fabs fraction 1

Percentage Accurate: 91.8% → 99.8%

Time: 7.5s

Alternatives: 9

Speedup: 1.6×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% 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, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.8 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.8e-40)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.8e-40) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.8e-40)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.8e-40], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8e-40

   1. Initial program 89.5%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 96.5%

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

   if 1.8e-40 < y

   1. Initial program 97.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. sub-neg N/A

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

      10. sub-div N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 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

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

      16. sub-neg N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-/l* N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lift-/.f64 N/A

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

      24. frac-2neg N/A

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

   6. Applied rewrites 99.9%

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

Alternative 2: 94.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(z, x, -x\right)}{y\_m}\right|\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (fma z x (- x)) y_m))))
   (if (<= x -5.5e+15) t_0 (if (<= x 4.0) (fabs (/ (fma z 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(z, x, -x) / y_m));
	double tmp;
	if (x <= -5.5e+15) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((fma(z, 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(z, x, Float64(-x)) / y_m))
	tmp = 0.0
	if (x <= -5.5e+15)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(fma(z, 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[(z * x + (-x)), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5.5e+15], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5e15 or 4 < x

   1. Initial program 85.3%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 91.5%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if -5.5e15 < x < 4

   1. Initial program 97.3%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.4%

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

Alternative 3: 95.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (fma z x -4.0) y_m))))
   (if (<= z -1.0) t_0 (if (<= z 2.5) (fabs (/ (- -4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((fma(z, x, -4.0) / y_m));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.5) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(fma(z, x, -4.0) / y_m))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.5)
		tmp = abs(Float64(Float64(-4.0 - x) / 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[(z * x + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 2.5], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 2.5 < z

   1. Initial program 85.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 92.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 92.1%

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

   if -1 < z < 2.5

   1. Initial program 98.4%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 98.5

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

   7. Applied rewrites 98.5%

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

Alternative 4: 86.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.42e+32)
   (fabs (* (/ x y_m) z))
   (if (<= z 3.3e+34) (fabs (/ (- -4.0 x) y_m)) (fabs (* (/ z y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.42e+32) {
		tmp = fabs(((x / y_m) * z));
	} else if (z <= 3.3e+34) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((z / y_m) * x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.42d+32)) then
        tmp = abs(((x / y_m) * z))
    else if (z <= 3.3d+34) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((z / y_m) * x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.42e+32) {
		tmp = Math.abs(((x / y_m) * z));
	} else if (z <= 3.3e+34) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((z / y_m) * x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.42e+32:
		tmp = math.fabs(((x / y_m) * z))
	elif z <= 3.3e+34:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((z / y_m) * x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.42e+32)
		tmp = abs(Float64(Float64(x / y_m) * z));
	elseif (z <= 3.3e+34)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(z / y_m) * x));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.42e+32)
		tmp = abs(((x / y_m) * z));
	elseif (z <= 3.3e+34)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((z / y_m) * x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.42e+32], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.3e+34], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.41999999999999992e32

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 85.3%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -1.41999999999999992e32 < z < 3.29999999999999988e34

   1. Initial program 98.4%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   if 3.29999999999999988e34 < z

   1. Initial program 73.5%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 96.1%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.4

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 79.1%

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

Alternative 5: 86.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{if}\;z \leq -1.42 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) z))))
   (if (<= z -1.42e+32) t_0 (if (<= z 4e+34) (fabs (/ (- -4.0 x) 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) * z));
	double tmp;
	if (z <= -1.42e+32) {
		tmp = t_0;
	} else if (z <= 4e+34) {
		tmp = fabs(((-4.0 - x) / 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) * z))
    if (z <= (-1.42d+32)) then
        tmp = t_0
    else if (z <= 4d+34) then
        tmp = abs((((-4.0d0) - x) / 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) * z));
	double tmp;
	if (z <= -1.42e+32) {
		tmp = t_0;
	} else if (z <= 4e+34) {
		tmp = Math.abs(((-4.0 - x) / 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) * z))
	tmp = 0
	if z <= -1.42e+32:
		tmp = t_0
	elif z <= 4e+34:
		tmp = math.fabs(((-4.0 - x) / 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 / y_m) * z))
	tmp = 0.0
	if (z <= -1.42e+32)
		tmp = t_0;
	elseif (z <= 4e+34)
		tmp = abs(Float64(Float64(-4.0 - x) / 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) * z));
	tmp = 0.0;
	if (z <= -1.42e+32)
		tmp = t_0;
	elseif (z <= 4e+34)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.42e+32], t$95$0, If[LessEqual[z, 4e+34], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.41999999999999992e32 or 3.99999999999999978e34 < z

   1. Initial program 84.4%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 91.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.4

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

   7. Applied rewrites 80.4%

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

   if -1.41999999999999992e32 < z < 3.99999999999999978e34

   1. Initial program 98.4%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

Alternative 6: 68.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{-x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{-4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- x) y_m))))
   (if (<= x -1.6) 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 <= -1.6) {
		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 <= (-1.6d0)) 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 <= -1.6) {
		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 <= -1.6:
		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(Float64(-x) / y_m))
	tmp = 0.0
	if (x <= -1.6)
		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 <= -1.6)
		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, -1.6], 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 < -1.6000000000000001 or 4 < x

   1. Initial program 85.5%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 91.7%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 63.4%

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

   if -1.6000000000000001 < x < 4

   1. Initial program 97.3%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. /-rgt-identity N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

      19. distribute-neg-out N/A

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

      20. remove-double-neg N/A

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

   3. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.2%

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

Alternative 7: 96.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. lift-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. div-sub N/A

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

   10. sub-neg N/A

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

   11. remove-double-neg N/A

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

   12. distribute-neg-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. sub-neg N/A

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

   16. /-rgt-identity N/A

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

   17. metadata-eval N/A

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

   18. distribute-neg-frac2 N/A

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

   19. distribute-neg-out N/A

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

   20. remove-double-neg N/A

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

3. Applied rewrites 95.9%

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

Alternative 8: 69.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. lift-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. div-sub N/A

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

   10. sub-neg N/A

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

   11. remove-double-neg N/A

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

   12. distribute-neg-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. sub-neg N/A

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

   16. /-rgt-identity N/A

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

   17. metadata-eval N/A

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

   18. distribute-neg-frac2 N/A

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

   19. distribute-neg-out N/A

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

   20. remove-double-neg N/A

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

3. Applied rewrites 95.9%

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

5. Taylor expanded in z around 0

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

   1. distribute-lft-in N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 68.9

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

7. Applied rewrites 68.9%

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

Alternative 9: 40.4% 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 91.5%

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

   1. lift-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. div-sub N/A

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

   10. sub-neg N/A

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

   11. remove-double-neg N/A

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

   12. distribute-neg-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. sub-neg N/A

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

   16. /-rgt-identity N/A

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

   17. metadata-eval N/A

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

   18. distribute-neg-frac2 N/A

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

   19. distribute-neg-out N/A

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

   20. remove-double-neg N/A

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

3. Applied rewrites 95.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 40.0%

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

Reproduce

herbie shell --seed 2024272 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))