fabs fraction 1

Percentage Accurate: 91.9% → 99.7%

Time: 8.2s

Alternatives: 9

Speedup: 1.6×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.8 \cdot 10^{+64}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{x + 4}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4.8e+64)
   (fabs (/ (fma x z (- -4.0 x)) y_m))
   (fabs (fma (- 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 (y_m <= 4.8e+64) {
		tmp = fabs((fma(x, z, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(fma(-x, (z / y_m), ((x + 4.0) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4.8e+64)
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(fma(Float64(-x), Float64(z / y_m), Float64(Float64(x + 4.0) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 4.8e+64], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-x) * N[(z / y$95$m), $MachinePrecision] + N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.79999999999999999e64

   1. Initial program 89.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. neg-fabs N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. sub-div N/A

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

      19. lower-/.f64 N/A

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

   4. Applied egg-rr 97.5%

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

   if 4.79999999999999999e64 < y

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* z (/ x y_m)))))
   (if (<= t_0 -1e+66) (fabs t_0) (fabs (/ (fma x z (- -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) - (z * (x / y_m));
	double tmp;
	if (t_0 <= -1e+66) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((fma(x, z, (-4.0 - x)) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_0 <= -1e+66)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+66], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -9.99999999999999945e65

   1. Initial program 99.9%

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

   if -9.99999999999999945e65 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. neg-fabs N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. sub-div N/A

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

      19. lower-/.f64 N/A

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

   4. Applied egg-rr 97.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;\frac{x + 4}{y\_m} - t\_0 \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x}{y\_m} - t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (* z (/ x y_m))))
   (if (<= (- (/ (+ x 4.0) y_m) t_0) -1e+66)
     (fabs (- (/ x y_m) t_0))
     (fabs (/ (fma x z (- -4.0 x)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = z * (x / y_m);
	double tmp;
	if ((((x + 4.0) / y_m) - t_0) <= -1e+66) {
		tmp = fabs(((x / y_m) - t_0));
	} else {
		tmp = fabs((fma(x, z, (-4.0 - x)) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(z * Float64(x / y_m))
	tmp = 0.0
	if (Float64(Float64(Float64(x + 4.0) / y_m) - t_0) <= -1e+66)
		tmp = abs(Float64(Float64(x / y_m) - t_0));
	else
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / y_m));
	end
	return 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]}, If[LessEqual[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - t$95$0), $MachinePrecision], -1e+66], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -9.99999999999999945e65

   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 inf

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

      1. lower-/.f64 73.5

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

   5. Simplified 73.5%

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

   if -9.99999999999999945e65 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. neg-fabs N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. sub-div N/A

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

      19. lower-/.f64 N/A

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

   4. Applied egg-rr 97.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (fma x z -4.0) y_m))))
   (if (<= z -4.8e+14) t_0 (if (<= z 3.6e-11) (fabs (/ (+ x 4.0) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((fma(x, z, -4.0) / y_m));
	double tmp;
	if (z <= -4.8e+14) {
		tmp = t_0;
	} else if (z <= 3.6e-11) {
		tmp = fabs(((x + 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(fma(x, z, -4.0) / y_m))
	tmp = 0.0
	if (z <= -4.8e+14)
		tmp = t_0;
	elseif (z <= 3.6e-11)
		tmp = abs(Float64(Float64(x + 4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -4.8e+14], t$95$0, If[LessEqual[z, 3.6e-11], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e14 or 3.59999999999999985e-11 < z

   1. Initial program 86.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. neg-fabs N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. sub-div N/A

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

      19. lower-/.f64 N/A

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

   4. Applied egg-rr 92.1%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 92.1%

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

   if -4.8e14 < z < 3.59999999999999985e-11

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-frac-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\left|x \cdot z\right|}{y\_m}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (fabs (* x z)) y_m)))
   (if (<= z -3.4e+124)
     t_0
     (if (<= z 1.22e+70) (fabs (/ (+ x 4.0) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x * z)) / y_m;
	double tmp;
	if (z <= -3.4e+124) {
		tmp = t_0;
	} else if (z <= 1.22e+70) {
		tmp = fabs(((x + 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x * z)) / y_m
    if (z <= (-3.4d+124)) then
        tmp = t_0
    else if (z <= 1.22d+70) then
        tmp = abs(((x + 4.0d0) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x * z)) / y_m;
	double tmp;
	if (z <= -3.4e+124) {
		tmp = t_0;
	} else if (z <= 1.22e+70) {
		tmp = Math.abs(((x + 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x * z)) / y_m
	tmp = 0
	if z <= -3.4e+124:
		tmp = t_0
	elif z <= 1.22e+70:
		tmp = math.fabs(((x + 4.0) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(abs(Float64(x * z)) / y_m)
	tmp = 0.0
	if (z <= -3.4e+124)
		tmp = t_0;
	elseif (z <= 1.22e+70)
		tmp = abs(Float64(Float64(x + 4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x * z)) / y_m;
	tmp = 0.0;
	if (z <= -3.4e+124)
		tmp = t_0;
	elseif (z <= 1.22e+70)
		tmp = abs(((x + 4.0) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Abs[N[(x * z), $MachinePrecision]], $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[z, -3.4e+124], t$95$0, If[LessEqual[z, 1.22e+70], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4e124 or 1.22e70 < z

   1. Initial program 85.1%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 75.3

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

   5. Simplified 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. div-fabs N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. /-rgt-identity N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. fabs-div N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. inv-pow N/A

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

      13. sqr-pow N/A

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

      14. fabs-sqr N/A

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

      15. sqr-pow N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

      18. /-rgt-identity N/A

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

      19. lower-/.f64 N/A

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

      20. lower-fabs.f64 45.8

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

   7. Applied egg-rr 45.8%

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

   if -3.4e124 < z < 1.22e70

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-frac-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 92.1

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

   5. Simplified 92.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left|x \cdot z\right|}{y}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot z\right|}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{y\_m}\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (fabs x) y_m)))
   (if (<= x -10.5) t_0 (if (<= x 4.0) (/ 4.0 y_m) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(x) / y_m;
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(x) / y_m
    if (x <= (-10.5d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(x) / y_m;
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(x) / y_m
	tmp = 0
	if x <= -10.5:
		tmp = t_0
	elif x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(abs(x) / y_m)
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(x) / y_m;
	tmp = 0.0;
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$0, If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 85.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. neg-fabs N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. sub-div N/A

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

      19. lower-/.f64 N/A

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.6

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

   7. Simplified 90.6%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left|\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{y}\right| \]

      2. lower-neg.f64 63.6

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

   10. Simplified 63.6%

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

       1. distribute-frac-neg N/A

          \[\leadsto \left|\color{blue}{\mathsf{neg}\left(\frac{x}{y}\right)}\right| \]

       2. fabs-neg N/A

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

       3. div-inv N/A

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

       4. lift-/.f64 N/A

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

       5. mul-fabs N/A

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

       6. lift-fabs.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. inv-pow N/A

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

       9. sqr-pow N/A

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

       10. fabs-sqr N/A

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

       11. sqr-pow N/A

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

       12. inv-pow N/A

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

       13. div-inv N/A

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

       14. lower-/.f64 32.1

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

   12. Applied egg-rr 32.1%

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

   if -10.5 < x < 4

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 74.6

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

   5. Simplified 74.6%

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

      1. div-inv N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-mul N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. sqr-pow N/A

         \[\leadsto 4 \cdot \left|\color{blue}{{y}^{\left(\frac{-1}{2}\right)} \cdot {y}^{\left(\frac{-1}{2}\right)}}\right| \]

      8. fabs-sqr N/A

         \[\leadsto 4 \cdot \color{blue}{\left({y}^{\left(\frac{-1}{2}\right)} \cdot {y}^{\left(\frac{-1}{2}\right)}\right)} \]

      9. sqr-pow N/A

         \[\leadsto 4 \cdot \color{blue}{{y}^{-1}} \]

      10. inv-pow N/A

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

      11. div-inv N/A

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

      12. lift-/.f64 40.2

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

   7. Applied egg-rr 40.2%

      \[\leadsto \color{blue}{\frac{4}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. neg-fabs N/A

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

   7. lower-fabs.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. remove-double-neg N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. lift-/.f64 N/A

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

   18. sub-div N/A

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

   19. lower-/.f64 N/A

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

4. Applied egg-rr 95.8%

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

Alternative 8: 69.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs(((x + 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 = abs(((x + 4.0d0) / y_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in z around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. associate-*r/ N/A

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

   8. neg-mul-1 N/A

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

   9. mul-1-neg N/A

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

   10. distribute-frac-neg N/A

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

   11. remove-double-neg N/A

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

   12. lower-/.f64 N/A

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

   13. lower-+.f64 70.2

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

5. Simplified 70.2%

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

6. Final simplification 70.2%

   \[\leadsto \left|\frac{x + 4}{y}\right| \]
7. Add Preprocessing

Alternative 9: 40.3% accurate, 3.0× 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 90.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 40.0

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

5. Simplified 40.0%

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

   1. div-inv N/A

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

   2. lift-/.f64 N/A

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

   3. fabs-mul N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. inv-pow N/A

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

   7. sqr-pow N/A

      \[\leadsto 4 \cdot \left|\color{blue}{{y}^{\left(\frac{-1}{2}\right)} \cdot {y}^{\left(\frac{-1}{2}\right)}}\right| \]

   8. fabs-sqr N/A

      \[\leadsto 4 \cdot \color{blue}{\left({y}^{\left(\frac{-1}{2}\right)} \cdot {y}^{\left(\frac{-1}{2}\right)}\right)} \]

   9. sqr-pow N/A

      \[\leadsto 4 \cdot \color{blue}{{y}^{-1}} \]

   10. inv-pow N/A

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

   11. div-inv N/A

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

   12. lift-/.f64 21.9

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

7. Applied egg-rr 21.9%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))