z y)))) z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))));
	double tmp;
	if (z <= -1e+154) {
		tmp = x - (y / z);
	} else if (z <= -6e-166) {
		tmp = t_1;
	} else if (z <= 1.1e-162) {
		tmp = x - ((z * -2.0) / t);
	} else if (z <= 7.2e+141) {
		tmp = t_1;
	} else {
		tmp = x - ((y - ((t * -0.5) * ((y / z) / (z / 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 = x + ((y * 2.0d0) * (z / ((y * t) - (z * (z * 2.0d0)))))
    if (z <= (-1d+154)) then
        tmp = x - (y / z)
    else if (z <= (-6d-166)) then
        tmp = t_1
    else if (z <= 1.1d-162) then
        tmp = x - ((z * (-2.0d0)) / t)
    else if (z <= 7.2d+141) then
        tmp = t_1
    else
        tmp = x - ((y - ((t * (-0.5d0)) * ((y / z) / (z / y)))) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))));
	double tmp;
	if (z <= -1e+154) {
		tmp = x - (y / z);
	} else if (z <= -6e-166) {
		tmp = t_1;
	} else if (z <= 1.1e-162) {
		tmp = x - ((z * -2.0) / t);
	} else if (z <= 7.2e+141) {
		tmp = t_1;
	} else {
		tmp = x - ((y - ((t * -0.5) * ((y / z) / (z / y)))) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))))
	tmp = 0
	if z <= -1e+154:
		tmp = x - (y / z)
	elif z <= -6e-166:
		tmp = t_1
	elif z <= 1.1e-162:
		tmp = x - ((z * -2.0) / t)
	elif z <= 7.2e+141:
		tmp = t_1
	else:
		tmp = x - ((y - ((t * -0.5) * ((y / z) / (z / y)))) / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(y * 2.0) * Float64(z / Float64(Float64(y * t) - Float64(z * Float64(z * 2.0))))))
	tmp = 0.0
	if (z <= -1e+154)
		tmp = Float64(x - Float64(y / z));
	elseif (z <= -6e-166)
		tmp = t_1;
	elseif (z <= 1.1e-162)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	elseif (z <= 7.2e+141)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(Float64(y - Float64(Float64(t * -0.5) * Float64(Float64(y / z) / Float64(z / y)))) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))));
	tmp = 0.0;
	if (z <= -1e+154)
		tmp = x - (y / z);
	elseif (z <= -6e-166)
		tmp = t_1;
	elseif (z <= 1.1e-162)
		tmp = x - ((z * -2.0) / t);
	elseif (z <= 7.2e+141)
		tmp = t_1;
	else
		tmp = x - ((y - ((t * -0.5) * ((y / z) / (z / y)))) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(y * 2.0), $MachinePrecision] * N[(z / N[(N[(y * t), $MachinePrecision] - N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+154], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -6e-166], t$95$1, If[LessEqual[z, 1.1e-162], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e+141], t$95$1, N[(x - N[(N[(y - N[(N[(t * -0.5), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.00000000000000004e154

   1. Initial program 60.0%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in y around 0 100.0%

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

   if -1.00000000000000004e154 < z < -6.0000000000000005e-166 or 1.1e-162 < z < 7.2000000000000003e141

   1. Initial program 87.4%

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

   3. Simplified 94.0%

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

   if -6.0000000000000005e-166 < z < 1.1e-162

   1. Initial program 76.5%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in y around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if 7.2000000000000003e141 < z

   1. Initial program 61.6%

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

   3. Simplified 75.1%

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

      1. clear-num 75.1%

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

      2. un-div-inv 75.1%

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

      3. *-commutative 75.1%

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

      4. *-commutative 75.1%

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

      5. associate-*l* 75.1%

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

      6. pow2 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. *-commutative 75.1%

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

      4. *-commutative 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in z around -inf 77.4%

      \[\leadsto x - \color{blue}{-1 \cdot \frac{-1 \cdot y + -0.5 \cdot \frac{t \cdot {y}^{2}}{{z}^{2}}}{z}} \]
   10. Step-by-step derivation

       1. mul-1-neg 77.4%

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

       2. neg-mul-1 77.4%

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

       3. distribute-neg-frac2 77.4%

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

   11. Simplified 96.9%

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

       1. clear-num 96.9%

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

       2. pow2 96.9%

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

       3. un-div-inv 96.9%

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

       4. clear-num 96.9%

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

   13. Applied egg-rr 96.9%

       \[\leadsto x - \frac{\left(-0.5 \cdot t\right) \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{z}{y}}} - y}{-z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-166}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(z \cdot 2\right)}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-162}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+141}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \left(t \cdot -0.5\right) \cdot \frac{\frac{y}{z}}{\frac{z}{y}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+141}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - 2 \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \left(t \cdot -0.5\right) \cdot \frac{\frac{y}{z}}{\frac{z}{y}}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6e-166)
   (fma y (* z (/ 2.0 (fma z (* z -2.0) (* y t)))) x)
   (if (<= z 2.1e-138)
     (- x (/ (* z -2.0) t))
     (if (<= z 5.8e+141)
       (+ x (* z (/ (* y 2.0) (- (* y t) (* 2.0 (pow z 2.0))))))
       (- x (/ (- y (* (* t -0.5) (/ (/ y z) (/ z y)))) z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e-166) {
		tmp = fma(y, (z * (2.0 / fma(z, (z * -2.0), (y * t)))), x);
	} else if (z <= 2.1e-138) {
		tmp = x - ((z * -2.0) / t);
	} else if (z <= 5.8e+141) {
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * pow(z, 2.0)))));
	} else {
		tmp = x - ((y - ((t * -0.5) * ((y / z) / (z / y)))) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6e-166)
		tmp = fma(y, Float64(z * Float64(2.0 / fma(z, Float64(z * -2.0), Float64(y * t)))), x);
	elseif (z <= 2.1e-138)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	elseif (z <= 5.8e+141)
		tmp = Float64(x + Float64(z * Float64(Float64(y * 2.0) / Float64(Float64(y * t) - Float64(2.0 * (z ^ 2.0))))));
	else
		tmp = Float64(x - Float64(Float64(y - Float64(Float64(t * -0.5) * Float64(Float64(y / z) / Float64(z / y)))) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6e-166], N[(y * N[(z * N[(2.0 / N[(z * N[(z * -2.0), $MachinePrecision] + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.1e-138], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e+141], N[(x + N[(z * N[(N[(y * 2.0), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(2.0 * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y - N[(N[(t * -0.5), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.0000000000000005e-166

   1. Initial program 75.2%

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

   3. Simplified 90.1%

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

   if -6.0000000000000005e-166 < z < 2.09999999999999986e-138

   1. Initial program 78.5%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in y around inf 98.3%

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

      1. associate-*r/ 98.3%

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

      2. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   if 2.09999999999999986e-138 < z < 5.80000000000000013e141

   1. Initial program 94.5%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   if 5.80000000000000013e141 < z

   1. Initial program 61.6%

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

   3. Simplified 75.1%

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

      1. clear-num 75.1%

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

      2. un-div-inv 75.1%

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

      3. *-commutative 75.1%

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

      4. *-commutative 75.1%

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

      5. associate-*l* 75.1%

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

      6. pow2 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. *-commutative 75.1%

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

      4. *-commutative 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in z around -inf 77.4%

      \[\leadsto x - \color{blue}{-1 \cdot \frac{-1 \cdot y + -0.5 \cdot \frac{t \cdot {y}^{2}}{{z}^{2}}}{z}} \]
   10. Step-by-step derivation

       1. mul-1-neg 77.4%

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

       2. neg-mul-1 77.4%

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

       3. distribute-neg-frac2 77.4%

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

   11. Simplified 96.9%

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

       1. clear-num 96.9%

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

       2. pow2 96.9%

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

       3. un-div-inv 96.9%

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

       4. clear-num 96.9%

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

   13. Applied egg-rr 96.9%

       \[\leadsto x - \frac{\left(-0.5 \cdot t\right) \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{z}{y}}} - y}{-z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{2}{\mathsf{fma}\left(z, z \cdot -2, y \cdot t\right)}, x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+141}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - 2 \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \left(t \cdot -0.5\right) \cdot \frac{\frac{y}{z}}{\frac{z}{y}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{z \cdot \left(y \cdot 2\right)}{y \cdot t - z \cdot \left(z \cdot 2\right)} \leq 2 \cdot 10^{+264}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - 2 \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x (/ (* z (* y 2.0)) (- (* y t) (* z (* z 2.0))))) 2e+264)
   (+ x (* z (/ (* y 2.0) (- (* y t) (* 2.0 (pow z 2.0))))))
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + ((z * (y * 2.0)) / ((y * t) - (z * (z * 2.0))))) <= 2e+264) {
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * pow(z, 2.0)))));
	} 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 ((x + ((z * (y * 2.0d0)) / ((y * t) - (z * (z * 2.0d0))))) <= 2d+264) then
        tmp = x + (z * ((y * 2.0d0) / ((y * t) - (2.0d0 * (z ** 2.0d0)))))
    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 ((x + ((z * (y * 2.0)) / ((y * t) - (z * (z * 2.0))))) <= 2e+264) {
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * Math.pow(z, 2.0)))));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + ((z * (y * 2.0)) / ((y * t) - (z * (z * 2.0))))) <= 2e+264:
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * math.pow(z, 2.0)))))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + Float64(Float64(z * Float64(y * 2.0)) / Float64(Float64(y * t) - Float64(z * Float64(z * 2.0))))) <= 2e+264)
		tmp = Float64(x + Float64(z * Float64(Float64(y * 2.0) / Float64(Float64(y * t) - Float64(2.0 * (z ^ 2.0))))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + ((z * (y * 2.0)) / ((y * t) - (z * (z * 2.0))))) <= 2e+264)
		tmp = x + (z * ((y * 2.0) / ((y * t) - (2.0 * (z ^ 2.0)))));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + N[(N[(z * N[(y * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+264], N[(x + N[(z * N[(N[(y * 2.0), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] - N[(2.0 * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)))) < 2.00000000000000009e264

   1. Initial program 92.1%

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

   3. Simplified 92.4%

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

      1. clear-num 92.1%

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

      2. un-div-inv 92.1%

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

      3. *-commutative 92.1%

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

      4. *-commutative 92.1%

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

      5. associate-*l* 92.1%

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

      6. pow2 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. associate-/r/ 94.8%

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

      2. *-commutative 94.8%

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

      3. *-commutative 94.8%

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

      4. *-commutative 94.8%

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

   8. Simplified 94.8%

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

   if 2.00000000000000009e264 < (-.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 17.4%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in y around 0 77.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{z \cdot \left(y \cdot 2\right)}{y \cdot t - z \cdot \left(z \cdot 2\right)} \leq 2 \cdot 10^{+264}:\\ \;\;\;\;x + z \cdot \frac{y \cdot 2}{y \cdot t - 2 \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ t_2 := x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(z \cdot 2\right)}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-166}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 10^{-162}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z)))
        (t_2 (+ x (* (* y 2.0) (/ z (- (* y t) (* z (* z 2.0))))))))
   (if (<= z -1e+154)
     t_1
     (if (<= z -6e-166)
       t_2
       (if (<= z 1e-162)
         (- x (/ (* z -2.0) t))
         (if (<= z 8.5e+138) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double t_2 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))));
	double tmp;
	if (z <= -1e+154) {
		tmp = t_1;
	} else if (z <= -6e-166) {
		tmp = t_2;
	} else if (z <= 1e-162) {
		tmp = x - ((z * -2.0) / t);
	} else if (z <= 8.5e+138) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = x - (y / z)
    t_2 = x + ((y * 2.0d0) * (z / ((y * t) - (z * (z * 2.0d0)))))
    if (z <= (-1d+154)) then
        tmp = t_1
    else if (z <= (-6d-166)) then
        tmp = t_2
    else if (z <= 1d-162) then
        tmp = x - ((z * (-2.0d0)) / t)
    else if (z <= 8.5d+138) then
        tmp = t_2
    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 t_2 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))));
	double tmp;
	if (z <= -1e+154) {
		tmp = t_1;
	} else if (z <= -6e-166) {
		tmp = t_2;
	} else if (z <= 1e-162) {
		tmp = x - ((z * -2.0) / t);
	} else if (z <= 8.5e+138) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	t_2 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))))
	tmp = 0
	if z <= -1e+154:
		tmp = t_1
	elif z <= -6e-166:
		tmp = t_2
	elif z <= 1e-162:
		tmp = x - ((z * -2.0) / t)
	elif z <= 8.5e+138:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	t_2 = Float64(x + Float64(Float64(y * 2.0) * Float64(z / Float64(Float64(y * t) - Float64(z * Float64(z * 2.0))))))
	tmp = 0.0
	if (z <= -1e+154)
		tmp = t_1;
	elseif (z <= -6e-166)
		tmp = t_2;
	elseif (z <= 1e-162)
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	elseif (z <= 8.5e+138)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	t_2 = x + ((y * 2.0) * (z / ((y * t) - (z * (z * 2.0)))));
	tmp = 0.0;
	if (z <= -1e+154)
		tmp = t_1;
	elseif (z <= -6e-166)
		tmp = t_2;
	elseif (z <= 1e-162)
		tmp = x - ((z * -2.0) / t);
	elseif (z <= 8.5e+138)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x + N[(N[(y * 2.0), $MachinePrecision] * N[(z / N[(N[(y * t), $MachinePrecision] - N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+154], t$95$1, If[LessEqual[z, -6e-166], t$95$2, If[LessEqual[z, 1e-162], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+138], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000004e154 or 8.5000000000000006e138 < z

   1. Initial program 60.8%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in y around 0 98.4%

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

   if -1.00000000000000004e154 < z < -6.0000000000000005e-166 or 9.99999999999999954e-163 < z < 8.5000000000000006e138

   1. Initial program 87.4%

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

   3. Simplified 94.0%

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

   if -6.0000000000000005e-166 < z < 9.99999999999999954e-163

   1. Initial program 76.5%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in y around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. *-commutative 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-166}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(z \cdot 2\right)}\\ \mathbf{elif}\;z \leq 10^{-162}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+138}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-215}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-232}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-133}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.6e-215)
   x
   (if (<= x 1.36e-232) (* 2.0 (/ z t)) (if (<= x 2.2e-133) (- x (/ y z)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.6e-215) {
		tmp = x;
	} else if (x <= 1.36e-232) {
		tmp = 2.0 * (z / t);
	} else if (x <= 2.2e-133) {
		tmp = x - (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 <= (-4.6d-215)) then
        tmp = x
    else if (x <= 1.36d-232) then
        tmp = 2.0d0 * (z / t)
    else if (x <= 2.2d-133) then
        tmp = x - (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 <= -4.6e-215) {
		tmp = x;
	} else if (x <= 1.36e-232) {
		tmp = 2.0 * (z / t);
	} else if (x <= 2.2e-133) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.6e-215:
		tmp = x
	elif x <= 1.36e-232:
		tmp = 2.0 * (z / t)
	elif x <= 2.2e-133:
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.6e-215)
		tmp = x;
	elseif (x <= 1.36e-232)
		tmp = Float64(2.0 * Float64(z / t));
	elseif (x <= 2.2e-133)
		tmp = Float64(x - Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.6e-215)
		tmp = x;
	elseif (x <= 1.36e-232)
		tmp = 2.0 * (z / t);
	elseif (x <= 2.2e-133)
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.6e-215], x, If[LessEqual[x, 1.36e-232], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-133], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999998e-215 or 2.2000000000000001e-133 < x

   1. Initial program 81.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around inf 85.8%

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

   if -4.5999999999999998e-215 < x < 1.3600000000000001e-232

   1. Initial program 67.4%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in y around inf 74.0%

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

      1. associate-*r/ 74.0%

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

      2. *-commutative 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in x around 0 59.8%

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

   if 1.3600000000000001e-232 < x < 2.2000000000000001e-133

   1. Initial program 72.4%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in y around 0 67.7%

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

Alternative 6: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-211}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-232}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-207}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.6e-211)
   x
   (if (<= x 3e-232) (* 2.0 (/ z t)) (if (<= x 2.25e-207) (/ (- y) z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.6e-211) {
		tmp = x;
	} else if (x <= 3e-232) {
		tmp = 2.0 * (z / t);
	} else if (x <= 2.25e-207) {
		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 <= (-6.6d-211)) then
        tmp = x
    else if (x <= 3d-232) then
        tmp = 2.0d0 * (z / t)
    else if (x <= 2.25d-207) 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 <= -6.6e-211) {
		tmp = x;
	} else if (x <= 3e-232) {
		tmp = 2.0 * (z / t);
	} else if (x <= 2.25e-207) {
		tmp = -y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6.6e-211:
		tmp = x
	elif x <= 3e-232:
		tmp = 2.0 * (z / t)
	elif x <= 2.25e-207:
		tmp = -y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.6e-211)
		tmp = x;
	elseif (x <= 3e-232)
		tmp = Float64(2.0 * Float64(z / t));
	elseif (x <= 2.25e-207)
		tmp = Float64(Float64(-y) / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -6.6e-211)
		tmp = x;
	elseif (x <= 3e-232)
		tmp = 2.0 * (z / t);
	elseif (x <= 2.25e-207)
		tmp = -y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.6e-211], x, If[LessEqual[x, 3e-232], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.25e-207], N[((-y) / z), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6000000000000004e-211 or 2.24999999999999996e-207 < x

   1. Initial program 81.2%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in x around inf 82.4%

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

   if -6.6000000000000004e-211 < x < 2.9999999999999999e-232

   1. Initial program 67.4%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in y around inf 74.0%

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

      1. associate-*r/ 74.0%

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

      2. *-commutative 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in x around 0 59.8%

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

   if 2.9999999999999999e-232 < x < 2.24999999999999996e-207

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in y around 0 96.4%

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

   6. Taylor expanded in x around 0 83.8%

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

      1. mul-1-neg 83.8%

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

      2. distribute-frac-neg2 83.8%

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

   8. Simplified 83.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-211}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-232}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-207}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+14} \lor \neg \left(z \leq 5.2 \cdot 10^{+86}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.8e+14) (not (<= z 5.2e+86)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.8e+14) || !(z <= 5.2e+86)) {
		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 <= (-5.8d+14)) .or. (.not. (z <= 5.2d+86))) 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 <= -5.8e+14) || !(z <= 5.2e+86)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.8e+14) or not (z <= 5.2e+86):
		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 <= -5.8e+14) || !(z <= 5.2e+86))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.8e+14) || ~((z <= 5.2e+86)))
		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, -5.8e+14], N[Not[LessEqual[z, 5.2e+86]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.8e14 or 5.1999999999999995e86 < z

   1. Initial program 66.1%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in y around 0 91.9%

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

   if -5.8e14 < z < 5.1999999999999995e86

   1. Initial program 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in y around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

   7. Simplified 88.9%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+14} \lor \neg \left(z \leq 5.2 \cdot 10^{+86}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 78.3%

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

3. Simplified 85.4%

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

5. Taylor expanded in x around inf 70.9%

   \[\leadsto \color{blue}{x} \]
6. 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 2024106 
(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)))))