fabs fraction 1

Percentage Accurate: 91.7% → 99.9%

Time: 16.0s

Alternatives: 19

Speedup: 2.0×

• 21.5% 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: 91.7% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 0.002:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 0.002)
   (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x))))
   (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 <= 0.002) {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	} 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 <= 0.002)
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	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, 0.002], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $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 < 2e-3

   1. Initial program 87.9%

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

   3. Simplified 97.3%

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

   if 2e-3 < y

   1. Initial program 94.2%

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

      1. fabs-sub 94.2%

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

      2. associate-*l/ 92.4%

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

      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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := \left|t\_0 - z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\left|t\_0 - \frac{x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (fabs (- t_0 (* z (/ x y_m))))))
   (if (<= t_1 2e+30)
     (fabs (- t_0 (/ (* x z) y_m)))
     (if (<= t_1 2e+296) t_1 (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = fabs((t_0 - (z * (x / y_m))));
	double tmp;
	if (t_1 <= 2e+30) {
		tmp = fabs((t_0 - ((x * z) / y_m)));
	} else if (t_1 <= 2e+296) {
		tmp = t_1;
	} else {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = abs(Float64(t_0 - Float64(z * Float64(x / y_m))))
	tmp = 0.0
	if (t_1 <= 2e+30)
		tmp = abs(Float64(t_0 - Float64(Float64(x * z) / y_m)));
	elseif (t_1 <= 2e+296)
		tmp = t_1;
	else
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	end
	return 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]}, Block[{t$95$1 = N[Abs[N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e+30], N[Abs[N[(t$95$0 - N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+296], t$95$1, N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2e30

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0 99.9%

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

   if 2e30 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1.99999999999999996e296

   1. Initial program 99.9%

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

   if 1.99999999999999996e296 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 66.7%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := z \cdot \frac{x}{y\_m}\\ t_2 := \left|t\_0 - t\_1\right|\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\left|t\_0 - \frac{x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (* z (/ x y_m))) (t_2 (fabs (- t_0 t_1))))
   (if (<= t_2 2e+30)
     (fabs (- t_0 (/ (* x z) y_m)))
     (if (<= t_2 2e+306) t_2 (fabs t_1)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double t_2 = fabs((t_0 - t_1));
	double tmp;
	if (t_2 <= 2e+30) {
		tmp = fabs((t_0 - ((x * z) / y_m)));
	} else if (t_2 <= 2e+306) {
		tmp = t_2;
	} else {
		tmp = fabs(t_1);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    t_1 = z * (x / y_m)
    t_2 = abs((t_0 - t_1))
    if (t_2 <= 2d+30) then
        tmp = abs((t_0 - ((x * z) / y_m)))
    else if (t_2 <= 2d+306) then
        tmp = t_2
    else
        tmp = abs(t_1)
    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 t_1 = z * (x / y_m);
	double t_2 = Math.abs((t_0 - t_1));
	double tmp;
	if (t_2 <= 2e+30) {
		tmp = Math.abs((t_0 - ((x * z) / y_m)));
	} else if (t_2 <= 2e+306) {
		tmp = t_2;
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = z * (x / y_m)
	t_2 = math.fabs((t_0 - t_1))
	tmp = 0
	if t_2 <= 2e+30:
		tmp = math.fabs((t_0 - ((x * z) / y_m)))
	elif t_2 <= 2e+306:
		tmp = t_2
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(z * Float64(x / y_m))
	t_2 = abs(Float64(t_0 - t_1))
	tmp = 0.0
	if (t_2 <= 2e+30)
		tmp = abs(Float64(t_0 - Float64(Float64(x * z) / y_m)));
	elseif (t_2 <= 2e+306)
		tmp = t_2;
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	t_1 = z * (x / y_m);
	t_2 = abs((t_0 - t_1));
	tmp = 0.0;
	if (t_2 <= 2e+30)
		tmp = abs((t_0 - ((x * z) / y_m)));
	elseif (t_2 <= 2e+306)
		tmp = t_2;
	else
		tmp = abs(t_1);
	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]}, Block[{t$95$1 = N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e+30], N[Abs[N[(t$95$0 - N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], t$95$2, N[Abs[t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2e30

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0 99.9%

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

   if 2e30 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2.00000000000000003e306

   1. Initial program 99.9%

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

   if 2.00000000000000003e306 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 64.8%

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

   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. Taylor expanded in z around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. distribute-frac-neg2 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := z \cdot \frac{x}{y\_m}\\ t_1 := \left|\frac{x + 4}{y\_m} - t\_0\right|\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (* z (/ x y_m))) (t_1 (fabs (- (/ (+ x 4.0) y_m) t_0))))
   (if (<= t_1 2e+306) t_1 (fabs t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = z * (x / y_m);
	double t_1 = fabs((((x + 4.0) / y_m) - t_0));
	double tmp;
	if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = fabs(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 = z * (x / y_m)
    t_1 = abs((((x + 4.0d0) / y_m) - t_0))
    if (t_1 <= 2d+306) then
        tmp = t_1
    else
        tmp = abs(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 = z * (x / y_m);
	double t_1 = Math.abs((((x + 4.0) / y_m) - t_0));
	double tmp;
	if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = z * (x / y_m)
	t_1 = math.fabs((((x + 4.0) / y_m) - t_0))
	tmp = 0
	if t_1 <= 2e+306:
		tmp = t_1
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(z * Float64(x / y_m))
	t_1 = abs(Float64(Float64(Float64(x + 4.0) / y_m) - t_0))
	tmp = 0.0
	if (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = z * (x / y_m);
	t_1 = abs((((x + 4.0) / y_m) - t_0));
	tmp = 0.0;
	if (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = abs(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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e+306], t$95$1, N[Abs[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2.00000000000000003e306

   1. Initial program 96.4%

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

   if 2.00000000000000003e306 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 64.8%

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

   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. Taylor expanded in z around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. distribute-frac-neg2 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := z \cdot \frac{x}{y\_m}\\ t_2 := t\_0 - t\_1\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;t\_2 \leq 10^{-33}:\\ \;\;\;\;t\_0 - \frac{x}{\frac{y\_m}{z}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (* z (/ x y_m))) (t_2 (- t_0 t_1)))
   (if (<= t_2 -4e-223)
     (- (* x (/ z y_m)) t_0)
     (if (<= t_2 1e-33)
       (- t_0 (/ x (/ y_m z)))
       (if (<= t_2 2e+306) t_2 (fabs t_1))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double t_2 = t_0 - t_1;
	double tmp;
	if (t_2 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_2 <= 1e-33) {
		tmp = t_0 - (x / (y_m / z));
	} else if (t_2 <= 2e+306) {
		tmp = t_2;
	} else {
		tmp = fabs(t_1);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    t_1 = z * (x / y_m)
    t_2 = t_0 - t_1
    if (t_2 <= (-4d-223)) then
        tmp = (x * (z / y_m)) - t_0
    else if (t_2 <= 1d-33) then
        tmp = t_0 - (x / (y_m / z))
    else if (t_2 <= 2d+306) then
        tmp = t_2
    else
        tmp = abs(t_1)
    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 t_1 = z * (x / y_m);
	double t_2 = t_0 - t_1;
	double tmp;
	if (t_2 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_2 <= 1e-33) {
		tmp = t_0 - (x / (y_m / z));
	} else if (t_2 <= 2e+306) {
		tmp = t_2;
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = z * (x / y_m)
	t_2 = t_0 - t_1
	tmp = 0
	if t_2 <= -4e-223:
		tmp = (x * (z / y_m)) - t_0
	elif t_2 <= 1e-33:
		tmp = t_0 - (x / (y_m / z))
	elif t_2 <= 2e+306:
		tmp = t_2
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(z * Float64(x / y_m))
	t_2 = Float64(t_0 - t_1)
	tmp = 0.0
	if (t_2 <= -4e-223)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (t_2 <= 1e-33)
		tmp = Float64(t_0 - Float64(x / Float64(y_m / z)));
	elseif (t_2 <= 2e+306)
		tmp = t_2;
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	t_1 = z * (x / y_m);
	t_2 = t_0 - t_1;
	tmp = 0.0;
	if (t_2 <= -4e-223)
		tmp = (x * (z / y_m)) - t_0;
	elseif (t_2 <= 1e-33)
		tmp = t_0 - (x / (y_m / z));
	elseif (t_2 <= 2e+306)
		tmp = t_2;
	else
		tmp = abs(t_1);
	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]}, Block[{t$95$1 = N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-223], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 1e-33], N[(t$95$0 - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], t$95$2, N[Abs[t$95$1], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -3.9999999999999999e-223

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. div-inv 99.9%

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

      3. cancel-sign-sub-inv 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. div-inv 100.0%

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

      9. associate-*l/ 95.4%

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

      10. associate-*r/ 95.4%

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

      11. fma-undefine 95.4%

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

      12. add-sqr-sqrt 94.9%

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

      13. fabs-sqr 94.9%

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

      14. add-sqr-sqrt 95.4%

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

      15. fma-undefine 95.4%

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

      16. associate-*r/ 95.4%

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

      17. associate-*l/ 100.0%

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

      18. div-inv 99.9%

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

   4. Applied egg-rr 95.4%

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

   if -3.9999999999999999e-223 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1.0000000000000001e-33

   1. Initial program 86.7%

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

      1. add-sqr-sqrt 78.2%

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

      2. fabs-sqr 78.2%

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

      3. add-sqr-sqrt 79.4%

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

      4. associate-*l/ 85.5%

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

      5. associate-*r/ 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. clear-num 85.6%

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

      2. un-div-inv 85.6%

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

   6. Applied egg-rr 85.6%

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

   if 1.0000000000000001e-33 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 2.00000000000000003e306

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 91.1%

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

      1. add-sqr-sqrt 90.5%

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

      2. fabs-sqr 90.5%

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

      3. add-sqr-sqrt 91.1%

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

      4. div-inv 91.1%

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

      5. *-commutative 91.1%

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

      6. associate-*l* 99.8%

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

      7. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 2.00000000000000003e306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 47.2%

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

   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. Taylor expanded in z around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. distribute-frac-neg2 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 10^{-33}:\\ \;\;\;\;\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := t\_0 - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-33}:\\ \;\;\;\;t\_0 - \frac{x}{\frac{y\_m}{z}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (- t_0 (* z (/ x y_m)))))
   (if (<= t_1 -4e-223)
     (- (* x (/ z y_m)) t_0)
     (if (<= t_1 1e-33)
       (- t_0 (/ x (/ y_m z)))
       (if (<= t_1 INFINITY) t_1 (fabs (/ (- -4.0 x) 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 t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_1 <= 1e-33) {
		tmp = t_0 - (x / (y_m / z));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

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 t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_1 <= 1e-33) {
		tmp = t_0 - (x / (y_m / z));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = t_0 - (z * (x / y_m))
	tmp = 0
	if t_1 <= -4e-223:
		tmp = (x * (z / y_m)) - t_0
	elif t_1 <= 1e-33:
		tmp = t_0 - (x / (y_m / z))
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(t_0 - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_1 <= -4e-223)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (t_1 <= 1e-33)
		tmp = Float64(t_0 - Float64(x / Float64(y_m / z)));
	elseif (t_1 <= Inf)
		tmp = t_1;
	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)
	t_0 = (x + 4.0) / y_m;
	t_1 = t_0 - (z * (x / y_m));
	tmp = 0.0;
	if (t_1 <= -4e-223)
		tmp = (x * (z / y_m)) - t_0;
	elseif (t_1 <= 1e-33)
		tmp = t_0 - (x / (y_m / z));
	elseif (t_1 <= Inf)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-223], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-33], N[(t$95$0 - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -3.9999999999999999e-223

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. div-inv 99.9%

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

      3. cancel-sign-sub-inv 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. div-inv 100.0%

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

      9. associate-*l/ 95.4%

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

      10. associate-*r/ 95.4%

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

      11. fma-undefine 95.4%

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

      12. add-sqr-sqrt 94.9%

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

      13. fabs-sqr 94.9%

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

      14. add-sqr-sqrt 95.4%

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

      15. fma-undefine 95.4%

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

      16. associate-*r/ 95.4%

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

      17. associate-*l/ 100.0%

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

      18. div-inv 99.9%

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

   4. Applied egg-rr 95.4%

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

   if -3.9999999999999999e-223 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1.0000000000000001e-33

   1. Initial program 86.7%

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

      1. add-sqr-sqrt 78.2%

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

      2. fabs-sqr 78.2%

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

      3. add-sqr-sqrt 79.4%

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

      4. associate-*l/ 85.5%

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

      5. associate-*r/ 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. clear-num 85.6%

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

      2. un-div-inv 85.6%

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

   6. Applied egg-rr 85.6%

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

   if 1.0000000000000001e-33 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 93.0%

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

      1. add-sqr-sqrt 92.5%

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

      2. fabs-sqr 92.5%

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

      3. add-sqr-sqrt 93.0%

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

      4. div-inv 93.0%

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

      5. *-commutative 93.0%

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

      6. associate-*l* 99.8%

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

      7. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

      1. fabs-sub 0.0%

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

      2. associate-*l/ 31.6%

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

      3. associate-*r/ 31.6%

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

      4. fma-neg 68.4%

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

      5. distribute-neg-frac 68.4%

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

      6. +-commutative 68.4%

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

      7. distribute-neg-in 68.4%

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

      8. unsub-neg 68.4%

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

      9. metadata-eval 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. sub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 10^{-33}:\\ \;\;\;\;\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq \infty:\\ \;\;\;\;\frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := t\_0 - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-33}:\\ \;\;\;\;t\_0 - \frac{x}{\frac{y\_m}{z}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (- t_0 (* z (/ x y_m)))))
   (if (<= t_1 -4e-223)
     (- (* x (/ z y_m)) t_0)
     (if (<= t_1 1e-33)
       (- t_0 (/ x (/ y_m z)))
       (if (<= t_1 2e+306) t_1 (* z (/ (- x) 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 t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_1 <= 1e-33) {
		tmp = t_0 - (x / (y_m / z));
	} else if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = z * (-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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    t_1 = t_0 - (z * (x / y_m))
    if (t_1 <= (-4d-223)) then
        tmp = (x * (z / y_m)) - t_0
    else if (t_1 <= 1d-33) then
        tmp = t_0 - (x / (y_m / z))
    else if (t_1 <= 2d+306) then
        tmp = t_1
    else
        tmp = z * (-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 t_0 = (x + 4.0) / y_m;
	double t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_1 <= 1e-33) {
		tmp = t_0 - (x / (y_m / z));
	} else if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = z * (-x / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = t_0 - (z * (x / y_m))
	tmp = 0
	if t_1 <= -4e-223:
		tmp = (x * (z / y_m)) - t_0
	elif t_1 <= 1e-33:
		tmp = t_0 - (x / (y_m / z))
	elif t_1 <= 2e+306:
		tmp = t_1
	else:
		tmp = z * (-x / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(t_0 - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_1 <= -4e-223)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (t_1 <= 1e-33)
		tmp = Float64(t_0 - Float64(x / Float64(y_m / z)));
	elseif (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(Float64(-x) / 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;
	t_1 = t_0 - (z * (x / y_m));
	tmp = 0.0;
	if (t_1 <= -4e-223)
		tmp = (x * (z / y_m)) - t_0;
	elseif (t_1 <= 1e-33)
		tmp = t_0 - (x / (y_m / z));
	elseif (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = z * (-x / 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]}, Block[{t$95$1 = N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-223], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-33], N[(t$95$0 - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], t$95$1, N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -3.9999999999999999e-223

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. div-inv 99.9%

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

      3. cancel-sign-sub-inv 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. div-inv 100.0%

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

      9. associate-*l/ 95.4%

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

      10. associate-*r/ 95.4%

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

      11. fma-undefine 95.4%

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

      12. add-sqr-sqrt 94.9%

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

      13. fabs-sqr 94.9%

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

      14. add-sqr-sqrt 95.4%

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

      15. fma-undefine 95.4%

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

      16. associate-*r/ 95.4%

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

      17. associate-*l/ 100.0%

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

      18. div-inv 99.9%

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

   4. Applied egg-rr 95.4%

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

   if -3.9999999999999999e-223 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1.0000000000000001e-33

   1. Initial program 86.7%

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

      1. add-sqr-sqrt 78.2%

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

      2. fabs-sqr 78.2%

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

      3. add-sqr-sqrt 79.4%

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

      4. associate-*l/ 85.5%

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

      5. associate-*r/ 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. clear-num 85.6%

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

      2. un-div-inv 85.6%

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

   6. Applied egg-rr 85.6%

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

   if 1.0000000000000001e-33 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 2.00000000000000003e306

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 91.1%

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

      1. add-sqr-sqrt 90.5%

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

      2. fabs-sqr 90.5%

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

      3. add-sqr-sqrt 91.1%

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

      4. div-inv 91.1%

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

      5. *-commutative 91.1%

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

      6. associate-*l* 99.8%

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

      7. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 2.00000000000000003e306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 47.2%

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

      1. add-sqr-sqrt 47.2%

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

      2. fabs-sqr 47.2%

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

      3. add-sqr-sqrt 47.2%

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

      4. sub-neg 47.2%

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

      5. +-commutative 47.2%

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

      6. associate-*l/ 50.0%

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

      7. associate-*r/ 50.0%

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

      8. frac-2neg 50.0%

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

      9. +-commutative 50.0%

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

      10. distribute-neg-in 50.0%

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

      11. metadata-eval 50.0%

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

      12. sub-neg 50.0%

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

      13. distribute-frac-neg2 50.0%

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

      14. distribute-neg-in 50.0%

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

      15. fma-undefine 61.1%

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

      16. neg-sub0 61.1%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 50.0%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 50.0%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 47.2%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 59.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 59.6%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 59.6%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 59.6%

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

      2. *-un-lft-identity 59.6%

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

      3. associate-*l/ 59.6%

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

      4. associate-*r* 77.8%

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

      5. distribute-rgt-neg-in 77.8%

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

      6. un-div-inv 77.8%

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

   9. Applied egg-rr 77.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 10^{-33}:\\ \;\;\;\;\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := t\_0 - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (- t_0 (* z (/ x y_m)))))
   (if (<= t_1 -4e-223)
     (- (* x (/ z y_m)) t_0)
     (if (<= t_1 5e-89)
       (/ (- (+ x 4.0) (* x z)) y_m)
       (if (<= t_1 2e+306) t_1 (* z (/ (- x) 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 t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_1 <= 5e-89) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = z * (-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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    t_1 = t_0 - (z * (x / y_m))
    if (t_1 <= (-4d-223)) then
        tmp = (x * (z / y_m)) - t_0
    else if (t_1 <= 5d-89) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else if (t_1 <= 2d+306) then
        tmp = t_1
    else
        tmp = z * (-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 t_0 = (x + 4.0) / y_m;
	double t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -4e-223) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (t_1 <= 5e-89) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = z * (-x / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = t_0 - (z * (x / y_m))
	tmp = 0
	if t_1 <= -4e-223:
		tmp = (x * (z / y_m)) - t_0
	elif t_1 <= 5e-89:
		tmp = ((x + 4.0) - (x * z)) / y_m
	elif t_1 <= 2e+306:
		tmp = t_1
	else:
		tmp = z * (-x / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(t_0 - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_1 <= -4e-223)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (t_1 <= 5e-89)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	elseif (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(Float64(-x) / 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;
	t_1 = t_0 - (z * (x / y_m));
	tmp = 0.0;
	if (t_1 <= -4e-223)
		tmp = (x * (z / y_m)) - t_0;
	elseif (t_1 <= 5e-89)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	elseif (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = z * (-x / 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]}, Block[{t$95$1 = N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-223], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-89], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], t$95$1, N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -3.9999999999999999e-223

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. div-inv 99.9%

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

      3. cancel-sign-sub-inv 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. div-inv 100.0%

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

      9. associate-*l/ 95.4%

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

      10. associate-*r/ 95.4%

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

      11. fma-undefine 95.4%

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

      12. add-sqr-sqrt 94.9%

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

      13. fabs-sqr 94.9%

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

      14. add-sqr-sqrt 95.4%

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

      15. fma-undefine 95.4%

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

      16. associate-*r/ 95.4%

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

      17. associate-*l/ 100.0%

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

      18. div-inv 99.9%

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

   4. Applied egg-rr 95.4%

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

   if -3.9999999999999999e-223 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.99999999999999967e-89

   1. Initial program 83.3%

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

      1. add-sqr-sqrt 72.8%

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

      2. fabs-sqr 72.8%

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

      3. add-sqr-sqrt 74.1%

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

      4. associate-*l/ 81.8%

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

      5. associate-*r/ 81.8%

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

   4. Applied egg-rr 81.8%

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

      1. clear-num 81.8%

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

      2. associate-/r/ 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in y around 0 81.8%

      \[\leadsto \color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}} \]
   8. Step-by-step derivation

      1. div-sub 81.8%

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

      2. +-commutative 81.8%

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

      3. div-sub 81.8%

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

   9. Simplified 81.8%

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

   if 4.99999999999999967e-89 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 2.00000000000000003e306

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 92.4%

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

      1. add-sqr-sqrt 91.8%

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

      2. fabs-sqr 91.8%

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

      3. add-sqr-sqrt 92.4%

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

      4. div-inv 92.4%

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

      5. *-commutative 92.4%

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

      6. associate-*l* 99.8%

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

      7. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 2.00000000000000003e306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 47.2%

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

      1. add-sqr-sqrt 47.2%

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

      2. fabs-sqr 47.2%

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

      3. add-sqr-sqrt 47.2%

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

      4. sub-neg 47.2%

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

      5. +-commutative 47.2%

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

      6. associate-*l/ 50.0%

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

      7. associate-*r/ 50.0%

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

      8. frac-2neg 50.0%

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

      9. +-commutative 50.0%

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

      10. distribute-neg-in 50.0%

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

      11. metadata-eval 50.0%

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

      12. sub-neg 50.0%

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

      13. distribute-frac-neg2 50.0%

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

      14. distribute-neg-in 50.0%

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

      15. fma-undefine 61.1%

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

      16. neg-sub0 61.1%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 50.0%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 50.0%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 47.2%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 59.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 59.6%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 59.6%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 59.6%

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

      2. *-un-lft-identity 59.6%

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

      3. associate-*l/ 59.6%

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

      4. associate-*r* 77.8%

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

      5. distribute-rgt-neg-in 77.8%

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

      6. un-div-inv 77.8%

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

   9. Applied egg-rr 77.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := x \cdot \frac{z}{y\_m}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+64}:\\ \;\;\;\;\frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (* x (/ z y_m))))
   (if (<= x -1.45e+64)
     (/ (- x) y_m)
     (if (<= x -6.4e-6)
       t_0
       (if (<= x 3.6e-16) (/ 4.0 y_m) (if (<= x 1.9e+45) t_0 (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = x * (z / y_m);
	double tmp;
	if (x <= -1.45e+64) {
		tmp = -x / y_m;
	} else if (x <= -6.4e-6) {
		tmp = t_0;
	} else if (x <= 3.6e-16) {
		tmp = 4.0 / y_m;
	} else if (x <= 1.9e+45) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * (z / y_m)
    if (x <= (-1.45d+64)) then
        tmp = -x / y_m
    else if (x <= (-6.4d-6)) then
        tmp = t_0
    else if (x <= 3.6d-16) then
        tmp = 4.0d0 / y_m
    else if (x <= 1.9d+45) then
        tmp = t_0
    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 t_0 = x * (z / y_m);
	double tmp;
	if (x <= -1.45e+64) {
		tmp = -x / y_m;
	} else if (x <= -6.4e-6) {
		tmp = t_0;
	} else if (x <= 3.6e-16) {
		tmp = 4.0 / y_m;
	} else if (x <= 1.9e+45) {
		tmp = t_0;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = x * (z / y_m)
	tmp = 0
	if x <= -1.45e+64:
		tmp = -x / y_m
	elif x <= -6.4e-6:
		tmp = t_0
	elif x <= 3.6e-16:
		tmp = 4.0 / y_m
	elif x <= 1.9e+45:
		tmp = t_0
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(x * Float64(z / y_m))
	tmp = 0.0
	if (x <= -1.45e+64)
		tmp = Float64(Float64(-x) / y_m);
	elseif (x <= -6.4e-6)
		tmp = t_0;
	elseif (x <= 3.6e-16)
		tmp = Float64(4.0 / y_m);
	elseif (x <= 1.9e+45)
		tmp = t_0;
	else
		tmp = Float64(x / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = x * (z / y_m);
	tmp = 0.0;
	if (x <= -1.45e+64)
		tmp = -x / y_m;
	elseif (x <= -6.4e-6)
		tmp = t_0;
	elseif (x <= 3.6e-16)
		tmp = 4.0 / y_m;
	elseif (x <= 1.9e+45)
		tmp = t_0;
	else
		tmp = x / 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[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+64], N[((-x) / y$95$m), $MachinePrecision], If[LessEqual[x, -6.4e-6], t$95$0, If[LessEqual[x, 3.6e-16], N[(4.0 / y$95$m), $MachinePrecision], If[LessEqual[x, 1.9e+45], t$95$0, N[(x / y$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.44999999999999997e64

   1. Initial program 74.4%

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

      1. fabs-sub 74.4%

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

      2. associate-*l/ 81.0%

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

      3. associate-*r/ 85.0%

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

      4. fma-neg 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. unsub-neg 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. distribute-lft-in 71.5%

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

      3. metadata-eval 71.5%

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

      4. neg-mul-1 71.5%

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

      5. sub-neg 71.5%

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

   7. Simplified 71.5%

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

      1. add-sqr-sqrt 39.4%

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

      2. fabs-sqr 39.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-4 - x}{y}} \cdot \sqrt{\frac{-4 - x}{y}}} \]

      3. add-sqr-sqrt 39.8%

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

      4. div-inv 39.7%

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

   9. Applied egg-rr 39.7%

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

   10. Taylor expanded in x around inf 39.8%

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

       1. associate-*r/ 39.8%

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

       2. mul-1-neg 39.8%

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

   12. Simplified 39.8%

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

   if -1.44999999999999997e64 < x < -6.3999999999999997e-6 or 3.59999999999999983e-16 < x < 1.9000000000000001e45

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 62.0%

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

      2. fabs-sqr 62.0%

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

      3. add-sqr-sqrt 62.8%

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

      4. associate-*l/ 62.8%

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

      5. associate-*r/ 62.9%

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

   4. Applied egg-rr 62.9%

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

      1. associate-*r/ 62.8%

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

      2. add-sqr-sqrt 21.0%

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

      3. sqrt-unprod 14.7%

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

      4. sqr-neg 14.7%

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

      5. sqrt-unprod 17.1%

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

      6. add-sqr-sqrt 38.2%

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

      7. associate-/l* 38.1%

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

      8. distribute-frac-neg 38.1%

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

      9. distribute-rgt-neg-in 38.1%

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

      10. distribute-lft-neg-in 38.1%

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

      11. clear-num 38.2%

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

      12. un-div-inv 38.3%

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

   6. Applied egg-rr 38.3%

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

   7. Taylor expanded in z around inf 30.4%

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

      1. associate-*r/ 30.3%

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

   9. Simplified 30.3%

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

   if -6.3999999999999997e-6 < x < 3.59999999999999983e-16

   1. Initial program 94.4%

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

      1. add-sqr-sqrt 48.7%

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

      2. fabs-sqr 48.7%

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

      3. add-sqr-sqrt 49.9%

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

      4. associate-*l/ 52.5%

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

      5. associate-*r/ 51.0%

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

   4. Applied egg-rr 51.0%

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

      1. associate-*r/ 52.5%

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

      2. add-sqr-sqrt 24.2%

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

      3. sqrt-unprod 44.9%

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

      4. sqr-neg 44.9%

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

      5. sqrt-unprod 27.0%

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

      6. add-sqr-sqrt 51.2%

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

      7. associate-/l* 49.1%

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

      8. distribute-frac-neg 49.1%

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

      9. distribute-rgt-neg-in 49.1%

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

      10. distribute-lft-neg-in 49.1%

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

      11. clear-num 49.1%

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

      12. un-div-inv 49.1%

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

   6. Applied egg-rr 49.1%

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

   7. Taylor expanded in x around 0 44.3%

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

   if 1.9000000000000001e45 < x

   1. Initial program 88.1%

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

      1. fabs-sub 88.1%

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

      2. associate-*l/ 75.5%

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

      3. associate-*r/ 89.7%

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

      4. fma-neg 93.1%

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

      5. distribute-neg-frac 93.1%

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

      6. +-commutative 93.1%

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

      7. distribute-neg-in 93.1%

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

      8. unsub-neg 93.1%

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

      9. metadata-eval 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in z around 0 64.8%

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

      1. associate-*r/ 64.8%

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

      2. distribute-lft-in 64.8%

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

      3. metadata-eval 64.8%

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

      4. neg-mul-1 64.8%

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

      5. sub-neg 64.8%

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

   7. Simplified 64.8%

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

      1. add-sqr-sqrt 64.5%

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

      2. sqrt-unprod 56.8%

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

      3. neg-fabs 56.8%

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

      4. neg-fabs 56.8%

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

      5. sqr-abs 56.8%

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

      6. distribute-frac-neg2 56.8%

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

      7. sub-neg 56.8%

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

      8. metadata-eval 56.8%

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

      9. distribute-neg-in 56.8%

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

      10. +-commutative 56.8%

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

      11. frac-2neg 56.8%

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

      12. distribute-frac-neg2 56.8%

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

      13. sub-neg 56.8%

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

      14. metadata-eval 56.8%

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

      15. distribute-neg-in 56.8%

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

      16. +-commutative 56.8%

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

      17. frac-2neg 56.8%

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

      18. sqrt-unprod 34.8%

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

      19. add-sqr-sqrt 35.1%

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

      20. clear-num 35.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 35.1%

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

Alternative 10: 77.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right) - 4}{y\_m}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} + \frac{x}{\frac{y\_m}{z}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.5e+255)
   (* z (/ (- x) y_m))
   (if (<= x -2.7e-5)
     (/ (- (* x (+ -1.0 z)) 4.0) y_m)
     (if (<= x 8e+153)
       (/ (- (+ x 4.0) (* x z)) y_m)
       (+ (/ x y_m) (/ x (/ y_m z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.5e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	} else if (x <= 8e+153) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = (x / y_m) + (x / (y_m / z));
	}
	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 <= (-4.5d+255)) then
        tmp = z * (-x / y_m)
    else if (x <= (-2.7d-5)) then
        tmp = ((x * ((-1.0d0) + z)) - 4.0d0) / y_m
    else if (x <= 8d+153) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else
        tmp = (x / y_m) + (x / (y_m / z))
    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 <= -4.5e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	} else if (x <= 8e+153) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = (x / y_m) + (x / (y_m / z));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.5e+255:
		tmp = z * (-x / y_m)
	elif x <= -2.7e-5:
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m
	elif x <= 8e+153:
		tmp = ((x + 4.0) - (x * z)) / y_m
	else:
		tmp = (x / y_m) + (x / (y_m / z))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.5e+255)
		tmp = Float64(z * Float64(Float64(-x) / y_m));
	elseif (x <= -2.7e-5)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + z)) - 4.0) / y_m);
	elseif (x <= 8e+153)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	else
		tmp = Float64(Float64(x / y_m) + Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -4.5e+255)
		tmp = z * (-x / y_m);
	elseif (x <= -2.7e-5)
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	elseif (x <= 8e+153)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	else
		tmp = (x / y_m) + (x / (y_m / z));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -4.5e+255], N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.7e-5], N[(N[(N[(x * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 8e+153], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] + N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.49999999999999964e255

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 25.0%

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

      2. fabs-sqr 25.0%

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

      3. add-sqr-sqrt 25.1%

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

      4. sub-neg 25.1%

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

      5. +-commutative 25.1%

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

      6. associate-*l/ 25.1%

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

      7. associate-*r/ 25.1%

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

      8. frac-2neg 25.1%

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

      9. +-commutative 25.1%

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

      10. distribute-neg-in 25.1%

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

      11. metadata-eval 25.1%

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

      12. sub-neg 25.1%

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

      13. distribute-frac-neg2 25.1%

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

      14. distribute-neg-in 25.1%

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

      15. fma-undefine 37.6%

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

      16. neg-sub0 37.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 25.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 39.5%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 39.3%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 39.3%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r* 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. un-div-inv 63.2%

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

   9. Applied egg-rr 63.2%

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

   if -4.49999999999999964e255 < x < -2.6999999999999999e-5

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. fabs-sub 90.8%

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

      2. div-sub 98.2%

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

      3. add-sqr-sqrt 50.0%

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

      4. fabs-sqr 50.0%

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

      5. add-sqr-sqrt 50.6%

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

      6. associate--r+ 50.6%

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

      7. div-sub 50.6%

         \[\leadsto \color{blue}{\frac{x \cdot z - x}{y} - \frac{4}{y}} \]

      8. *-commutative 50.6%

         \[\leadsto \frac{\color{blue}{z \cdot x} - x}{y} - \frac{4}{y} \]

      9. *-un-lft-identity 50.6%

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

      10. distribute-rgt-out-- 50.6%

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

   5. Applied egg-rr 50.6%

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

   6. Taylor expanded in y around 0 50.6%

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

   if -2.6999999999999999e-5 < x < 8e153

   1. Initial program 93.7%

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

      1. add-sqr-sqrt 50.4%

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

      2. fabs-sqr 50.4%

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

      3. add-sqr-sqrt 51.5%

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

      4. associate-*l/ 52.4%

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

      5. associate-*r/ 52.4%

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

   4. Applied egg-rr 52.4%

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

      1. clear-num 52.4%

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

      2. associate-/r/ 52.4%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in y around 0 53.0%

      \[\leadsto \color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}} \]
   8. Step-by-step derivation

      1. div-sub 52.4%

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

      2. +-commutative 52.4%

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

      3. div-sub 53.0%

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

   9. Simplified 53.0%

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

   if 8e153 < x

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 48.1%

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

      2. fabs-sqr 48.1%

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

      3. add-sqr-sqrt 48.6%

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

      4. associate-*l/ 37.3%

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

      5. associate-*r/ 48.6%

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

   4. Applied egg-rr 48.6%

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

      1. associate-*r/ 37.3%

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

      2. add-sqr-sqrt 18.6%

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

      3. sqrt-unprod 37.1%

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

      4. sqr-neg 37.1%

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

      5. sqrt-unprod 15.4%

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

      6. add-sqr-sqrt 40.2%

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

      7. associate-/l* 48.6%

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

      8. distribute-frac-neg 48.6%

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

      9. distribute-rgt-neg-in 48.6%

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

      10. distribute-lft-neg-in 48.6%

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

      11. clear-num 48.5%

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

      12. un-div-inv 48.4%

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

   6. Applied egg-rr 48.4%

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

   7. Taylor expanded in x around inf 48.4%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right) - 4}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{x}{\frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right) - 4}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} - x \cdot \frac{z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.6e+255)
   (* z (/ (- x) y_m))
   (if (<= x -5.6e-6)
     (/ (- (* x (+ -1.0 z)) 4.0) y_m)
     (if (<= x 4.0) (/ (- 4.0 (* x z)) y_m) (- (/ x y_m) (* x (/ z y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.6e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -5.6e-6) {
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x / 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 <= (-1.6d+255)) then
        tmp = z * (-x / y_m)
    else if (x <= (-5.6d-6)) then
        tmp = ((x * ((-1.0d0) + z)) - 4.0d0) / y_m
    else if (x <= 4.0d0) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = (x / 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 <= -1.6e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -5.6e-6) {
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x / y_m) - (x * (z / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.6e+255:
		tmp = z * (-x / y_m)
	elif x <= -5.6e-6:
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m
	elif x <= 4.0:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = (x / y_m) - (x * (z / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.6e+255)
		tmp = Float64(z * Float64(Float64(-x) / y_m));
	elseif (x <= -5.6e-6)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + z)) - 4.0) / y_m);
	elseif (x <= 4.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = Float64(Float64(x / 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 <= -1.6e+255)
		tmp = z * (-x / y_m);
	elseif (x <= -5.6e-6)
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	elseif (x <= 4.0)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = (x / 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, -1.6e+255], N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.6e-6], N[(N[(N[(x * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 4.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] - N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.5999999999999999e255

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 25.0%

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

      2. fabs-sqr 25.0%

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

      3. add-sqr-sqrt 25.1%

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

      4. sub-neg 25.1%

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

      5. +-commutative 25.1%

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

      6. associate-*l/ 25.1%

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

      7. associate-*r/ 25.1%

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

      8. frac-2neg 25.1%

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

      9. +-commutative 25.1%

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

      10. distribute-neg-in 25.1%

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

      11. metadata-eval 25.1%

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

      12. sub-neg 25.1%

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

      13. distribute-frac-neg2 25.1%

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

      14. distribute-neg-in 25.1%

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

      15. fma-undefine 37.6%

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

      16. neg-sub0 37.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 25.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 39.5%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 39.3%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 39.3%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r* 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. un-div-inv 63.2%

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

   9. Applied egg-rr 63.2%

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

   if -1.5999999999999999e255 < x < -5.59999999999999975e-6

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. fabs-sub 90.8%

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

      2. div-sub 98.2%

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

      3. add-sqr-sqrt 50.0%

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

      4. fabs-sqr 50.0%

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

      5. add-sqr-sqrt 50.6%

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

      6. associate--r+ 50.6%

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

      7. div-sub 50.6%

         \[\leadsto \color{blue}{\frac{x \cdot z - x}{y} - \frac{4}{y}} \]

      8. *-commutative 50.6%

         \[\leadsto \frac{\color{blue}{z \cdot x} - x}{y} - \frac{4}{y} \]

      9. *-un-lft-identity 50.6%

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

      10. distribute-rgt-out-- 50.6%

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

   5. Applied egg-rr 50.6%

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

   6. Taylor expanded in y around 0 50.6%

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

   if -5.59999999999999975e-6 < x < 4

   1. Initial program 94.5%

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

      1. add-sqr-sqrt 48.7%

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

      2. fabs-sqr 48.7%

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

      3. add-sqr-sqrt 49.9%

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

      4. sub-neg 49.9%

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

      5. +-commutative 49.9%

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

      6. associate-*l/ 52.4%

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

      7. associate-*r/ 51.0%

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

      8. frac-2neg 51.0%

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

      9. +-commutative 51.0%

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

      10. distribute-neg-in 51.0%

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

      11. metadata-eval 51.0%

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

      12. sub-neg 51.0%

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

      13. distribute-frac-neg2 51.0%

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

      14. distribute-neg-in 51.0%

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

      15. fma-undefine 51.0%

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

      16. neg-sub0 51.0%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 51.0%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 52.4%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 49.9%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 52.5%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in x around 0 51.6%

      \[\leadsto 0 - \frac{x \cdot z - \color{blue}{4}}{y} \]

   6. Taylor expanded in y around 0 51.6%

      \[\leadsto \color{blue}{\frac{4 - x \cdot z}{y}} \]

   if 4 < x

   1. Initial program 89.3%

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

      1. add-sqr-sqrt 52.7%

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

      2. fabs-sqr 52.7%

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

      3. add-sqr-sqrt 53.2%

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

      4. associate-*l/ 44.7%

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

      5. associate-*r/ 53.2%

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

   4. Applied egg-rr 53.2%

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

   5. Taylor expanded in x around inf 53.2%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right) - 4}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right)}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} - x \cdot \frac{z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.2e+255)
   (* z (/ (- x) y_m))
   (if (<= x -2.7e-5)
     (/ (* x (+ -1.0 z)) y_m)
     (if (<= x 4.0) (/ (- 4.0 (* x z)) y_m) (- (/ x y_m) (* x (/ z y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.2e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = (x * (-1.0 + z)) / y_m;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x / 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 <= (-4.2d+255)) then
        tmp = z * (-x / y_m)
    else if (x <= (-2.7d-5)) then
        tmp = (x * ((-1.0d0) + z)) / y_m
    else if (x <= 4.0d0) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = (x / 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 <= -4.2e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = (x * (-1.0 + z)) / y_m;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x / y_m) - (x * (z / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.2e+255:
		tmp = z * (-x / y_m)
	elif x <= -2.7e-5:
		tmp = (x * (-1.0 + z)) / y_m
	elif x <= 4.0:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = (x / y_m) - (x * (z / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.2e+255)
		tmp = Float64(z * Float64(Float64(-x) / y_m));
	elseif (x <= -2.7e-5)
		tmp = Float64(Float64(x * Float64(-1.0 + z)) / y_m);
	elseif (x <= 4.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = Float64(Float64(x / 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 <= -4.2e+255)
		tmp = z * (-x / y_m);
	elseif (x <= -2.7e-5)
		tmp = (x * (-1.0 + z)) / y_m;
	elseif (x <= 4.0)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = (x / 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, -4.2e+255], N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.7e-5], N[(N[(x * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 4.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] - N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.2e255

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 25.0%

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

      2. fabs-sqr 25.0%

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

      3. add-sqr-sqrt 25.1%

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

      4. sub-neg 25.1%

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

      5. +-commutative 25.1%

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

      6. associate-*l/ 25.1%

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

      7. associate-*r/ 25.1%

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

      8. frac-2neg 25.1%

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

      9. +-commutative 25.1%

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

      10. distribute-neg-in 25.1%

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

      11. metadata-eval 25.1%

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

      12. sub-neg 25.1%

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

      13. distribute-frac-neg2 25.1%

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

      14. distribute-neg-in 25.1%

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

      15. fma-undefine 37.6%

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

      16. neg-sub0 37.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 25.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 39.5%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 39.3%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 39.3%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r* 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. un-div-inv 63.2%

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

   9. Applied egg-rr 63.2%

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

   if -4.2e255 < x < -2.6999999999999999e-5

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. fabs-sub 90.8%

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

      2. div-sub 98.2%

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

      3. add-sqr-sqrt 50.0%

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

      4. fabs-sqr 50.0%

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

      5. add-sqr-sqrt 50.6%

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

      6. associate--r+ 50.6%

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

      7. div-sub 50.6%

         \[\leadsto \color{blue}{\frac{x \cdot z - x}{y} - \frac{4}{y}} \]

      8. *-commutative 50.6%

         \[\leadsto \frac{\color{blue}{z \cdot x} - x}{y} - \frac{4}{y} \]

      9. *-un-lft-identity 50.6%

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

      10. distribute-rgt-out-- 50.6%

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

   5. Applied egg-rr 50.6%

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

   6. Taylor expanded in x around -inf 50.4%

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

   if -2.6999999999999999e-5 < x < 4

   1. Initial program 94.5%

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

      1. add-sqr-sqrt 48.7%

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

      2. fabs-sqr 48.7%

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

      3. add-sqr-sqrt 49.9%

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

      4. sub-neg 49.9%

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

      5. +-commutative 49.9%

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

      6. associate-*l/ 52.4%

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

      7. associate-*r/ 51.0%

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

      8. frac-2neg 51.0%

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

      9. +-commutative 51.0%

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

      10. distribute-neg-in 51.0%

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

      11. metadata-eval 51.0%

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

      12. sub-neg 51.0%

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

      13. distribute-frac-neg2 51.0%

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

      14. distribute-neg-in 51.0%

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

      15. fma-undefine 51.0%

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

      16. neg-sub0 51.0%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 51.0%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 52.4%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 49.9%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 52.5%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in x around 0 51.6%

      \[\leadsto 0 - \frac{x \cdot z - \color{blue}{4}}{y} \]

   6. Taylor expanded in y around 0 51.6%

      \[\leadsto \color{blue}{\frac{4 - x \cdot z}{y}} \]

   if 4 < x

   1. Initial program 89.3%

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

      1. add-sqr-sqrt 52.7%

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

      2. fabs-sqr 52.7%

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

      3. add-sqr-sqrt 53.2%

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

      4. associate-*l/ 44.7%

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

      5. associate-*r/ 53.2%

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

   4. Applied egg-rr 53.2%

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

   5. Taylor expanded in x around inf 53.2%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right)}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right)}{y\_m}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -3.8e+255)
   (* z (/ (- x) y_m))
   (if (<= x -2.7e-5)
     (/ (* x (+ -1.0 z)) y_m)
     (if (<= x 4.8e+49) (/ (- 4.0 (* x z)) 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 <= -3.8e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = (x * (-1.0 + z)) / y_m;
	} else if (x <= 4.8e+49) {
		tmp = (4.0 - (x * z)) / 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 <= (-3.8d+255)) then
        tmp = z * (-x / y_m)
    else if (x <= (-2.7d-5)) then
        tmp = (x * ((-1.0d0) + z)) / y_m
    else if (x <= 4.8d+49) then
        tmp = (4.0d0 - (x * z)) / 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 <= -3.8e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = (x * (-1.0 + z)) / y_m;
	} else if (x <= 4.8e+49) {
		tmp = (4.0 - (x * z)) / 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 <= -3.8e+255:
		tmp = z * (-x / y_m)
	elif x <= -2.7e-5:
		tmp = (x * (-1.0 + z)) / y_m
	elif x <= 4.8e+49:
		tmp = (4.0 - (x * z)) / 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 <= -3.8e+255)
		tmp = Float64(z * Float64(Float64(-x) / y_m));
	elseif (x <= -2.7e-5)
		tmp = Float64(Float64(x * Float64(-1.0 + z)) / y_m);
	elseif (x <= 4.8e+49)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / 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 <= -3.8e+255)
		tmp = z * (-x / y_m);
	elseif (x <= -2.7e-5)
		tmp = (x * (-1.0 + z)) / y_m;
	elseif (x <= 4.8e+49)
		tmp = (4.0 - (x * z)) / 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, -3.8e+255], N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.7e-5], N[(N[(x * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 4.8e+49], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.7999999999999999e255

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 25.0%

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

      2. fabs-sqr 25.0%

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

      3. add-sqr-sqrt 25.1%

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

      4. sub-neg 25.1%

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

      5. +-commutative 25.1%

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

      6. associate-*l/ 25.1%

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

      7. associate-*r/ 25.1%

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

      8. frac-2neg 25.1%

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

      9. +-commutative 25.1%

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

      10. distribute-neg-in 25.1%

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

      11. metadata-eval 25.1%

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

      12. sub-neg 25.1%

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

      13. distribute-frac-neg2 25.1%

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

      14. distribute-neg-in 25.1%

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

      15. fma-undefine 37.6%

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

      16. neg-sub0 37.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 25.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 39.5%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 39.3%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 39.3%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r* 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. un-div-inv 63.2%

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

   9. Applied egg-rr 63.2%

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

   if -3.7999999999999999e255 < x < -2.6999999999999999e-5

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. fabs-sub 90.8%

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

      2. div-sub 98.2%

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

      3. add-sqr-sqrt 50.0%

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

      4. fabs-sqr 50.0%

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

      5. add-sqr-sqrt 50.6%

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

      6. associate--r+ 50.6%

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

      7. div-sub 50.6%

         \[\leadsto \color{blue}{\frac{x \cdot z - x}{y} - \frac{4}{y}} \]

      8. *-commutative 50.6%

         \[\leadsto \frac{\color{blue}{z \cdot x} - x}{y} - \frac{4}{y} \]

      9. *-un-lft-identity 50.6%

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

      10. distribute-rgt-out-- 50.6%

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

   5. Applied egg-rr 50.6%

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

   6. Taylor expanded in x around -inf 50.4%

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

   if -2.6999999999999999e-5 < x < 4.8e49

   1. Initial program 94.9%

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

      1. add-sqr-sqrt 50.2%

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

      2. fabs-sqr 50.2%

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

      3. add-sqr-sqrt 51.4%

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

      4. sub-neg 51.4%

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

      5. +-commutative 51.4%

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

      6. associate-*l/ 53.7%

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

      7. associate-*r/ 52.4%

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

      8. frac-2neg 52.4%

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

      9. +-commutative 52.4%

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

      10. distribute-neg-in 52.4%

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

      11. metadata-eval 52.4%

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

      12. sub-neg 52.4%

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

      13. distribute-frac-neg2 52.4%

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

      14. distribute-neg-in 52.4%

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

      15. fma-undefine 52.4%

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

      16. neg-sub0 52.4%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 52.4%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 53.7%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 51.4%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in x around 0 52.5%

      \[\leadsto 0 - \frac{x \cdot z - \color{blue}{4}}{y} \]

   6. Taylor expanded in y around 0 52.5%

      \[\leadsto \color{blue}{\frac{4 - x \cdot z}{y}} \]

   if 4.8e49 < x

   1. Initial program 87.4%

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

      1. fabs-sub 87.4%

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

      2. associate-*l/ 74.3%

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

      3. associate-*r/ 89.2%

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

      4. fma-neg 92.7%

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

      5. distribute-neg-frac 92.7%

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

      6. +-commutative 92.7%

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

      7. distribute-neg-in 92.7%

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

      8. unsub-neg 92.7%

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

      9. metadata-eval 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around 0 66.3%

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

      1. associate-*r/ 66.3%

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

      2. distribute-lft-in 66.3%

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

      3. metadata-eval 66.3%

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

      4. neg-mul-1 66.3%

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

      5. sub-neg 66.3%

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

   7. Simplified 66.3%

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

      1. add-sqr-sqrt 66.0%

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

      2. sqrt-unprod 57.9%

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

      3. neg-fabs 57.9%

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

      4. neg-fabs 57.9%

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

      5. sqr-abs 57.9%

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

      6. distribute-frac-neg2 57.9%

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

      7. sub-neg 57.9%

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

      8. metadata-eval 57.9%

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

      9. distribute-neg-in 57.9%

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

      10. +-commutative 57.9%

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

      11. frac-2neg 57.9%

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

      12. distribute-frac-neg2 57.9%

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

      13. sub-neg 57.9%

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

      14. metadata-eval 57.9%

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

      15. distribute-neg-in 57.9%

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

      16. +-commutative 57.9%

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

      17. frac-2neg 57.9%

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

      18. sqrt-unprod 36.5%

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

      19. add-sqr-sqrt 36.8%

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

   9. Applied egg-rr 36.8%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right)}{y}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right) - 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.5e+255)
   (* z (/ (- x) y_m))
   (if (<= x -2.7e-5)
     (/ (- (* x (+ -1.0 z)) 4.0) y_m)
     (/ (- (+ x 4.0) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.5e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	} else {
		tmp = ((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) :: tmp
    if (x <= (-4.5d+255)) then
        tmp = z * (-x / y_m)
    else if (x <= (-2.7d-5)) then
        tmp = ((x * ((-1.0d0) + z)) - 4.0d0) / y_m
    else
        tmp = ((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 tmp;
	if (x <= -4.5e+255) {
		tmp = z * (-x / y_m);
	} else if (x <= -2.7e-5) {
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.5e+255:
		tmp = z * (-x / y_m)
	elif x <= -2.7e-5:
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m
	else:
		tmp = ((x + 4.0) - (x * z)) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.5e+255)
		tmp = Float64(z * Float64(Float64(-x) / y_m));
	elseif (x <= -2.7e-5)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + z)) - 4.0) / y_m);
	else
		tmp = 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)
	tmp = 0.0;
	if (x <= -4.5e+255)
		tmp = z * (-x / y_m);
	elseif (x <= -2.7e-5)
		tmp = ((x * (-1.0 + z)) - 4.0) / y_m;
	else
		tmp = ((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_] := If[LessEqual[x, -4.5e+255], N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.7e-5], N[(N[(N[(x * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.49999999999999964e255

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 25.0%

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

      2. fabs-sqr 25.0%

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

      3. add-sqr-sqrt 25.1%

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

      4. sub-neg 25.1%

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

      5. +-commutative 25.1%

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

      6. associate-*l/ 25.1%

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

      7. associate-*r/ 25.1%

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

      8. frac-2neg 25.1%

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

      9. +-commutative 25.1%

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

      10. distribute-neg-in 25.1%

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

      11. metadata-eval 25.1%

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

      12. sub-neg 25.1%

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

      13. distribute-frac-neg2 25.1%

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

      14. distribute-neg-in 25.1%

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

      15. fma-undefine 37.6%

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

      16. neg-sub0 37.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 25.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 39.5%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 39.3%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 39.3%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r* 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. un-div-inv 63.2%

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

   9. Applied egg-rr 63.2%

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

   if -4.49999999999999964e255 < x < -2.6999999999999999e-5

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. fabs-sub 90.8%

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

      2. div-sub 98.2%

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

      3. add-sqr-sqrt 50.0%

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

      4. fabs-sqr 50.0%

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

      5. add-sqr-sqrt 50.6%

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

      6. associate--r+ 50.6%

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

      7. div-sub 50.6%

         \[\leadsto \color{blue}{\frac{x \cdot z - x}{y} - \frac{4}{y}} \]

      8. *-commutative 50.6%

         \[\leadsto \frac{\color{blue}{z \cdot x} - x}{y} - \frac{4}{y} \]

      9. *-un-lft-identity 50.6%

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

      10. distribute-rgt-out-- 50.6%

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

   5. Applied egg-rr 50.6%

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

   6. Taylor expanded in y around 0 50.6%

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

   if -2.6999999999999999e-5 < x

   1. Initial program 92.7%

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

      1. add-sqr-sqrt 50.0%

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

      2. fabs-sqr 50.0%

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

      3. add-sqr-sqrt 51.0%

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

      4. associate-*l/ 49.8%

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

      5. associate-*r/ 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. clear-num 51.7%

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

      2. associate-/r/ 51.7%

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

   6. Applied egg-rr 51.7%

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

   7. Taylor expanded in y around 0 51.9%

      \[\leadsto \color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}} \]
   8. Step-by-step derivation

      1. div-sub 49.8%

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

      2. +-commutative 49.8%

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

      3. div-sub 51.9%

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

   9. Simplified 51.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+255}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right) - 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+253}:\\ \;\;\;\;z \cdot \frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -7e+253)
   (* z (/ (- x) y_m))
   (if (<= x -1.55e-5) (/ (* x (+ -1.0 z)) 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 <= -7e+253) {
		tmp = z * (-x / y_m);
	} else if (x <= -1.55e-5) {
		tmp = (x * (-1.0 + z)) / 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 <= (-7d+253)) then
        tmp = z * (-x / y_m)
    else if (x <= (-1.55d-5)) then
        tmp = (x * ((-1.0d0) + z)) / 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 <= -7e+253) {
		tmp = z * (-x / y_m);
	} else if (x <= -1.55e-5) {
		tmp = (x * (-1.0 + z)) / 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 <= -7e+253:
		tmp = z * (-x / y_m)
	elif x <= -1.55e-5:
		tmp = (x * (-1.0 + z)) / 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 <= -7e+253)
		tmp = Float64(z * Float64(Float64(-x) / y_m));
	elseif (x <= -1.55e-5)
		tmp = Float64(Float64(x * Float64(-1.0 + z)) / 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 <= -7e+253)
		tmp = z * (-x / y_m);
	elseif (x <= -1.55e-5)
		tmp = (x * (-1.0 + z)) / 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, -7e+253], N[(z * N[((-x) / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-5], N[(N[(x * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.99999999999999955e253

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 25.0%

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

      2. fabs-sqr 25.0%

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

      3. add-sqr-sqrt 25.1%

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

      4. sub-neg 25.1%

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

      5. +-commutative 25.1%

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

      6. associate-*l/ 25.1%

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

      7. associate-*r/ 25.1%

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

      8. frac-2neg 25.1%

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

      9. +-commutative 25.1%

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

      10. distribute-neg-in 25.1%

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

      11. metadata-eval 25.1%

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

      12. sub-neg 25.1%

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

      13. distribute-frac-neg2 25.1%

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

      14. distribute-neg-in 25.1%

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

      15. fma-undefine 37.6%

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

      16. neg-sub0 37.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 25.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 25.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 39.5%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 39.3%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]

   7. Simplified 39.3%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r* 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. un-div-inv 63.2%

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

   9. Applied egg-rr 63.2%

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

   if -6.99999999999999955e253 < x < -1.55000000000000007e-5

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. fabs-sub 90.8%

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

      2. div-sub 98.2%

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

      3. add-sqr-sqrt 50.0%

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

      4. fabs-sqr 50.0%

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

      5. add-sqr-sqrt 50.6%

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

      6. associate--r+ 50.6%

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

      7. div-sub 50.6%

         \[\leadsto \color{blue}{\frac{x \cdot z - x}{y} - \frac{4}{y}} \]

      8. *-commutative 50.6%

         \[\leadsto \frac{\color{blue}{z \cdot x} - x}{y} - \frac{4}{y} \]

      9. *-un-lft-identity 50.6%

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

      10. distribute-rgt-out-- 50.6%

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

   5. Applied egg-rr 50.6%

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

   6. Taylor expanded in x around -inf 50.4%

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

   if -1.55000000000000007e-5 < x

   1. Initial program 92.7%

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

      1. fabs-sub 92.7%

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

      2. associate-*l/ 92.5%

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

      3. associate-*r/ 93.5%

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

      4. fma-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. +-commutative 94.5%

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

      7. distribute-neg-in 94.5%

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

      8. unsub-neg 94.5%

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

      9. metadata-eval 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in z around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. distribute-lft-in 75.7%

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

      3. metadata-eval 75.7%

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

      4. neg-mul-1 75.7%

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

      5. sub-neg 75.7%

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

   7. Simplified 75.7%

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

      1. add-sqr-sqrt 75.3%

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

      2. sqrt-unprod 50.7%

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

      3. neg-fabs 50.7%

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

      4. neg-fabs 50.7%

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

      5. sqr-abs 50.7%

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

      6. distribute-frac-neg2 50.7%

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

      7. sub-neg 50.7%

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

      8. metadata-eval 50.7%

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

      9. distribute-neg-in 50.7%

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

      10. +-commutative 50.7%

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

      11. frac-2neg 50.7%

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

      12. distribute-frac-neg2 50.7%

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

      13. sub-neg 50.7%

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

      14. metadata-eval 50.7%

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

      15. distribute-neg-in 50.7%

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

      16. +-commutative 50.7%

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

      17. frac-2neg 50.7%

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

      18. sqrt-unprod 39.5%

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

      19. add-sqr-sqrt 40.4%

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

   9. Applied egg-rr 40.4%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+253}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(-1 + z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq -1.24 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.5e+63)
   (/ (- x) y_m)
   (if (<= x -1.24e-5) (* x (/ z 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+63) {
		tmp = -x / y_m;
	} else if (x <= -1.24e-5) {
		tmp = x * (z / 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+63)) then
        tmp = -x / y_m
    else if (x <= (-1.24d-5)) then
        tmp = x * (z / 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+63) {
		tmp = -x / y_m;
	} else if (x <= -1.24e-5) {
		tmp = x * (z / 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+63:
		tmp = -x / y_m
	elif x <= -1.24e-5:
		tmp = x * (z / 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+63)
		tmp = Float64(Float64(-x) / y_m);
	elseif (x <= -1.24e-5)
		tmp = Float64(x * Float64(z / 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+63)
		tmp = -x / y_m;
	elseif (x <= -1.24e-5)
		tmp = x * (z / 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+63], N[((-x) / y$95$m), $MachinePrecision], If[LessEqual[x, -1.24e-5], N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000004e63

   1. Initial program 74.4%

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

      1. fabs-sub 74.4%

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

      2. associate-*l/ 81.0%

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

      3. associate-*r/ 85.0%

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

      4. fma-neg 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. unsub-neg 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. distribute-lft-in 71.5%

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

      3. metadata-eval 71.5%

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

      4. neg-mul-1 71.5%

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

      5. sub-neg 71.5%

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

   7. Simplified 71.5%

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

      1. add-sqr-sqrt 39.4%

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

      2. fabs-sqr 39.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-4 - x}{y}} \cdot \sqrt{\frac{-4 - x}{y}}} \]

      3. add-sqr-sqrt 39.8%

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

      4. div-inv 39.7%

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

   9. Applied egg-rr 39.7%

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

   10. Taylor expanded in x around inf 39.8%

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

       1. associate-*r/ 39.8%

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

       2. mul-1-neg 39.8%

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

   12. Simplified 39.8%

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

   if -8.5000000000000004e63 < x < -1.24e-5

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 59.8%

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

      2. fabs-sqr 59.8%

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

      3. add-sqr-sqrt 60.5%

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

      4. associate-*l/ 60.5%

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

      5. associate-*r/ 60.5%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

   4. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
   5. Step-by-step derivation

      1. associate-*r/ 60.5%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      2. add-sqr-sqrt 13.8%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}{y} \]

      3. sqrt-unprod 15.7%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\sqrt{z \cdot z}}}{y} \]

      4. sqr-neg 15.7%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}{y} \]

      5. sqrt-unprod 27.1%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)}}{y} \]

      6. add-sqr-sqrt 34.2%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(-z\right)}}{y} \]

      7. associate-/l* 34.1%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{-z}{y}} \]

      8. distribute-frac-neg 34.1%

         \[\leadsto \frac{x + 4}{y} - x \cdot \color{blue}{\left(-\frac{z}{y}\right)} \]

      9. distribute-rgt-neg-in 34.1%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x \cdot \frac{z}{y}\right)} \]

      10. distribute-lft-neg-in 34.1%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x\right) \cdot \frac{z}{y}} \]

      11. clear-num 34.2%

          \[\leadsto \frac{x + 4}{y} - \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}} \]

      12. un-div-inv 34.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   6. Applied egg-rr 34.3%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   7. Taylor expanded in z around inf 28.1%

      \[\leadsto \color{blue}{\frac{x \cdot z}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 28.0%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y}} \]

   9. Simplified 28.0%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y}} \]

   if -1.24e-5 < x

   1. Initial program 92.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 92.7%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 92.5%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 93.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 94.5%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 94.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 94.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 94.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 94.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 94.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.7%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 75.7%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 75.7%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 75.7%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 75.7%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 75.7%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 75.7%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 75.3%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right|} \cdot \sqrt{\left|\frac{-4 - x}{y}\right|}} \]

      2. sqrt-unprod 50.7%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right| \cdot \left|\frac{-4 - x}{y}\right|}} \]

      3. neg-fabs 50.7%

         \[\leadsto \sqrt{\color{blue}{\left|-\frac{-4 - x}{y}\right|} \cdot \left|\frac{-4 - x}{y}\right|} \]

      4. neg-fabs 50.7%

         \[\leadsto \sqrt{\left|-\frac{-4 - x}{y}\right| \cdot \color{blue}{\left|-\frac{-4 - x}{y}\right|}} \]

      5. sqr-abs 50.7%

         \[\leadsto \sqrt{\color{blue}{\left(-\frac{-4 - x}{y}\right) \cdot \left(-\frac{-4 - x}{y}\right)}} \]

      6. distribute-frac-neg2 50.7%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 - x}{-y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      7. sub-neg 50.7%

         \[\leadsto \sqrt{\frac{\color{blue}{-4 + \left(-x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      8. metadata-eval 50.7%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      9. distribute-neg-in 50.7%

         \[\leadsto \sqrt{\frac{\color{blue}{-\left(4 + x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      10. +-commutative 50.7%

          \[\leadsto \sqrt{\frac{-\color{blue}{\left(x + 4\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      11. frac-2neg 50.7%

          \[\leadsto \sqrt{\color{blue}{\frac{x + 4}{y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      12. distribute-frac-neg2 50.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 50.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-4 + \left(-x\right)}}{-y}} \]

      14. metadata-eval 50.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y}} \]

      15. distribute-neg-in 50.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-\left(4 + x\right)}}{-y}} \]

      16. +-commutative 50.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{-\color{blue}{\left(x + 4\right)}}{-y}} \]

      17. frac-2neg 50.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{x + 4}{y}}} \]

      18. sqrt-unprod 39.5%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y}} \cdot \sqrt{\frac{x + 4}{y}}} \]

      19. add-sqr-sqrt 40.4%

          \[\leadsto \color{blue}{\frac{x + 4}{y}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\frac{x + 4}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 68.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -10.2) (/ (- x) y_m) (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -10.2) {
		tmp = -x / y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / 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 <= (-10.2d0)) then
        tmp = -x / y_m
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / 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 <= -10.2) {
		tmp = -x / y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / 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 <= -10.2:
		tmp = -x / y_m
	elif x <= 4.0:
		tmp = 4.0 / 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 <= -10.2)
		tmp = Float64(Float64(-x) / y_m);
	elseif (x <= 4.0)
		tmp = Float64(4.0 / 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 <= -10.2)
		tmp = -x / y_m;
	elseif (x <= 4.0)
		tmp = 4.0 / 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, -10.2], N[((-x) / y$95$m), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -10.199999999999999

   1. Initial program 80.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 80.2%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 85.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 88.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 96.6%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 96.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 96.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 96.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 96.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 96.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 64.0%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 64.0%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 64.0%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 64.0%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 64.0%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 64.0%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 64.0%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 35.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{-4 - x}{y}} \cdot \sqrt{\frac{-4 - x}{y}}}\right| \]

      2. fabs-sqr 35.4%

         \[\leadsto \color{blue}{\sqrt{\frac{-4 - x}{y}} \cdot \sqrt{\frac{-4 - x}{y}}} \]

      3. add-sqr-sqrt 36.0%

         \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

      4. div-inv 35.9%

         \[\leadsto \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

   9. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

   10. Taylor expanded in x around inf 35.8%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   11. Step-by-step derivation

       1. associate-*r/ 35.8%

          \[\leadsto \color{blue}{\frac{-1 \cdot x}{y}} \]

       2. mul-1-neg 35.8%

          \[\leadsto \frac{\color{blue}{-x}}{y} \]

   12. Simplified 35.8%

       \[\leadsto \color{blue}{\frac{-x}{y}} \]

   if -10.199999999999999 < x < 4

   1. Initial program 94.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 48.3%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 48.3%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 49.5%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. associate-*l/ 52.0%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      5. associate-*r/ 50.6%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

   4. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
   5. Step-by-step derivation

      1. associate-*r/ 52.0%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      2. add-sqr-sqrt 24.4%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}{y} \]

      3. sqrt-unprod 44.6%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\sqrt{z \cdot z}}}{y} \]

      4. sqr-neg 44.6%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}{y} \]

      5. sqrt-unprod 27.1%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)}}{y} \]

      6. add-sqr-sqrt 52.3%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(-z\right)}}{y} \]

      7. associate-/l* 50.3%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{-z}{y}} \]

      8. distribute-frac-neg 50.3%

         \[\leadsto \frac{x + 4}{y} - x \cdot \color{blue}{\left(-\frac{z}{y}\right)} \]

      9. distribute-rgt-neg-in 50.3%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x \cdot \frac{z}{y}\right)} \]

      10. distribute-lft-neg-in 50.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x\right) \cdot \frac{z}{y}} \]

      11. clear-num 50.3%

          \[\leadsto \frac{x + 4}{y} - \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}} \]

      12. un-div-inv 50.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   6. Applied egg-rr 50.3%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   7. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 89.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 89.3%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 78.1%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 90.8%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 93.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 58.6%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 58.6%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 58.6%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 58.6%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 58.6%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 58.6%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 58.6%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 58.4%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right|} \cdot \sqrt{\left|\frac{-4 - x}{y}\right|}} \]

      2. sqrt-unprod 52.8%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right| \cdot \left|\frac{-4 - x}{y}\right|}} \]

      3. neg-fabs 52.8%

         \[\leadsto \sqrt{\color{blue}{\left|-\frac{-4 - x}{y}\right|} \cdot \left|\frac{-4 - x}{y}\right|} \]

      4. neg-fabs 52.8%

         \[\leadsto \sqrt{\left|-\frac{-4 - x}{y}\right| \cdot \color{blue}{\left|-\frac{-4 - x}{y}\right|}} \]

      5. sqr-abs 52.8%

         \[\leadsto \sqrt{\color{blue}{\left(-\frac{-4 - x}{y}\right) \cdot \left(-\frac{-4 - x}{y}\right)}} \]

      6. distribute-frac-neg2 52.8%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 - x}{-y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      7. sub-neg 52.8%

         \[\leadsto \sqrt{\frac{\color{blue}{-4 + \left(-x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      8. metadata-eval 52.8%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      9. distribute-neg-in 52.8%

         \[\leadsto \sqrt{\frac{\color{blue}{-\left(4 + x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      10. +-commutative 52.8%

          \[\leadsto \sqrt{\frac{-\color{blue}{\left(x + 4\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      11. frac-2neg 52.8%

          \[\leadsto \sqrt{\color{blue}{\frac{x + 4}{y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      12. distribute-frac-neg2 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-4 + \left(-x\right)}}{-y}} \]

      14. metadata-eval 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y}} \]

      15. distribute-neg-in 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-\left(4 + x\right)}}{-y}} \]

      16. +-commutative 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{-\color{blue}{\left(x + 4\right)}}{-y}} \]

      17. frac-2neg 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{x + 4}{y}}} \]

      18. sqrt-unprod 31.7%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y}} \cdot \sqrt{\frac{x + 4}{y}}} \]

      19. add-sqr-sqrt 32.0%

          \[\leadsto \color{blue}{\frac{x + 4}{y}} \]

      20. clear-num 32.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 32.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 32.0%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 54.5% accurate, 13.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / 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 <= 4.0d0) then
        tmp = 4.0d0 / 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 <= 4.0) {
		tmp = 4.0 / 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 <= 4.0:
		tmp = 4.0 / 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 <= 4.0)
		tmp = Float64(4.0 / 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 <= 4.0)
		tmp = 4.0 / 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, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 89.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 45.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 45.9%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 46.8%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. associate-*l/ 49.1%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      5. associate-*r/ 48.1%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

   4. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
   5. Step-by-step derivation

      1. associate-*r/ 49.1%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      2. add-sqr-sqrt 22.4%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}{y} \]

      3. sqrt-unprod 41.5%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\sqrt{z \cdot z}}}{y} \]

      4. sqr-neg 41.5%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}{y} \]

      5. sqrt-unprod 25.8%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)}}{y} \]

      6. add-sqr-sqrt 48.7%

         \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(-z\right)}}{y} \]

      7. associate-/l* 48.3%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{-z}{y}} \]

      8. distribute-frac-neg 48.3%

         \[\leadsto \frac{x + 4}{y} - x \cdot \color{blue}{\left(-\frac{z}{y}\right)} \]

      9. distribute-rgt-neg-in 48.3%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x \cdot \frac{z}{y}\right)} \]

      10. distribute-lft-neg-in 48.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x\right) \cdot \frac{z}{y}} \]

      11. clear-num 48.3%

          \[\leadsto \frac{x + 4}{y} - \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}} \]

      12. un-div-inv 48.4%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   6. Applied egg-rr 48.4%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   7. Taylor expanded in x around 0 30.7%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 89.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 89.3%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 78.1%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 90.8%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 93.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 93.8%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 58.6%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 58.6%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 58.6%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 58.6%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 58.6%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 58.6%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 58.6%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 58.4%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right|} \cdot \sqrt{\left|\frac{-4 - x}{y}\right|}} \]

      2. sqrt-unprod 52.8%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right| \cdot \left|\frac{-4 - x}{y}\right|}} \]

      3. neg-fabs 52.8%

         \[\leadsto \sqrt{\color{blue}{\left|-\frac{-4 - x}{y}\right|} \cdot \left|\frac{-4 - x}{y}\right|} \]

      4. neg-fabs 52.8%

         \[\leadsto \sqrt{\left|-\frac{-4 - x}{y}\right| \cdot \color{blue}{\left|-\frac{-4 - x}{y}\right|}} \]

      5. sqr-abs 52.8%

         \[\leadsto \sqrt{\color{blue}{\left(-\frac{-4 - x}{y}\right) \cdot \left(-\frac{-4 - x}{y}\right)}} \]

      6. distribute-frac-neg2 52.8%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 - x}{-y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      7. sub-neg 52.8%

         \[\leadsto \sqrt{\frac{\color{blue}{-4 + \left(-x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      8. metadata-eval 52.8%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      9. distribute-neg-in 52.8%

         \[\leadsto \sqrt{\frac{\color{blue}{-\left(4 + x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      10. +-commutative 52.8%

          \[\leadsto \sqrt{\frac{-\color{blue}{\left(x + 4\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      11. frac-2neg 52.8%

          \[\leadsto \sqrt{\color{blue}{\frac{x + 4}{y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      12. distribute-frac-neg2 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-4 + \left(-x\right)}}{-y}} \]

      14. metadata-eval 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y}} \]

      15. distribute-neg-in 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-\left(4 + x\right)}}{-y}} \]

      16. +-commutative 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{-\color{blue}{\left(x + 4\right)}}{-y}} \]

      17. frac-2neg 52.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{x + 4}{y}}} \]

      18. sqrt-unprod 31.7%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y}} \cdot \sqrt{\frac{x + 4}{y}}} \]

      19. add-sqr-sqrt 32.0%

          \[\leadsto \color{blue}{\frac{x + 4}{y}} \]

      20. clear-num 32.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 32.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 32.0%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 40.2% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ 4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return 4.0 / 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 = 4.0d0 / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return 4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return 4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = 4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 89.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 47.6%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   2. fabs-sqr 47.6%

      \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   3. add-sqr-sqrt 48.5%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   4. associate-*l/ 47.9%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   5. associate-*r/ 49.4%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

4. Applied egg-rr 49.4%

   \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
5. Step-by-step derivation

   1. associate-*r/ 47.9%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   2. add-sqr-sqrt 21.4%

      \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}{y} \]

   3. sqrt-unprod 38.5%

      \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\sqrt{z \cdot z}}}{y} \]

   4. sqr-neg 38.5%

      \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}{y} \]

   5. sqrt-unprod 22.8%

      \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)}}{y} \]

   6. add-sqr-sqrt 45.7%

      \[\leadsto \frac{x + 4}{y} - \frac{x \cdot \color{blue}{\left(-z\right)}}{y} \]

   7. associate-/l* 46.5%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{-z}{y}} \]

   8. distribute-frac-neg 46.5%

      \[\leadsto \frac{x + 4}{y} - x \cdot \color{blue}{\left(-\frac{z}{y}\right)} \]

   9. distribute-rgt-neg-in 46.5%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x \cdot \frac{z}{y}\right)} \]

   10. distribute-lft-neg-in 46.5%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x\right) \cdot \frac{z}{y}} \]

   11. clear-num 46.5%

       \[\leadsto \frac{x + 4}{y} - \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}} \]

   12. un-div-inv 46.5%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

6. Applied egg-rr 46.5%

   \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

7. Taylor expanded in x around 0 23.6%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))