fabs fraction 1

Percentage Accurate: 91.7% → 99.8%

Time: 8.9s

Alternatives: 13

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: 91.7% 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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, z \cdot \frac{-1}{y\_m}, \frac{x + 4}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 7.2e-25)
   (fabs (/ (fma x z (- -4.0 x)) y_m))
   (fabs (fma x (* z (/ -1.0 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 <= 7.2e-25) {
		tmp = fabs((fma(x, z, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(fma(x, (z * (-1.0 / 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 <= 7.2e-25)
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(fma(x, Float64(z * Float64(-1.0 / 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, 7.2e-25], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.1999999999999998e-25

   1. Initial program 89.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.2%

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

   if 7.1999999999999998e-25 < y

   1. Initial program 97.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. 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. distribute-lft-neg-in N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. distribute-frac-neg2 N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{-1}{y} \cdot z, \frac{x + 4}{y}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, z \cdot \frac{-1}{y}, \frac{x + 4}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;\frac{x + 4}{y\_m} - t\_0 \leq -20000000000:\\ \;\;\;\;\left|\frac{x}{y\_m} - t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

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) -20000000000.0)
     (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) <= -20000000000.0) {
		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) <= -20000000000.0)
		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], -20000000000.0], 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)) < -2e10

   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 70.9

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

   5. Simplified 70.9%

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

   if -2e10 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 88.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 95.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -20000000000:\\ \;\;\;\;\left|\frac{x}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_0 - t\_1 \leq -1 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (* z (/ x y_m))))
   (if (<= (- t_0 t_1) -1e+207) t_1 (fabs t_0))))

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 tmp;
	if ((t_0 - t_1) <= -1e+207) {
		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 = (x + 4.0d0) / y_m
    t_1 = z * (x / y_m)
    if ((t_0 - t_1) <= (-1d+207)) 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 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= -1e+207) {
		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 = (x + 4.0) / y_m
	t_1 = z * (x / y_m)
	tmp = 0
	if (t_0 - t_1) <= -1e+207:
		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(Float64(x + 4.0) / y_m)
	t_1 = Float64(z * Float64(x / y_m))
	tmp = 0.0
	if (Float64(t_0 - t_1) <= -1e+207)
		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 = (x + 4.0) / y_m;
	t_1 = z * (x / y_m);
	tmp = 0.0;
	if ((t_0 - t_1) <= -1e+207)
		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[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - t$95$1), $MachinePrecision], -1e+207], t$95$1, N[Abs[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -1e207

   1. Initial program 100.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 58.0

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

   5. Simplified 58.0%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{-y}}\right| \]
   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. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fabs-neg N/A

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

      10. lower-fabs.f64 68.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.5

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

   7. Applied egg-rr 68.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.7

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

   9. Applied egg-rr 60.7%

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

       1. lift-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. associate-/l* N/A

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

       5. lift-*.f64 N/A

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

       6. clear-num N/A

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

       7. inv-pow N/A

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

       8. sqr-pow N/A

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

       9. fabs-sqr N/A

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

       10. sqr-pow N/A

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

       11. inv-pow N/A

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

       12. clear-num N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l/ N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 67.9

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

   11. Applied egg-rr 67.9%

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

   if -1e207 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 90.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 74.8

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

   5. Simplified 74.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -1 \cdot 10^{+207}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{x + 4}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e+56)
   (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 <= 5e+56) {
		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 <= 5e+56)
		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, 5e+56], 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 < 5.00000000000000024e56

   1. Initial program 90.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.7%

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

   if 5.00000000000000024e56 < y

   1. Initial program 96.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.8

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

   4. Applied egg-rr 99.8%

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

Alternative 5: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.8 \cdot 10^{+103}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.8e+103)
   (fabs (/ (fma x z (- -4.0 x)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.8e+103) {
		tmp = fabs((fma(x, z, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (z * (x / y_m))));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.8e+103)
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(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, 1.8e+103], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000008e103

   1. Initial program 91.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.8%

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

   if 1.80000000000000008e103 < y

   1. Initial program 95.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+103}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -140:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -140.0) t_0 (if (<= z 4.5e-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 <= -140.0) {
		tmp = t_0;
	} else if (z <= 4.5e-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 <= -140.0)
		tmp = t_0;
	elseif (z <= 4.5e-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, -140.0], t$95$0, If[LessEqual[z, 4.5e-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 < -140 or 4.5e-11 < z

   1. Initial program 89.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 87.7%

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

   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 87.3%

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

   if -140 < z < 4.5e-11

   1. Initial program 93.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -51000000000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 70000000000:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -51000000000000.0)
   (fabs (* z (/ x y_m)))
   (if (<= z 70000000000.0) (fabs (/ (+ x 4.0) y_m)) (fabs (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -51000000000000.0) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 70000000000.0) {
		tmp = fabs(((x + 4.0) / y_m));
	} else {
		tmp = fabs((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 (z <= (-51000000000000.0d0)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 70000000000.0d0) then
        tmp = abs(((x + 4.0d0) / y_m))
    else
        tmp = abs((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 (z <= -51000000000000.0) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 70000000000.0) {
		tmp = Math.abs(((x + 4.0) / y_m));
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -51000000000000.0:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 70000000000.0:
		tmp = math.fabs(((x + 4.0) / y_m))
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -51000000000000.0)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 70000000000.0)
		tmp = abs(Float64(Float64(x + 4.0) / y_m));
	else
		tmp = abs(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 (z <= -51000000000000.0)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 70000000000.0)
		tmp = abs(((x + 4.0) / y_m));
	else
		tmp = abs((x * (z / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -51000000000000.0], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 70000000000.0], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1e13

   1. Initial program 99.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 65.1

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

   5. Simplified 65.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{-y}}\right| \]
   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. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fabs-neg N/A

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

      10. lower-fabs.f64 74.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.6

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

   7. Applied egg-rr 74.6%

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

   if -5.1e13 < z < 7e10

   1. Initial program 93.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.1

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

   5. Simplified 99.1%

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

   if 7e10 < z

   1. Initial program 80.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 61.1

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

   5. Simplified 61.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{-y}}\right| \]
   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. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fabs-neg N/A

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

      10. lower-fabs.f64 69.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.8

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

   7. Applied egg-rr 69.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 72.8

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

   9. Applied egg-rr 72.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -51000000000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 70000000000:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;z \leq -51000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 70000000000:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))))
   (if (<= z -51000000000000.0)
     t_0
     (if (<= z 70000000000.0) (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((z * (x / y_m)));
	double tmp;
	if (z <= -51000000000000.0) {
		tmp = t_0;
	} else if (z <= 70000000000.0) {
		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((z * (x / y_m)))
    if (z <= (-51000000000000.0d0)) then
        tmp = t_0
    else if (z <= 70000000000.0d0) 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((z * (x / y_m)));
	double tmp;
	if (z <= -51000000000000.0) {
		tmp = t_0;
	} else if (z <= 70000000000.0) {
		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((z * (x / y_m)))
	tmp = 0
	if z <= -51000000000000.0:
		tmp = t_0
	elif z <= 70000000000.0:
		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 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (z <= -51000000000000.0)
		tmp = t_0;
	elseif (z <= 70000000000.0)
		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((z * (x / y_m)));
	tmp = 0.0;
	if (z <= -51000000000000.0)
		tmp = t_0;
	elseif (z <= 70000000000.0)
		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[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -51000000000000.0], t$95$0, If[LessEqual[z, 70000000000.0], 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 < -5.1e13 or 7e10 < z

   1. Initial program 89.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 62.8

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

   5. Simplified 62.8%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{-y}}\right| \]
   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. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fabs-neg N/A

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

      10. lower-fabs.f64 71.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.9

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

   7. Applied egg-rr 71.9%

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

   if -5.1e13 < z < 7e10

   1. Initial program 93.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.1

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

   5. Simplified 99.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -51000000000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 70000000000:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.2% accurate, 1.4× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.0)
   (/ (- -4.0 x) y_m)
   (if (<= x 4.0) (fabs (/ 4.0 y_m)) (fabs (/ 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 - x) / y_m;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((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) - x) / y_m
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((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 - x) / y_m;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((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 - x) / y_m
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((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(Float64(-4.0 - x) / y_m);
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(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 - x) / y_m;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((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[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 89.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 91.2%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 64.9

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

   7. Simplified 64.9%

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

      1. lift--.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 42.0

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

   9. Applied egg-rr 42.0%

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

   if -4 < x < 4

   1. Initial program 96.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 80.0

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

   5. Simplified 80.0%

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

   if 4 < x

   1. Initial program 83.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 60.1

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

   8. Simplified 60.1%

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

Alternative 10: 68.9% accurate, 1.4× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -10.5) t_0 (if (<= x 4.0) (fabs (/ 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 = fabs((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 = abs((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 = Math.abs((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 = math.fabs((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(x / y_m))
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(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 = abs((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[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 86.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 62.3

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

   8. Simplified 62.3%

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

   if -10.5 < x < 4

   1. Initial program 96.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 80.0

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

   5. Simplified 80.0%

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

Alternative 11: 96.3% 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 91.7%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 94.4%

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

Alternative 12: 69.9% 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 91.7%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 72.8

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

5. Simplified 72.8%

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

6. Final simplification 72.8%

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

Alternative 13: 40.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 43.1

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

5. Simplified 43.1%

   \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))