fabs fraction 1

Percentage Accurate: 91.9% → 99.8%

Time: 10.4s

Alternatives: 17

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

C

double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

Julia

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

C

double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

Julia

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-39}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e-39)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-39) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-39)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 5e-39], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e-39

   1. Initial program 86.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 86.4%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 89.7%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.7%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 89.4%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 89.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 89.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 89.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 89.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 89.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.6%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. sqrt-unprod 62.0%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right) \cdot \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      3. sqr-abs 62.0%

         \[\leadsto \left|\sqrt{\color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

      4. neg-fabs 62.0%

         \[\leadsto \left|\sqrt{\color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}\right| \]

      5. neg-fabs 62.0%

         \[\leadsto \left|\sqrt{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

      6. sqr-abs 62.0%

         \[\leadsto \left|\sqrt{\color{blue}{\left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right) \cdot \left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right)}}\right| \]

   6. Applied egg-rr 98.4%

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]

   if 4.9999999999999998e-39 < y

   1. Initial program 97.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 97.3%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 96.0%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 99.9%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 99.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 99.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 99.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 99.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 99.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 99.9%

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

Alternative 2: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+174}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y\_m} + x \cdot \frac{z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* x z) y_m))))
   (if (<= z -1.65e+220)
     t_0
     (if (<= z -7.5e+87)
       (/ (- (+ x 4.0) (* x z)) y_m)
       (if (<= z -2.6e+14)
         t_0
         (if (<= z 5.8e-6)
           (fabs (/ (- -4.0 x) y_m))
           (if (<= z 3.1e+174)
             (+ (* (- -4.0 x) (/ -1.0 y_m)) (* x (/ z y_m)))
             t_0)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x * z) / y_m));
	double tmp;
	if (z <= -1.65e+220) {
		tmp = t_0;
	} else if (z <= -7.5e+87) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else if (z <= -2.6e+14) {
		tmp = t_0;
	} else if (z <= 5.8e-6) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else if (z <= 3.1e+174) {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x * z) / y_m))
    if (z <= (-1.65d+220)) then
        tmp = t_0
    else if (z <= (-7.5d+87)) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else if (z <= (-2.6d+14)) then
        tmp = t_0
    else if (z <= 5.8d-6) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else if (z <= 3.1d+174) then
        tmp = (((-4.0d0) - x) * ((-1.0d0) / y_m)) + (x * (z / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x * z) / y_m));
	double tmp;
	if (z <= -1.65e+220) {
		tmp = t_0;
	} else if (z <= -7.5e+87) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else if (z <= -2.6e+14) {
		tmp = t_0;
	} else if (z <= 5.8e-6) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else if (z <= 3.1e+174) {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x * z) / y_m))
	tmp = 0
	if z <= -1.65e+220:
		tmp = t_0
	elif z <= -7.5e+87:
		tmp = ((x + 4.0) - (x * z)) / y_m
	elif z <= -2.6e+14:
		tmp = t_0
	elif z <= 5.8e-6:
		tmp = math.fabs(((-4.0 - x) / y_m))
	elif z <= 3.1e+174:
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x * z) / y_m))
	tmp = 0.0
	if (z <= -1.65e+220)
		tmp = t_0;
	elseif (z <= -7.5e+87)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	elseif (z <= -2.6e+14)
		tmp = t_0;
	elseif (z <= 5.8e-6)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	elseif (z <= 3.1e+174)
		tmp = Float64(Float64(Float64(-4.0 - x) * Float64(-1.0 / y_m)) + Float64(x * Float64(z / y_m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x * z) / y_m));
	tmp = 0.0;
	if (z <= -1.65e+220)
		tmp = t_0;
	elseif (z <= -7.5e+87)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	elseif (z <= -2.6e+14)
		tmp = t_0;
	elseif (z <= 5.8e-6)
		tmp = abs(((-4.0 - x) / y_m));
	elseif (z <= 3.1e+174)
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.65e+220], t$95$0, If[LessEqual[z, -7.5e+87], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[z, -2.6e+14], t$95$0, If[LessEqual[z, 5.8e-6], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.1e+174], N[(N[(N[(-4.0 - x), $MachinePrecision] * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.65000000000000011e220 or -7.50000000000000014e87 < z < -2.6e14 or 3.1e174 < z

   1. Initial program 78.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 86.2%

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

   if -1.65000000000000011e220 < z < -7.50000000000000014e87

   1. Initial program 96.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 68.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 68.9%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 69.4%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 69.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in y around 0 69.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - \left(4 + x\right)}{y}} \]

   if -2.6e14 < z < 5.8000000000000004e-6

   1. Initial program 94.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 94.9%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 99.2%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 99.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 100.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 98.2%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 98.2%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 98.2%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 98.2%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 98.2%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 98.2%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 98.2%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]

   if 5.8000000000000004e-6 < z < 3.1e174

   1. Initial program 82.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 45.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 45.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 46.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 46.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]
   7. Step-by-step derivation

      1. *-commutative 46.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      2. fma-undefine 46.2%

         \[\leadsto \frac{-1}{y} \cdot \color{blue}{\left(x \cdot z + \left(-4 - x\right)\right)} \]

      3. distribute-rgt-in 43.2%

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot \frac{-1}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]

      4. div-inv 43.2%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-1 \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      5. mul-1-neg 43.2%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-\frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      6. distribute-rgt-neg-in 43.2%

         \[\leadsto \color{blue}{\left(-\left(x \cdot z\right) \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      7. div-inv 43.3%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      8. mul-1-neg 43.3%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      9. add-sqr-sqrt 42.8%

         \[\leadsto \color{blue}{\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      10. fabs-sqr 42.8%

          \[\leadsto \color{blue}{\left|\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      11. add-sqr-sqrt 73.2%

          \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      12. mul-1-neg 73.2%

          \[\leadsto \left|\color{blue}{-\frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      13. associate-*r/ 76.5%

          \[\leadsto \left|-\color{blue}{x \cdot \frac{z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      14. neg-fabs 76.5%

          \[\leadsto \color{blue}{\left|x \cdot \frac{z}{y}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      15. add-sqr-sqrt 42.3%

          \[\leadsto \left|\color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      16. fabs-sqr 42.3%

          \[\leadsto \color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      17. add-sqr-sqrt 55.0%

          \[\leadsto \color{blue}{x \cdot \frac{z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

   8. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+220}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+174}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y} + x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z + -1}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-207}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-121}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y\_m} + x \cdot \frac{z}{y\_m}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x + 4}{y\_m} + \left(x \cdot z\right) \cdot \frac{-1}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ (+ z -1.0) y_m)))))
   (if (<= x -1.7e-7)
     t_0
     (if (<= x 1e-207)
       (/ (- (+ x 4.0) (* x z)) y_m)
       (if (<= x 6.3e-121)
         (+ (* (- -4.0 x) (/ -1.0 y_m)) (* x (/ z y_m)))
         (if (<= x 7.8e+15)
           (+ (/ (+ x 4.0) y_m) (* (* x z) (/ -1.0 y_m)))
           t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x * ((z + -1.0) / y_m)));
	double tmp;
	if (x <= -1.7e-7) {
		tmp = t_0;
	} else if (x <= 1e-207) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else if (x <= 6.3e-121) {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	} else if (x <= 7.8e+15) {
		tmp = ((x + 4.0) / y_m) + ((x * z) * (-1.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x * ((z + (-1.0d0)) / y_m)))
    if (x <= (-1.7d-7)) then
        tmp = t_0
    else if (x <= 1d-207) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else if (x <= 6.3d-121) then
        tmp = (((-4.0d0) - x) * ((-1.0d0) / y_m)) + (x * (z / y_m))
    else if (x <= 7.8d+15) then
        tmp = ((x + 4.0d0) / y_m) + ((x * z) * ((-1.0d0) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x * ((z + -1.0) / y_m)));
	double tmp;
	if (x <= -1.7e-7) {
		tmp = t_0;
	} else if (x <= 1e-207) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else if (x <= 6.3e-121) {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	} else if (x <= 7.8e+15) {
		tmp = ((x + 4.0) / y_m) + ((x * z) * (-1.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x * ((z + -1.0) / y_m)))
	tmp = 0
	if x <= -1.7e-7:
		tmp = t_0
	elif x <= 1e-207:
		tmp = ((x + 4.0) - (x * z)) / y_m
	elif x <= 6.3e-121:
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m))
	elif x <= 7.8e+15:
		tmp = ((x + 4.0) / y_m) + ((x * z) * (-1.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x * Float64(Float64(z + -1.0) / y_m)))
	tmp = 0.0
	if (x <= -1.7e-7)
		tmp = t_0;
	elseif (x <= 1e-207)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	elseif (x <= 6.3e-121)
		tmp = Float64(Float64(Float64(-4.0 - x) * Float64(-1.0 / y_m)) + Float64(x * Float64(z / y_m)));
	elseif (x <= 7.8e+15)
		tmp = Float64(Float64(Float64(x + 4.0) / y_m) + Float64(Float64(x * z) * Float64(-1.0 / y_m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x * ((z + -1.0) / y_m)));
	tmp = 0.0;
	if (x <= -1.7e-7)
		tmp = t_0;
	elseif (x <= 1e-207)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	elseif (x <= 6.3e-121)
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	elseif (x <= 7.8e+15)
		tmp = ((x + 4.0) / y_m) + ((x * z) * (-1.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.7e-7], t$95$0, If[LessEqual[x, 1e-207], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 6.3e-121], N[(N[(N[(-4.0 - x), $MachinePrecision] * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e+15], N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] + N[(N[(x * z), $MachinePrecision] * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.69999999999999987e-7 or 7.8e15 < x

   1. Initial program 82.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 82.3%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 82.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.1%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 89.5%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 89.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 89.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 89.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 89.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 89.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 89.5%

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

   5. Taylor expanded in x around inf 99.4%

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)}\right| \]
   6. Step-by-step derivation

      1. div-sub 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if -1.69999999999999987e-7 < x < 9.99999999999999925e-208

   1. Initial program 96.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 41.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 41.7%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 43.1%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 43.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in y around 0 43.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - \left(4 + x\right)}{y}} \]

   if 9.99999999999999925e-208 < x < 6.29999999999999961e-121

   1. Initial program 95.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 54.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 54.7%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 56.0%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 56.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]
   7. Step-by-step derivation

      1. *-commutative 56.0%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      2. fma-undefine 56.0%

         \[\leadsto \frac{-1}{y} \cdot \color{blue}{\left(x \cdot z + \left(-4 - x\right)\right)} \]

      3. distribute-rgt-in 56.0%

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot \frac{-1}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]

      4. div-inv 56.0%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-1 \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      5. mul-1-neg 56.0%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-\frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      6. distribute-rgt-neg-in 56.0%

         \[\leadsto \color{blue}{\left(-\left(x \cdot z\right) \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      7. div-inv 56.0%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      8. mul-1-neg 56.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      9. add-sqr-sqrt 50.7%

         \[\leadsto \color{blue}{\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      10. fabs-sqr 50.7%

          \[\leadsto \color{blue}{\left|\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      11. add-sqr-sqrt 65.7%

          \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      12. mul-1-neg 65.7%

          \[\leadsto \left|\color{blue}{-\frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      13. associate-*r/ 65.7%

          \[\leadsto \left|-\color{blue}{x \cdot \frac{z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      14. neg-fabs 65.7%

          \[\leadsto \color{blue}{\left|x \cdot \frac{z}{y}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      15. add-sqr-sqrt 25.5%

          \[\leadsto \left|\color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      16. fabs-sqr 25.5%

          \[\leadsto \color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      17. add-sqr-sqrt 50.8%

          \[\leadsto \color{blue}{x \cdot \frac{z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

   8. Applied egg-rr 50.8%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]

   if 6.29999999999999961e-121 < x < 7.8e15

   1. Initial program 96.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.5%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 49.5%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 51.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 51.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]
   7. Step-by-step derivation

      1. *-commutative 51.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      2. fma-undefine 51.2%

         \[\leadsto \frac{-1}{y} \cdot \color{blue}{\left(x \cdot z + \left(-4 - x\right)\right)} \]

      3. distribute-lft-in 51.3%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{-1}{y} \cdot \left(-4 - x\right)} \]

      4. frac-2neg 51.3%

         \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \color{blue}{\frac{--1}{-y}} \cdot \left(-4 - x\right) \]

      5. metadata-eval 51.3%

         \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{\color{blue}{1}}{-y} \cdot \left(-4 - x\right) \]

      6. associate-/r/ 51.2%

         \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \color{blue}{\frac{1}{\frac{-y}{-4 - x}}} \]

      7. sub-neg 51.2%

         \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{1}{\frac{-y}{\color{blue}{-4 + \left(-x\right)}}} \]

      8. metadata-eval 51.2%

         \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{1}{\frac{-y}{\color{blue}{\left(-4\right)} + \left(-x\right)}} \]

      9. distribute-neg-in 51.2%

         \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{1}{\frac{-y}{\color{blue}{-\left(4 + x\right)}}} \]

      10. +-commutative 51.2%

          \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{1}{\frac{-y}{-\color{blue}{\left(x + 4\right)}}} \]

      11. frac-2neg 51.2%

          \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{1}{\color{blue}{\frac{y}{x + 4}}} \]

      12. clear-num 51.3%

          \[\leadsto \frac{-1}{y} \cdot \left(x \cdot z\right) + \color{blue}{\frac{x + 4}{y}} \]

   8. Applied egg-rr 51.3%

      \[\leadsto \color{blue}{\frac{-1}{y} \cdot \left(x \cdot z\right) + \frac{x + 4}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\left|x \cdot \frac{z + -1}{y}\right|\\ \mathbf{elif}\;x \leq 10^{-207}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-121}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y} + x \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x + 4}{y} + \left(x \cdot z\right) \cdot \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z + -1}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ \mathbf{if}\;t\_0 - z \cdot \frac{x}{y\_m} \leq -5 \cdot 10^{-242}:\\ \;\;\;\;\left|t\_0 - \frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)))
   (if (<= (- t_0 (* z (/ x y_m))) -5e-242)
     (fabs (- t_0 (/ z (/ y_m x))))
     (fabs (/ (- (+ x 4.0) (* x z)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if ((t_0 - (z * (x / y_m))) <= -5e-242) {
		tmp = fabs((t_0 - (z / (y_m / x))));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    if ((t_0 - (z * (x / y_m))) <= (-5d-242)) then
        tmp = abs((t_0 - (z / (y_m / x))))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if ((t_0 - (z * (x / y_m))) <= -5e-242) {
		tmp = Math.abs((t_0 - (z / (y_m / x))));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	tmp = 0
	if (t_0 - (z * (x / y_m))) <= -5e-242:
		tmp = math.fabs((t_0 - (z / (y_m / x))))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	tmp = 0.0
	if (Float64(t_0 - Float64(z * Float64(x / y_m))) <= -5e-242)
		tmp = abs(Float64(t_0 - Float64(z / Float64(y_m / x))));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	tmp = 0.0;
	if ((t_0 - (z * (x / y_m))) <= -5e-242)
		tmp = abs((t_0 - (z / (y_m / x))));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-242], N[Abs[N[(t$95$0 - N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -4.9999999999999998e-242

   1. Initial program 98.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{z \cdot \frac{x}{y}}\right| \]

      2. clear-num 98.2%

         \[\leadsto \left|\frac{x + 4}{y} - z \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right| \]

      3. un-div-inv 98.2%

         \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

   4. Applied egg-rr 98.2%

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

   if -4.9999999999999998e-242 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 83.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 83.0%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 86.5%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.1%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 89.2%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 6.1%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. sqrt-unprod 63.3%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right) \cdot \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      3. sqr-abs 63.3%

         \[\leadsto \left|\sqrt{\color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

      4. neg-fabs 63.3%

         \[\leadsto \left|\sqrt{\color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}\right| \]

      5. neg-fabs 63.3%

         \[\leadsto \left|\sqrt{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

      6. sqr-abs 63.3%

         \[\leadsto \left|\sqrt{\color{blue}{\left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right) \cdot \left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right)}}\right| \]

   6. Applied egg-rr 97.4%

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -5 \cdot 10^{-242}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-242}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* z (/ x y_m)))))
   (if (<= t_0 -5e-242) (fabs t_0) (fabs (/ (- (+ x 4.0) (* x z)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - (z * (x / y_m));
	double tmp;
	if (t_0 <= -5e-242) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y_m) - (z * (x / y_m))
    if (t_0 <= (-5d-242)) then
        tmp = abs(t_0)
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - (z * (x / y_m));
	double tmp;
	if (t_0 <= -5e-242) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = ((x + 4.0) / y_m) - (z * (x / y_m))
	tmp = 0
	if t_0 <= -5e-242:
		tmp = math.fabs(t_0)
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_0 <= -5e-242)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = ((x + 4.0) / y_m) - (z * (x / y_m));
	tmp = 0.0;
	if (t_0 <= -5e-242)
		tmp = abs(t_0);
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-242], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -4.9999999999999998e-242

   1. Initial program 98.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   if -4.9999999999999998e-242 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 83.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 83.0%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 86.5%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.1%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 89.2%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 89.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 6.1%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. sqrt-unprod 63.3%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right) \cdot \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      3. sqr-abs 63.3%

         \[\leadsto \left|\sqrt{\color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

      4. neg-fabs 63.3%

         \[\leadsto \left|\sqrt{\color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}\right| \]

      5. neg-fabs 63.3%

         \[\leadsto \left|\sqrt{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

      6. sqr-abs 63.3%

         \[\leadsto \left|\sqrt{\color{blue}{\left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right) \cdot \left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right)}}\right| \]

   6. Applied egg-rr 97.4%

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -5 \cdot 10^{-242}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := x \cdot \frac{z}{y\_m}\\ t_1 := \frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -0.00142:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y\_m} + t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (* x (/ z y_m))) (t_1 (/ (- (+ x 4.0) (* x z)) y_m)))
   (if (<= z -1.3e+67)
     t_1
     (if (<= z -8.5e+55)
       t_0
       (if (<= z -0.00142)
         t_1
         (if (<= z 5.8e-6)
           (fabs (/ (- -4.0 x) y_m))
           (+ (* (- -4.0 x) (/ -1.0 y_m)) t_0)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = x * (z / y_m);
	double t_1 = ((x + 4.0) - (x * z)) / y_m;
	double tmp;
	if (z <= -1.3e+67) {
		tmp = t_1;
	} else if (z <= -8.5e+55) {
		tmp = t_0;
	} else if (z <= -0.00142) {
		tmp = t_1;
	} else if (z <= 5.8e-6) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (z / y_m)
    t_1 = ((x + 4.0d0) - (x * z)) / y_m
    if (z <= (-1.3d+67)) then
        tmp = t_1
    else if (z <= (-8.5d+55)) then
        tmp = t_0
    else if (z <= (-0.00142d0)) then
        tmp = t_1
    else if (z <= 5.8d-6) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = (((-4.0d0) - x) * ((-1.0d0) / y_m)) + t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = x * (z / y_m);
	double t_1 = ((x + 4.0) - (x * z)) / y_m;
	double tmp;
	if (z <= -1.3e+67) {
		tmp = t_1;
	} else if (z <= -8.5e+55) {
		tmp = t_0;
	} else if (z <= -0.00142) {
		tmp = t_1;
	} else if (z <= 5.8e-6) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = x * (z / y_m)
	t_1 = ((x + 4.0) - (x * z)) / y_m
	tmp = 0
	if z <= -1.3e+67:
		tmp = t_1
	elif z <= -8.5e+55:
		tmp = t_0
	elif z <= -0.00142:
		tmp = t_1
	elif z <= 5.8e-6:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(x * Float64(z / y_m))
	t_1 = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m)
	tmp = 0.0
	if (z <= -1.3e+67)
		tmp = t_1;
	elseif (z <= -8.5e+55)
		tmp = t_0;
	elseif (z <= -0.00142)
		tmp = t_1;
	elseif (z <= 5.8e-6)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = Float64(Float64(Float64(-4.0 - x) * Float64(-1.0 / y_m)) + t_0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = x * (z / y_m);
	t_1 = ((x + 4.0) - (x * z)) / y_m;
	tmp = 0.0;
	if (z <= -1.3e+67)
		tmp = t_1;
	elseif (z <= -8.5e+55)
		tmp = t_0;
	elseif (z <= -0.00142)
		tmp = t_1;
	elseif (z <= 5.8e-6)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[z, -1.3e+67], t$95$1, If[LessEqual[z, -8.5e+55], t$95$0, If[LessEqual[z, -0.00142], t$95$1, If[LessEqual[z, 5.8e-6], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[(N[(N[(-4.0 - x), $MachinePrecision] * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.3e67 or -8.50000000000000002e55 < z < -0.00142000000000000004

   1. Initial program 96.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 61.1%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 61.7%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 61.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in y around 0 61.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - \left(4 + x\right)}{y}} \]

   if -1.3e67 < z < -8.50000000000000002e55

   1. Initial program 99.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 29.8%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 29.8%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 30.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 30.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 30.6%

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

      1. clear-num 30.3%

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

      2. un-div-inv 30.3%

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

      3. associate-/r* 43.2%

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

      4. metadata-eval 43.2%

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

      5. distribute-neg-frac 43.2%

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

      6. clear-num 43.2%

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

      7. distribute-neg-frac 43.2%

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

   9. Applied egg-rr 43.2%

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

       1. add-sqr-sqrt 43.2%

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

       2. sqrt-unprod 43.2%

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

       3. sqr-neg 43.2%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 57.3%

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

       6. associate-/r/ 57.3%

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

   11. Applied egg-rr 57.3%

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

   if -0.00142000000000000004 < z < 5.8000000000000004e-6

   1. Initial program 94.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 94.7%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 99.2%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 99.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 100.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 99.9%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 99.9%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 99.9%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 99.9%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 99.9%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 99.9%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]

   if 5.8000000000000004e-6 < z

   1. Initial program 70.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 39.9%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 40.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 40.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]
   7. Step-by-step derivation

      1. *-commutative 40.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      2. fma-undefine 40.6%

         \[\leadsto \frac{-1}{y} \cdot \color{blue}{\left(x \cdot z + \left(-4 - x\right)\right)} \]

      3. distribute-rgt-in 35.6%

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot \frac{-1}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]

      4. div-inv 35.6%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-1 \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      5. mul-1-neg 35.6%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-\frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      6. distribute-rgt-neg-in 35.6%

         \[\leadsto \color{blue}{\left(-\left(x \cdot z\right) \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      7. div-inv 35.7%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      8. mul-1-neg 35.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      9. add-sqr-sqrt 35.3%

         \[\leadsto \color{blue}{\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      10. fabs-sqr 35.3%

          \[\leadsto \color{blue}{\left|\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      11. add-sqr-sqrt 78.4%

          \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      12. mul-1-neg 78.4%

          \[\leadsto \left|\color{blue}{-\frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      13. associate-*r/ 80.7%

          \[\leadsto \left|-\color{blue}{x \cdot \frac{z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      14. neg-fabs 80.7%

          \[\leadsto \color{blue}{\left|x \cdot \frac{z}{y}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      15. add-sqr-sqrt 50.2%

          \[\leadsto \left|\color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      16. fabs-sqr 50.2%

          \[\leadsto \color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      17. add-sqr-sqrt 57.2%

          \[\leadsto \color{blue}{x \cdot \frac{z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

   8. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \frac{z}{y}\\ \mathbf{elif}\;z \leq -0.00142:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y} + x \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\frac{4}{y\_m} - \frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 5.8e-6)))
   (fabs (- (/ 4.0 y_m) (/ z (/ y_m x))))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-6)) {
		tmp = fabs(((4.0 / y_m) - (z / (y_m / x))));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 5.8d-6))) then
        tmp = abs(((4.0d0 / y_m) - (z / (y_m / x))))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-6)) {
		tmp = Math.abs(((4.0 / y_m) - (z / (y_m / x))));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 5.8e-6):
		tmp = math.fabs(((4.0 / y_m) - (z / (y_m / x))))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 5.8e-6))
		tmp = abs(Float64(Float64(4.0 / y_m) - Float64(z / Float64(y_m / x))));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 5.8e-6)))
		tmp = abs(((4.0 / y_m) - (z / (y_m / x))));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 5.8e-6]], $MachinePrecision]], N[Abs[N[(N[(4.0 / y$95$m), $MachinePrecision] - N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 5.8000000000000004e-6 < z

   1. Initial program 83.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 83.6%

         \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{z \cdot \frac{x}{y}}\right| \]

      2. clear-num 83.5%

         \[\leadsto \left|\frac{x + 4}{y} - z \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right| \]

      3. un-div-inv 83.6%

         \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

   4. Applied egg-rr 83.6%

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

   5. Taylor expanded in x around 0 95.4%

      \[\leadsto \left|\frac{\color{blue}{4}}{y} - \frac{z}{\frac{y}{x}}\right| \]

   if -1 < z < 5.8000000000000004e-6

   1. Initial program 94.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 94.8%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 99.2%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 99.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 100.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 100.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 98.9%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 98.9%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 98.9%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 98.9%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 98.9%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\frac{4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right| \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ (- (+ x 4.0) (* x z)) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs((((x + 4.0) - (x * z)) / y_m));
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) - (x * z)) / y_m))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return Math.abs((((x + 4.0) - (x * z)) / y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return math.fabs((((x + 4.0) - (x * z)) / y_m))

Julia

y_m = abs(y)
function code(x, y_m, z)
	return abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = abs((((x + 4.0) - (x * z)) / y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 89.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 89.5%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 91.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 91.2%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fma-neg 92.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 92.4%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 92.4%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 92.4%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 92.4%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 92.4%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 92.4%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.9%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

   2. sqrt-unprod 65.8%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right) \cdot \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

   3. sqr-abs 65.8%

      \[\leadsto \left|\sqrt{\color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

   4. neg-fabs 65.8%

      \[\leadsto \left|\sqrt{\color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \cdot \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}\right| \]

   5. neg-fabs 65.8%

      \[\leadsto \left|\sqrt{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right| \cdot \color{blue}{\left|-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|}}\right| \]

   6. sqr-abs 65.8%

      \[\leadsto \left|\sqrt{\color{blue}{\left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right) \cdot \left(-\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right)}}\right| \]

6. Applied egg-rr 97.7%

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

Alternative 9: 69.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y\_m}{x}}\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y\_m}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+227}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y\_m} + x \cdot \frac{z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.35e+249)
   (/ z (/ y_m x))
   (if (<= x -2.55e+178)
     (* z (/ x (- y_m)))
     (if (<= x -3e+26)
       (* z (/ x y_m))
       (if (<= x 1.3e+227)
         (/ (- (+ x 4.0) (* x z)) y_m)
         (+ (* (- -4.0 x) (/ -1.0 y_m)) (* x (/ z y_m))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.35e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -2.55e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -3e+26) {
		tmp = z * (x / y_m);
	} else if (x <= 1.3e+227) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.35d+249)) then
        tmp = z / (y_m / x)
    else if (x <= (-2.55d+178)) then
        tmp = z * (x / -y_m)
    else if (x <= (-3d+26)) then
        tmp = z * (x / y_m)
    else if (x <= 1.3d+227) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else
        tmp = (((-4.0d0) - x) * ((-1.0d0) / y_m)) + (x * (z / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.35e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -2.55e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -3e+26) {
		tmp = z * (x / y_m);
	} else if (x <= 1.3e+227) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.35e+249:
		tmp = z / (y_m / x)
	elif x <= -2.55e+178:
		tmp = z * (x / -y_m)
	elif x <= -3e+26:
		tmp = z * (x / y_m)
	elif x <= 1.3e+227:
		tmp = ((x + 4.0) - (x * z)) / y_m
	else:
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.35e+249)
		tmp = Float64(z / Float64(y_m / x));
	elseif (x <= -2.55e+178)
		tmp = Float64(z * Float64(x / Float64(-y_m)));
	elseif (x <= -3e+26)
		tmp = Float64(z * Float64(x / y_m));
	elseif (x <= 1.3e+227)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	else
		tmp = Float64(Float64(Float64(-4.0 - x) * Float64(-1.0 / y_m)) + Float64(x * Float64(z / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1.35e+249)
		tmp = z / (y_m / x);
	elseif (x <= -2.55e+178)
		tmp = z * (x / -y_m);
	elseif (x <= -3e+26)
		tmp = z * (x / y_m);
	elseif (x <= 1.3e+227)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	else
		tmp = ((-4.0 - x) * (-1.0 / y_m)) + (x * (z / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.35e+249], N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.55e+178], N[(z * N[(x / (-y$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e+26], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e+227], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(-4.0 - x), $MachinePrecision] * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.35000000000000009e249

   1. Initial program 86.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.8%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 33.8%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 33.9%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 33.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 34.2%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 80.7%

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

      3. mul-1-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. sqr-neg 80.7%

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

      6. sqrt-unprod 34.3%

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

      7. add-sqr-sqrt 34.5%

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

      8. clear-num 34.5%

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

      9. associate-/r* 40.5%

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

      10. clear-num 40.5%

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

   9. Applied egg-rr 40.5%

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

   if -1.35000000000000009e249 < x < -2.5499999999999998e178

   1. Initial program 25.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 14.8%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 88.1%

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

      3. mul-1-neg 88.1%

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

      4. mul-1-neg 88.1%

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

      5. sqr-neg 88.1%

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

      6. sqrt-unprod 26.5%

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

      7. add-sqr-sqrt 26.8%

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

      8. clear-num 26.8%

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

      9. associate-/r* 50.3%

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

      10. clear-num 50.3%

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

   9. Applied egg-rr 50.3%

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

       1. associate-/r/ 26.9%

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

       2. associate-*l/ 26.8%

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

       3. *-commutative 26.8%

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

       4. add-sqr-sqrt 26.5%

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

       5. sqrt-unprod 88.1%

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

       6. sqr-neg 88.1%

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

       7. mul-1-neg 88.1%

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

       8. mul-1-neg 88.1%

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

       9. sqrt-unprod 14.8%

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

       10. add-sqr-sqrt 14.8%

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

       11. mul-1-neg 14.8%

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

       12. div-inv 14.8%

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

       13. *-commutative 14.8%

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

       14. associate-*l* 38.1%

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

       15. div-inv 38.1%

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

       16. distribute-lft-neg-in 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -2.5499999999999998e178 < x < -2.99999999999999997e26

   1. Initial program 89.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.5%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 47.0%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 47.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 23.7%

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

      1. clear-num 23.7%

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

      2. un-div-inv 23.7%

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

      3. associate-/r* 26.9%

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

      4. metadata-eval 26.9%

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

      5. distribute-neg-frac 26.9%

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

      6. clear-num 26.9%

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

      7. distribute-neg-frac 26.9%

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

   9. Applied egg-rr 26.9%

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

       1. clear-num 26.9%

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

       2. associate-/r/ 30.3%

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

       3. clear-num 30.3%

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

       4. add-sqr-sqrt 22.5%

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

       5. sqrt-unprod 34.1%

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

       6. sqr-neg 34.1%

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

       7. sqrt-unprod 14.7%

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

       8. add-sqr-sqrt 33.9%

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

   11. Applied egg-rr 33.9%

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

   if -2.99999999999999997e26 < x < 1.29999999999999991e227

   1. Initial program 95.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.2%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 48.2%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 49.4%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 49.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 49.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in y around 0 49.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - \left(4 + x\right)}{y}} \]

   if 1.29999999999999991e227 < x

   1. Initial program 70.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 54.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 54.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 54.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 54.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]
   7. Step-by-step derivation

      1. *-commutative 54.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      2. fma-undefine 54.6%

         \[\leadsto \frac{-1}{y} \cdot \color{blue}{\left(x \cdot z + \left(-4 - x\right)\right)} \]

      3. distribute-rgt-in 46.3%

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot \frac{-1}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]

      4. div-inv 46.3%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-1 \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      5. mul-1-neg 46.3%

         \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{\left(-\frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      6. distribute-rgt-neg-in 46.3%

         \[\leadsto \color{blue}{\left(-\left(x \cdot z\right) \cdot \frac{1}{y}\right)} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      7. div-inv 46.3%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      8. mul-1-neg 46.3%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      9. add-sqr-sqrt 33.8%

         \[\leadsto \color{blue}{\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      10. fabs-sqr 33.8%

          \[\leadsto \color{blue}{\left|\sqrt{-1 \cdot \frac{x \cdot z}{y}} \cdot \sqrt{-1 \cdot \frac{x \cdot z}{y}}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      11. add-sqr-sqrt 71.4%

          \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      12. mul-1-neg 71.4%

          \[\leadsto \left|\color{blue}{-\frac{x \cdot z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      13. associate-*r/ 83.3%

          \[\leadsto \left|-\color{blue}{x \cdot \frac{z}{y}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      14. neg-fabs 83.3%

          \[\leadsto \color{blue}{\left|x \cdot \frac{z}{y}\right|} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      15. add-sqr-sqrt 45.9%

          \[\leadsto \left|\color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}}\right| + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      16. fabs-sqr 45.9%

          \[\leadsto \color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

      17. add-sqr-sqrt 58.5%

          \[\leadsto \color{blue}{x \cdot \frac{z}{y}} + \left(-4 - x\right) \cdot \frac{-1}{y} \]

   8. Applied egg-rr 58.5%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \left(-4 - x\right) \cdot \frac{-1}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+227}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 - x\right) \cdot \frac{-1}{y} + x \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y\_m}{x}}\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y\_m}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y\_m}{x \cdot \left(z + 1\right)}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.8e+249)
   (/ z (/ y_m x))
   (if (<= x -1.08e+178)
     (* z (/ x (- y_m)))
     (if (<= x -2.1e+24)
       (* z (/ x y_m))
       (if (<= x 5.5e+213)
         (/ (- (+ x 4.0) (* x z)) y_m)
         (/ 1.0 (/ y_m (* x (+ z 1.0)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.8e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -1.08e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -2.1e+24) {
		tmp = z * (x / y_m);
	} else if (x <= 5.5e+213) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = 1.0 / (y_m / (x * (z + 1.0)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.8d+249)) then
        tmp = z / (y_m / x)
    else if (x <= (-1.08d+178)) then
        tmp = z * (x / -y_m)
    else if (x <= (-2.1d+24)) then
        tmp = z * (x / y_m)
    else if (x <= 5.5d+213) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else
        tmp = 1.0d0 / (y_m / (x * (z + 1.0d0)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.8e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -1.08e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -2.1e+24) {
		tmp = z * (x / y_m);
	} else if (x <= 5.5e+213) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = 1.0 / (y_m / (x * (z + 1.0)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -5.8e+249:
		tmp = z / (y_m / x)
	elif x <= -1.08e+178:
		tmp = z * (x / -y_m)
	elif x <= -2.1e+24:
		tmp = z * (x / y_m)
	elif x <= 5.5e+213:
		tmp = ((x + 4.0) - (x * z)) / y_m
	else:
		tmp = 1.0 / (y_m / (x * (z + 1.0)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5.8e+249)
		tmp = Float64(z / Float64(y_m / x));
	elseif (x <= -1.08e+178)
		tmp = Float64(z * Float64(x / Float64(-y_m)));
	elseif (x <= -2.1e+24)
		tmp = Float64(z * Float64(x / y_m));
	elseif (x <= 5.5e+213)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	else
		tmp = Float64(1.0 / Float64(y_m / Float64(x * Float64(z + 1.0))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -5.8e+249)
		tmp = z / (y_m / x);
	elseif (x <= -1.08e+178)
		tmp = z * (x / -y_m);
	elseif (x <= -2.1e+24)
		tmp = z * (x / y_m);
	elseif (x <= 5.5e+213)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	else
		tmp = 1.0 / (y_m / (x * (z + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -5.8e+249], N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.08e+178], N[(z * N[(x / (-y$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e+24], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e+213], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(y$95$m / N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -5.80000000000000034e249

   1. Initial program 86.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.8%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 33.8%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 33.9%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 33.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 34.2%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 80.7%

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

      3. mul-1-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. sqr-neg 80.7%

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

      6. sqrt-unprod 34.3%

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

      7. add-sqr-sqrt 34.5%

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

      8. clear-num 34.5%

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

      9. associate-/r* 40.5%

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

      10. clear-num 40.5%

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

   9. Applied egg-rr 40.5%

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

   if -5.80000000000000034e249 < x < -1.07999999999999996e178

   1. Initial program 25.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 14.8%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 88.1%

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

      3. mul-1-neg 88.1%

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

      4. mul-1-neg 88.1%

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

      5. sqr-neg 88.1%

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

      6. sqrt-unprod 26.5%

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

      7. add-sqr-sqrt 26.8%

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

      8. clear-num 26.8%

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

      9. associate-/r* 50.3%

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

      10. clear-num 50.3%

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

   9. Applied egg-rr 50.3%

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

       1. associate-/r/ 26.9%

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

       2. associate-*l/ 26.8%

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

       3. *-commutative 26.8%

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

       4. add-sqr-sqrt 26.5%

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

       5. sqrt-unprod 88.1%

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

       6. sqr-neg 88.1%

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

       7. mul-1-neg 88.1%

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

       8. mul-1-neg 88.1%

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

       9. sqrt-unprod 14.8%

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

       10. add-sqr-sqrt 14.8%

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

       11. mul-1-neg 14.8%

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

       12. div-inv 14.8%

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

       13. *-commutative 14.8%

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

       14. associate-*l* 38.1%

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

       15. div-inv 38.1%

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

       16. distribute-lft-neg-in 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -1.07999999999999996e178 < x < -2.1000000000000001e24

   1. Initial program 89.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.5%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 47.0%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 47.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 23.7%

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

      1. clear-num 23.7%

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

      2. un-div-inv 23.7%

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

      3. associate-/r* 26.9%

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

      4. metadata-eval 26.9%

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

      5. distribute-neg-frac 26.9%

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

      6. clear-num 26.9%

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

      7. distribute-neg-frac 26.9%

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

   9. Applied egg-rr 26.9%

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

       1. clear-num 26.9%

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

       2. associate-/r/ 30.3%

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

       3. clear-num 30.3%

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

       4. add-sqr-sqrt 22.5%

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

       5. sqrt-unprod 34.1%

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

       6. sqr-neg 34.1%

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

       7. sqrt-unprod 14.7%

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

       8. add-sqr-sqrt 33.9%

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

   11. Applied egg-rr 33.9%

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

   if -2.1000000000000001e24 < x < 5.50000000000000059e213

   1. Initial program 96.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 49.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in y around 0 50.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - \left(4 + x\right)}{y}} \]

   if 5.50000000000000059e213 < x

   1. Initial program 68.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 90.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.5%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 48.5%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 48.7%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 48.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in x around inf 48.7%

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

      1. mul-1-neg 48.7%

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

      2. associate-/l* 55.3%

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

      3. distribute-rgt-neg-in 55.3%

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

      4. mul-1-neg 55.3%

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

      5. associate-*r/ 55.3%

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

      6. sub-neg 55.3%

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

      7. metadata-eval 55.3%

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

      8. distribute-lft-in 55.3%

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

      9. neg-mul-1 55.3%

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

      10. metadata-eval 55.3%

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

      11. +-commutative 55.3%

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

      12. unsub-neg 55.3%

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

   9. Simplified 55.3%

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

       1. associate-*r/ 48.7%

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

       2. clear-num 48.7%

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

       3. *-un-lft-identity 48.7%

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

       4. *-un-lft-identity 48.7%

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

       5. sub-neg 48.7%

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

       6. add-sqr-sqrt 27.7%

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

       7. sqrt-unprod 65.9%

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

       8. sqr-neg 65.9%

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

       9. sqrt-unprod 38.1%

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

       10. add-sqr-sqrt 55.5%

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

       11. +-commutative 55.5%

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

   11. Applied egg-rr 55.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot \left(z + 1\right)}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(z + 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+248}:\\ \;\;\;\;\frac{z}{\frac{y\_m}{x}}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y\_m}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y\_m}{x \cdot \left(z + 1\right)}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -7.5e+248)
   (/ z (/ y_m x))
   (if (<= x -2.2e+178)
     (* z (/ x (- y_m)))
     (if (<= x -8.5e-6)
       (* z (/ x y_m))
       (if (<= x 1.45e+16)
         (/ (+ x 4.0) y_m)
         (/ 1.0 (/ y_m (* x (+ z 1.0)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -7.5e+248) {
		tmp = z / (y_m / x);
	} else if (x <= -2.2e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -8.5e-6) {
		tmp = z * (x / y_m);
	} else if (x <= 1.45e+16) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = 1.0 / (y_m / (x * (z + 1.0)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-7.5d+248)) then
        tmp = z / (y_m / x)
    else if (x <= (-2.2d+178)) then
        tmp = z * (x / -y_m)
    else if (x <= (-8.5d-6)) then
        tmp = z * (x / y_m)
    else if (x <= 1.45d+16) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = 1.0d0 / (y_m / (x * (z + 1.0d0)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -7.5e+248) {
		tmp = z / (y_m / x);
	} else if (x <= -2.2e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -8.5e-6) {
		tmp = z * (x / y_m);
	} else if (x <= 1.45e+16) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = 1.0 / (y_m / (x * (z + 1.0)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -7.5e+248:
		tmp = z / (y_m / x)
	elif x <= -2.2e+178:
		tmp = z * (x / -y_m)
	elif x <= -8.5e-6:
		tmp = z * (x / y_m)
	elif x <= 1.45e+16:
		tmp = (x + 4.0) / y_m
	else:
		tmp = 1.0 / (y_m / (x * (z + 1.0)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -7.5e+248)
		tmp = Float64(z / Float64(y_m / x));
	elseif (x <= -2.2e+178)
		tmp = Float64(z * Float64(x / Float64(-y_m)));
	elseif (x <= -8.5e-6)
		tmp = Float64(z * Float64(x / y_m));
	elseif (x <= 1.45e+16)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(1.0 / Float64(y_m / Float64(x * Float64(z + 1.0))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -7.5e+248)
		tmp = z / (y_m / x);
	elseif (x <= -2.2e+178)
		tmp = z * (x / -y_m);
	elseif (x <= -8.5e-6)
		tmp = z * (x / y_m);
	elseif (x <= 1.45e+16)
		tmp = (x + 4.0) / y_m;
	else
		tmp = 1.0 / (y_m / (x * (z + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -7.5e+248], N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e+178], N[(z * N[(x / (-y$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-6], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e+16], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(y$95$m / N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.5000000000000004e248

   1. Initial program 86.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.8%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 33.8%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 33.9%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 33.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 34.2%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 80.7%

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

      3. mul-1-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. sqr-neg 80.7%

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

      6. sqrt-unprod 34.3%

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

      7. add-sqr-sqrt 34.5%

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

      8. clear-num 34.5%

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

      9. associate-/r* 40.5%

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

      10. clear-num 40.5%

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

   9. Applied egg-rr 40.5%

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

   if -7.5000000000000004e248 < x < -2.19999999999999997e178

   1. Initial program 25.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 14.8%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 88.1%

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

      3. mul-1-neg 88.1%

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

      4. mul-1-neg 88.1%

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

      5. sqr-neg 88.1%

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

      6. sqrt-unprod 26.5%

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

      7. add-sqr-sqrt 26.8%

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

      8. clear-num 26.8%

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

      9. associate-/r* 50.3%

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

      10. clear-num 50.3%

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

   9. Applied egg-rr 50.3%

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

       1. associate-/r/ 26.9%

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

       2. associate-*l/ 26.8%

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

       3. *-commutative 26.8%

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

       4. add-sqr-sqrt 26.5%

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

       5. sqrt-unprod 88.1%

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

       6. sqr-neg 88.1%

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

       7. mul-1-neg 88.1%

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

       8. mul-1-neg 88.1%

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

       9. sqrt-unprod 14.8%

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

       10. add-sqr-sqrt 14.8%

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

       11. mul-1-neg 14.8%

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

       12. div-inv 14.8%

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

       13. *-commutative 14.8%

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

       14. associate-*l* 38.1%

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

       15. div-inv 38.1%

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

       16. distribute-lft-neg-in 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -2.19999999999999997e178 < x < -8.4999999999999999e-6

   1. Initial program 90.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 30.3%

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

      1. clear-num 30.3%

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

      2. un-div-inv 30.3%

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

      3. associate-/r* 33.1%

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

      4. metadata-eval 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. clear-num 33.1%

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

      7. distribute-neg-frac 33.1%

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

   9. Applied egg-rr 33.1%

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

       1. clear-num 33.1%

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

       2. associate-/r/ 36.1%

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

       3. clear-num 36.2%

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

       4. add-sqr-sqrt 22.7%

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

       5. sqrt-unprod 30.3%

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

       6. sqr-neg 30.3%

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

       7. sqrt-unprod 13.2%

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

       8. add-sqr-sqrt 30.0%

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

   11. Applied egg-rr 30.0%

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

   if -8.4999999999999999e-6 < x < 1.45e16

   1. Initial program 96.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 45.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 45.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 46.8%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 46.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around 0 40.8%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]

   if 1.45e16 < x

   1. Initial program 84.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 54.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 54.1%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 54.5%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 54.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 54.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in x around inf 54.6%

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

      1. mul-1-neg 54.6%

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

      2. associate-/l* 57.2%

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

      3. distribute-rgt-neg-in 57.2%

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

      4. mul-1-neg 57.2%

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

      5. associate-*r/ 57.2%

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

      6. sub-neg 57.2%

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

      7. metadata-eval 57.2%

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

      8. distribute-lft-in 57.2%

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

      9. neg-mul-1 57.2%

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

      10. metadata-eval 57.2%

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

      11. +-commutative 57.2%

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

      12. unsub-neg 57.2%

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

   9. Simplified 57.2%

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

       1. associate-*r/ 54.6%

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

       2. clear-num 54.5%

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

       3. *-un-lft-identity 54.5%

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

       4. *-un-lft-identity 54.5%

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

       5. sub-neg 54.5%

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

       6. add-sqr-sqrt 30.0%

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

       7. sqrt-unprod 58.9%

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

       8. sqr-neg 58.9%

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

       9. sqrt-unprod 28.8%

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

       10. add-sqr-sqrt 50.8%

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

       11. +-commutative 50.8%

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

   11. Applied egg-rr 50.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot \left(z + 1\right)}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+248}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(z + 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y\_m}{x}}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{+177}:\\ \;\;\;\;z \cdot \frac{x}{-y\_m}\\ \mathbf{elif}\;x \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -9e+249)
   (/ z (/ y_m x))
   (if (<= x -9.8e+177)
     (* z (/ x (- y_m)))
     (if (<= x -6.9e-6)
       (* z (/ x y_m))
       (if (<= x 3.1e+35) (/ (+ x 4.0) y_m) (* x (/ (- 1.0 z) y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -9e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -9.8e+177) {
		tmp = z * (x / -y_m);
	} else if (x <= -6.9e-6) {
		tmp = z * (x / y_m);
	} else if (x <= 3.1e+35) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - z) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-9d+249)) then
        tmp = z / (y_m / x)
    else if (x <= (-9.8d+177)) then
        tmp = z * (x / -y_m)
    else if (x <= (-6.9d-6)) then
        tmp = z * (x / y_m)
    else if (x <= 3.1d+35) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = x * ((1.0d0 - z) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -9e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -9.8e+177) {
		tmp = z * (x / -y_m);
	} else if (x <= -6.9e-6) {
		tmp = z * (x / y_m);
	} else if (x <= 3.1e+35) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - z) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -9e+249:
		tmp = z / (y_m / x)
	elif x <= -9.8e+177:
		tmp = z * (x / -y_m)
	elif x <= -6.9e-6:
		tmp = z * (x / y_m)
	elif x <= 3.1e+35:
		tmp = (x + 4.0) / y_m
	else:
		tmp = x * ((1.0 - z) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -9e+249)
		tmp = Float64(z / Float64(y_m / x));
	elseif (x <= -9.8e+177)
		tmp = Float64(z * Float64(x / Float64(-y_m)));
	elseif (x <= -6.9e-6)
		tmp = Float64(z * Float64(x / y_m));
	elseif (x <= 3.1e+35)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(x * Float64(Float64(1.0 - z) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -9e+249)
		tmp = z / (y_m / x);
	elseif (x <= -9.8e+177)
		tmp = z * (x / -y_m);
	elseif (x <= -6.9e-6)
		tmp = z * (x / y_m);
	elseif (x <= 3.1e+35)
		tmp = (x + 4.0) / y_m;
	else
		tmp = x * ((1.0 - z) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -9e+249], N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.8e+177], N[(z * N[(x / (-y$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.9e-6], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+35], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.9999999999999993e249

   1. Initial program 86.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.8%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 33.8%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 33.9%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 33.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 34.2%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 80.7%

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

      3. mul-1-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. sqr-neg 80.7%

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

      6. sqrt-unprod 34.3%

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

      7. add-sqr-sqrt 34.5%

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

      8. clear-num 34.5%

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

      9. associate-/r* 40.5%

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

      10. clear-num 40.5%

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

   9. Applied egg-rr 40.5%

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

   if -8.9999999999999993e249 < x < -9.8000000000000003e177

   1. Initial program 25.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 14.8%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 88.1%

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

      3. mul-1-neg 88.1%

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

      4. mul-1-neg 88.1%

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

      5. sqr-neg 88.1%

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

      6. sqrt-unprod 26.5%

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

      7. add-sqr-sqrt 26.8%

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

      8. clear-num 26.8%

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

      9. associate-/r* 50.3%

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

      10. clear-num 50.3%

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

   9. Applied egg-rr 50.3%

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

       1. associate-/r/ 26.9%

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

       2. associate-*l/ 26.8%

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

       3. *-commutative 26.8%

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

       4. add-sqr-sqrt 26.5%

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

       5. sqrt-unprod 88.1%

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

       6. sqr-neg 88.1%

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

       7. mul-1-neg 88.1%

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

       8. mul-1-neg 88.1%

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

       9. sqrt-unprod 14.8%

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

       10. add-sqr-sqrt 14.8%

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

       11. mul-1-neg 14.8%

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

       12. div-inv 14.8%

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

       13. *-commutative 14.8%

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

       14. associate-*l* 38.1%

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

       15. div-inv 38.1%

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

       16. distribute-lft-neg-in 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -9.8000000000000003e177 < x < -6.9e-6

   1. Initial program 90.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 30.3%

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

      1. clear-num 30.3%

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

      2. un-div-inv 30.3%

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

      3. associate-/r* 33.1%

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

      4. metadata-eval 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. clear-num 33.1%

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

      7. distribute-neg-frac 33.1%

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

   9. Applied egg-rr 33.1%

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

       1. clear-num 33.1%

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

       2. associate-/r/ 36.1%

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

       3. clear-num 36.2%

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

       4. add-sqr-sqrt 22.7%

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

       5. sqrt-unprod 30.3%

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

       6. sqr-neg 30.3%

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

       7. sqrt-unprod 13.2%

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

       8. add-sqr-sqrt 30.0%

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

   11. Applied egg-rr 30.0%

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

   if -6.9e-6 < x < 3.09999999999999987e35

   1. Initial program 96.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.2%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 46.2%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 47.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 47.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around 0 41.1%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]

   if 3.09999999999999987e35 < x

   1. Initial program 83.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 94.1%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 52.9%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 53.3%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 53.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in x around inf 53.3%

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

      1. mul-1-neg 53.3%

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

      2. associate-/l* 56.1%

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

      3. distribute-rgt-neg-in 56.1%

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

      4. mul-1-neg 56.1%

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

      5. associate-*r/ 56.1%

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

      6. sub-neg 56.1%

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

      7. metadata-eval 56.1%

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

      8. distribute-lft-in 56.1%

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

      9. neg-mul-1 56.1%

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

      10. metadata-eval 56.1%

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

      11. +-commutative 56.1%

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

      12. unsub-neg 56.1%

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

   9. Simplified 56.1%

      \[\leadsto \color{blue}{x \cdot \frac{1 - z}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{+177}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;x \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y\_m}{x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y\_m}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -6.7e+249)
   (/ z (/ y_m x))
   (if (<= x -3e+178)
     (* z (/ x (- y_m)))
     (if (<= x -7.5e-6) (* z (/ x y_m)) (/ (+ x 4.0) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -6.7e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -3e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -7.5e-6) {
		tmp = z * (x / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-6.7d+249)) then
        tmp = z / (y_m / x)
    else if (x <= (-3d+178)) then
        tmp = z * (x / -y_m)
    else if (x <= (-7.5d-6)) then
        tmp = z * (x / y_m)
    else
        tmp = (x + 4.0d0) / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -6.7e+249) {
		tmp = z / (y_m / x);
	} else if (x <= -3e+178) {
		tmp = z * (x / -y_m);
	} else if (x <= -7.5e-6) {
		tmp = z * (x / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -6.7e+249:
		tmp = z / (y_m / x)
	elif x <= -3e+178:
		tmp = z * (x / -y_m)
	elif x <= -7.5e-6:
		tmp = z * (x / y_m)
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -6.7e+249)
		tmp = Float64(z / Float64(y_m / x));
	elseif (x <= -3e+178)
		tmp = Float64(z * Float64(x / Float64(-y_m)));
	elseif (x <= -7.5e-6)
		tmp = Float64(z * Float64(x / y_m));
	else
		tmp = Float64(Float64(x + 4.0) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -6.7e+249)
		tmp = z / (y_m / x);
	elseif (x <= -3e+178)
		tmp = z * (x / -y_m);
	elseif (x <= -7.5e-6)
		tmp = z * (x / y_m);
	else
		tmp = (x + 4.0) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -6.7e+249], N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e+178], N[(z * N[(x / (-y$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-6], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.70000000000000027e249

   1. Initial program 86.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.8%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 33.8%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 33.9%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 33.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 34.2%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 80.7%

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

      3. mul-1-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. sqr-neg 80.7%

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

      6. sqrt-unprod 34.3%

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

      7. add-sqr-sqrt 34.5%

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

      8. clear-num 34.5%

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

      9. associate-/r* 40.5%

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

      10. clear-num 40.5%

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

   9. Applied egg-rr 40.5%

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

   if -6.70000000000000027e249 < x < -3.00000000000000016e178

   1. Initial program 25.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 14.8%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 88.1%

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

      3. mul-1-neg 88.1%

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

      4. mul-1-neg 88.1%

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

      5. sqr-neg 88.1%

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

      6. sqrt-unprod 26.5%

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

      7. add-sqr-sqrt 26.8%

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

      8. clear-num 26.8%

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

      9. associate-/r* 50.3%

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

      10. clear-num 50.3%

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

   9. Applied egg-rr 50.3%

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

       1. associate-/r/ 26.9%

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

       2. associate-*l/ 26.8%

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

       3. *-commutative 26.8%

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

       4. add-sqr-sqrt 26.5%

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

       5. sqrt-unprod 88.1%

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

       6. sqr-neg 88.1%

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

       7. mul-1-neg 88.1%

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

       8. mul-1-neg 88.1%

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

       9. sqrt-unprod 14.8%

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

       10. add-sqr-sqrt 14.8%

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

       11. mul-1-neg 14.8%

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

       12. div-inv 14.8%

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

       13. *-commutative 14.8%

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

       14. associate-*l* 38.1%

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

       15. div-inv 38.1%

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

       16. distribute-lft-neg-in 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -3.00000000000000016e178 < x < -7.50000000000000019e-6

   1. Initial program 90.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 50.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 50.6%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 50.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 30.3%

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

      1. clear-num 30.3%

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

      2. un-div-inv 30.3%

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

      3. associate-/r* 33.1%

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

      4. metadata-eval 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. clear-num 33.1%

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

      7. distribute-neg-frac 33.1%

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

   9. Applied egg-rr 33.1%

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

       1. clear-num 33.1%

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

       2. associate-/r/ 36.1%

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

       3. clear-num 36.2%

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

       4. add-sqr-sqrt 22.7%

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

       5. sqrt-unprod 30.3%

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

       6. sqr-neg 30.3%

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

       7. sqrt-unprod 13.2%

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

       8. add-sqr-sqrt 30.0%

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

   11. Applied egg-rr 30.0%

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

   if -7.50000000000000019e-6 < x

   1. Initial program 92.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 48.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 49.5%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 49.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around 0 41.2%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+249}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+178}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -0.17 \lor \neg \left(z \leq 6.6 \cdot 10^{+18}\right):\\ \;\;\;\;x \cdot \frac{z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -0.17) (not (<= z 6.6e+18))) (* x (/ z y_m)) (/ x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -0.17) || !(z <= 6.6e+18)) {
		tmp = x * (z / y_m);
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-0.17d0)) .or. (.not. (z <= 6.6d+18))) then
        tmp = x * (z / y_m)
    else
        tmp = x / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -0.17) || !(z <= 6.6e+18)) {
		tmp = x * (z / y_m);
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -0.17) or not (z <= 6.6e+18):
		tmp = x * (z / y_m)
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -0.17) || !(z <= 6.6e+18))
		tmp = Float64(x * Float64(z / y_m));
	else
		tmp = Float64(x / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -0.17) || ~((z <= 6.6e+18)))
		tmp = x * (z / y_m);
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -0.17], N[Not[LessEqual[z, 6.6e+18]], $MachinePrecision]], N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.170000000000000012 or 6.6e18 < z

   1. Initial program 83.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 47.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 47.9%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 48.5%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 48.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 33.9%

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

      1. clear-num 33.8%

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

      2. un-div-inv 33.8%

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

      3. associate-/r* 35.3%

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

      4. metadata-eval 35.3%

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

      5. distribute-neg-frac 35.3%

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

      6. clear-num 35.3%

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

      7. distribute-neg-frac 35.3%

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

   9. Applied egg-rr 35.3%

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

       1. add-sqr-sqrt 20.4%

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

       2. sqrt-unprod 34.3%

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

       3. sqr-neg 34.3%

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

       4. sqrt-unprod 21.0%

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

       5. add-sqr-sqrt 35.4%

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

       6. associate-/r/ 36.4%

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

   11. Applied egg-rr 36.4%

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

   if -0.170000000000000012 < z < 6.6e18

   1. Initial program 94.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 47.8%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 47.8%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 48.9%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 48.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 48.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in x around inf 26.8%

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

      1. mul-1-neg 26.8%

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

      2. associate-/l* 26.7%

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

      3. distribute-rgt-neg-in 26.7%

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

      4. mul-1-neg 26.7%

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

      5. associate-*r/ 26.7%

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

      6. sub-neg 26.7%

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

      7. metadata-eval 26.7%

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

      8. distribute-lft-in 26.7%

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

      9. neg-mul-1 26.7%

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

      10. metadata-eval 26.7%

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

      11. +-commutative 26.7%

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

      12. unsub-neg 26.7%

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

   9. Simplified 26.7%

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

   10. Taylor expanded in z around 0 26.2%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.17 \lor \neg \left(z \leq 6.6 \cdot 10^{+18}\right):\\ \;\;\;\;x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.1% accurate, 11.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.5e-6) (* z (/ x y_m)) (/ (+ x 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.5e-6) {
		tmp = z * (x / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.5d-6)) then
        tmp = z * (x / y_m)
    else
        tmp = (x + 4.0d0) / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.5e-6) {
		tmp = z * (x / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -8.5e-6:
		tmp = z * (x / y_m)
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -8.5e-6)
		tmp = Float64(z * Float64(x / y_m));
	else
		tmp = Float64(Float64(x + 4.0) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -8.5e-6)
		tmp = z * (x / y_m);
	else
		tmp = (x + 4.0) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -8.5e-6], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4999999999999999e-6

   1. Initial program 79.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 45.6%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 45.6%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 46.0%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 46.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 46.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 29.1%

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

      1. clear-num 29.1%

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

      2. un-div-inv 29.1%

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

      3. associate-/r* 37.5%

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

      4. metadata-eval 37.5%

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

      5. distribute-neg-frac 37.5%

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

      6. clear-num 37.5%

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

      7. distribute-neg-frac 37.5%

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

   9. Applied egg-rr 37.5%

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

       1. clear-num 37.5%

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

       2. associate-/r/ 39.2%

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

       3. clear-num 39.2%

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

       4. add-sqr-sqrt 24.0%

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

       5. sqrt-unprod 32.1%

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

       6. sqr-neg 32.1%

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

       7. sqrt-unprod 16.8%

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

       8. add-sqr-sqrt 35.8%

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

   11. Applied egg-rr 35.8%

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

   if -8.4999999999999999e-6 < x

   1. Initial program 92.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 48.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 49.5%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 49.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around 0 41.2%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 2.4e-46) (* z (/ x y_m)) (/ x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 2.4e-46) {
		tmp = z * (x / y_m);
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.4d-46) then
        tmp = z * (x / y_m)
    else
        tmp = x / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 2.4e-46) {
		tmp = z * (x / y_m);
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 2.4e-46:
		tmp = z * (x / y_m)
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 2.4e-46)
		tmp = Float64(z * Float64(x / y_m));
	else
		tmp = Float64(x / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 2.4e-46)
		tmp = z * (x / y_m);
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 2.4e-46], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.40000000000000013e-46

   1. Initial program 90.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 45.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 45.1%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 46.2%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 46.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in z around inf 16.0%

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

      1. clear-num 15.9%

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

      2. un-div-inv 15.9%

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

      3. associate-/r* 17.5%

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

      4. metadata-eval 17.5%

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

      5. distribute-neg-frac 17.5%

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

      6. clear-num 17.5%

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

      7. distribute-neg-frac 17.5%

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

   9. Applied egg-rr 17.5%

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

       1. clear-num 17.5%

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

       2. associate-/r/ 18.0%

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

       3. clear-num 18.0%

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

       4. add-sqr-sqrt 10.2%

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

       5. sqrt-unprod 14.4%

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

       6. sqr-neg 14.4%

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

       7. sqrt-unprod 9.9%

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

       8. add-sqr-sqrt 17.8%

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

   11. Applied egg-rr 17.8%

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

   if 2.40000000000000013e-46 < x

   1. Initial program 86.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 53.5%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

      2. fabs-sqr 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

      3. add-sqr-sqrt 54.1%

         \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

      4. *-commutative 54.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   6. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

   7. Taylor expanded in x around inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. associate-/l* 52.4%

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

      3. distribute-rgt-neg-in 52.4%

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

      4. mul-1-neg 52.4%

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

      5. associate-*r/ 52.4%

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

      6. sub-neg 52.4%

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

      7. metadata-eval 52.4%

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

      8. distribute-lft-in 52.4%

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

      9. neg-mul-1 52.4%

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

      10. metadata-eval 52.4%

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

      11. +-commutative 52.4%

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

      12. unsub-neg 52.4%

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

   9. Simplified 52.4%

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

   10. Taylor expanded in z around 0 39.4%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 18.0% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{x}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ x y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return x / y_m;
}

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return x / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return x / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(x / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = x / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(x / y$95$m), $MachinePrecision]

Derivation

1. Initial program 89.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 47.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}}\right| \]

   2. fabs-sqr 47.8%

      \[\leadsto \color{blue}{\sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \cdot \sqrt{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)}} \]

   3. add-sqr-sqrt 48.7%

      \[\leadsto \color{blue}{\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)} \]

   4. *-commutative 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

6. Applied egg-rr 48.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{-1}{y}} \]

7. Taylor expanded in x around inf 30.7%

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

   1. mul-1-neg 30.7%

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

   2. associate-/l* 32.1%

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

   3. distribute-rgt-neg-in 32.1%

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

   4. mul-1-neg 32.1%

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

   5. associate-*r/ 32.1%

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

   6. sub-neg 32.1%

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

   7. metadata-eval 32.1%

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

   8. distribute-lft-in 32.1%

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

   9. neg-mul-1 32.1%

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

   10. metadata-eval 32.1%

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

   11. +-commutative 32.1%

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

   12. unsub-neg 32.1%

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

9. Simplified 32.1%

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

10. Taylor expanded in z around 0 21.6%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))