fabs fraction 1

Percentage Accurate: 91.5% → 99.7%

Time: 8.3s

Alternatives: 12

Speedup: 1.6×

• 21.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.5% 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

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{x + 4}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e+65)
   (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 <= 2e+65) {
		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 <= 2e+65)
		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, 2e+65], 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 < 2e65

   1. Initial program 92.8%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 97.5

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

   4. Applied egg-rr 97.5%

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

   if 2e65 < y

   1. Initial program 96.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-lowering-+.f64 99.8

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

   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 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

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 -5e-23) (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 <= -5e-23) {
		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 <= -5e-23)
		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, -5e-23], 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)) < -5.0000000000000002e-23

   1. Initial program 99.9%

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

   if -5.0000000000000002e-23 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   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. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 96.5

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

   4. Applied egg-rr 96.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;\frac{x + 4}{y\_m} - t\_0 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x}{y\_m} - t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

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) -5e-23)
     (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) <= -5e-23) {
		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) <= -5e-23)
		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], -5e-23], 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)) < -5.0000000000000002e-23

   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. /-lowering-/.f64 76.3

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

   5. Simplified 76.3%

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

   if -5.0000000000000002e-23 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   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. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 96.5

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

   4. Applied egg-rr 96.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1750000000000:\\ \;\;\;\;\left|\frac{x - x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(z + -1\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1750000000000.0)
   (fabs (/ (- x (* x z)) y_m))
   (if (<= x 4.0)
     (fabs (/ (fma x z -4.0) y_m))
     (fabs (/ (* x (+ z -1.0)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1750000000000.0) {
		tmp = fabs(((x - (x * z)) / y_m));
	} else if (x <= 4.0) {
		tmp = fabs((fma(x, z, -4.0) / y_m));
	} else {
		tmp = fabs(((x * (z + -1.0)) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1750000000000.0)
		tmp = abs(Float64(Float64(x - Float64(x * z)) / y_m));
	elseif (x <= 4.0)
		tmp = abs(Float64(fma(x, z, -4.0) / y_m));
	else
		tmp = abs(Float64(Float64(x * Float64(z + -1.0)) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1750000000000.0], N[Abs[N[(N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(N[(x * z + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e12

   1. Initial program 86.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 94.2

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

   5. Simplified 94.2%

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

   if -1.75e12 < x < 4

   1. Initial program 97.1%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   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 97.7%

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

   if 4 < x

   1. Initial program 94.6%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 88.2

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

   4. Applied egg-rr 88.2%

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

   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. neg-lowering-neg.f64 87.5

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

   7. Simplified 87.5%

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 87.6

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

   9. Applied egg-rr 87.6%

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

Alternative 5: 94.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x \cdot \left(z + -1\right)}{y\_m}\right|\\ \mathbf{if}\;x \leq -1750000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* x (+ z -1.0)) y_m))))
   (if (<= x -1750000000000.0)
     t_0
     (if (<= x 4.0) (fabs (/ (fma x z -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 + -1.0)) / y_m));
	double tmp;
	if (x <= -1750000000000.0) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((fma(x, z, -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(Float64(x * Float64(z + -1.0)) / y_m))
	tmp = 0.0
	if (x <= -1750000000000.0)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(fma(x, z, -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 * N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1750000000000.0], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(N[(x * z + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e12 or 4 < x

   1. Initial program 90.0%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 91.4

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

   4. Applied egg-rr 91.4%

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

   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. neg-lowering-neg.f64 91.1

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

   7. Simplified 91.1%

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 91.1

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

   9. Applied egg-rr 91.1%

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

   if -1.75e12 < x < 4

   1. Initial program 97.1%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   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 97.7%

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

Alternative 6: 95.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.38:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1.0) t_0 (if (<= z 0.38) (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 <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.38) {
		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 <= -1.0)
		tmp = t_0;
	elseif (z <= 0.38)
		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, -1.0], t$95$0, If[LessEqual[z, 0.38], 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 < -1 or 0.38 < z

   1. Initial program 91.9%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. sub-div N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. metadata-eval 90.5

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

   4. Applied egg-rr 90.5%

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

   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 89.5%

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

   if -1 < z < 0.38

   1. Initial program 95.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 0.38:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -3.6e+40)
   (fabs (* z (/ x y_m)))
   (if (<= z 6.5e+45) (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 <= -3.6e+40) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 6.5e+45) {
		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 <= (-3.6d+40)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 6.5d+45) 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 <= -3.6e+40) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 6.5e+45) {
		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 <= -3.6e+40:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 6.5e+45:
		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 <= -3.6e+40)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 6.5e+45)
		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 <= -3.6e+40)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 6.5e+45)
		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, -3.6e+40], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 6.5e+45], 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 < -3.59999999999999996e40

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 83.3

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

   5. Simplified 83.3%

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

      1. distribute-frac-neg2 N/A

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

      2. associate-*l/ N/A

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

      3. fabs-neg N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. associate-*l/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 83.3

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

   7. Applied egg-rr 83.3%

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

      1. associate-/l* N/A

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

      2. remove-double-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\left(0 - x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      4. flip3-- N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}}\right)\right) \cdot \frac{z}{y}\right| \]

      5. metadata-eval N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      6. neg-sub0 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      7. cube-neg N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      8. sqr-pow N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      9. pow-prod-down N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      10. sqr-neg N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(\frac{{\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      11. pow-prod-down N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      12. sqr-pow N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      13. distribute-neg-frac N/A

          \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left({x}^{3}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \cdot \frac{z}{y}\right| \]

      14. neg-sub0 N/A

          \[\leadsto \left|\frac{\color{blue}{0 - {x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot \frac{z}{y}\right| \]

      15. metadata-eval N/A

          \[\leadsto \left|\frac{\color{blue}{{0}^{3}} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot \frac{z}{y}\right| \]

      16. flip3-- N/A

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

      17. neg-sub0 N/A

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

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

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

      19. associate-/l* N/A

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

      20. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 83.4%

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

   if -3.59999999999999996e40 < z < 6.50000000000000034e45

   1. Initial program 95.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 96.2

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

   5. Simplified 96.2%

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

   if 6.50000000000000034e45 < z

   1. Initial program 89.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 66.7

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

   5. Simplified 66.7%

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

      1. distribute-frac-neg2 N/A

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

      2. associate-*l/ N/A

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

      3. fabs-neg N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. associate-*l/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 66.7

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

   7. Applied egg-rr 66.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 80.5

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

   9. Applied egg-rr 80.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))))
   (if (<= z -5.4e+42) t_0 (if (<= z 1.55e+45) (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 <= -5.4e+42) {
		tmp = t_0;
	} else if (z <= 1.55e+45) {
		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 <= (-5.4d+42)) then
        tmp = t_0
    else if (z <= 1.55d+45) 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 <= -5.4e+42) {
		tmp = t_0;
	} else if (z <= 1.55e+45) {
		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 <= -5.4e+42:
		tmp = t_0
	elif z <= 1.55e+45:
		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 <= -5.4e+42)
		tmp = t_0;
	elseif (z <= 1.55e+45)
		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 <= -5.4e+42)
		tmp = t_0;
	elseif (z <= 1.55e+45)
		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, -5.4e+42], t$95$0, If[LessEqual[z, 1.55e+45], 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 < -5.4000000000000001e42 or 1.54999999999999994e45 < z

   1. Initial program 91.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 74.3

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

   5. Simplified 74.3%

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

      1. distribute-frac-neg2 N/A

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

      2. associate-*l/ N/A

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

      3. fabs-neg N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. associate-*l/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 74.3

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

   7. Applied egg-rr 74.3%

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

      1. associate-/l* N/A

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

      2. remove-double-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\left(0 - x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      4. flip3-- N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}}\right)\right) \cdot \frac{z}{y}\right| \]

      5. metadata-eval N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      6. neg-sub0 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      7. cube-neg N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      8. sqr-pow N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      9. pow-prod-down N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      10. sqr-neg N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(\frac{{\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      11. pow-prod-down N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      12. sqr-pow N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(\frac{\color{blue}{{x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{z}{y}\right| \]

      13. distribute-neg-frac N/A

          \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left({x}^{3}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \cdot \frac{z}{y}\right| \]

      14. neg-sub0 N/A

          \[\leadsto \left|\frac{\color{blue}{0 - {x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot \frac{z}{y}\right| \]

      15. metadata-eval N/A

          \[\leadsto \left|\frac{\color{blue}{{0}^{3}} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot \frac{z}{y}\right| \]

      16. flip3-- N/A

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

      17. neg-sub0 N/A

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

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

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

      19. associate-/l* N/A

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

      20. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 81.7%

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

   if -5.4000000000000001e42 < z < 1.54999999999999994e45

   1. Initial program 95.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 96.2

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

   5. Simplified 96.2%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.5) 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 <= -1.5) {
		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 <= (-1.5d0)) 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 <= -1.5) {
		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 <= -1.5:
		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 <= -1.5)
		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 <= -1.5)
		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, -1.5], 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 < -1.5 or 4 < x

   1. Initial program 90.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. /-lowering-/.f64 89.9

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

   5. Simplified 89.9%

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

   6. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 64.7

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

   8. Simplified 64.7%

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

   if -1.5 < x < 4

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 77.1

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

   5. Simplified 77.1%

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

Alternative 10: 96.2% 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 93.8%

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

   1. neg-fabs N/A

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

   2. fabs-lowering-fabs.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-neg-in N/A

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

   6. remove-double-neg N/A

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

   7. sub-neg N/A

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

   8. associate-*l/ N/A

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

   9. sub-div N/A

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

   10. /-lowering-/.f64 N/A

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

   11. sub-neg N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. unsub-neg N/A

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

   16. --lowering--.f64 N/A

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

   17. metadata-eval 95.9

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

4. Applied egg-rr 95.9%

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

Alternative 11: 69.6% 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 93.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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. /-lowering-/.f64 N/A

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

   13. +-lowering-+.f64 72.2

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

5. Simplified 72.2%

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

6. Final simplification 72.2%

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

Alternative 12: 39.4% 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 93.8%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 42.1

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

5. Simplified 42.1%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))