fabs fraction 1

Percentage Accurate: 91.4% → 99.9%

Time: 7.6s

Alternatives: 9

Speedup: 1.2×

• 20.9% 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: 91.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{x + 4}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e+21)
   (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+21) {
		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+21)
		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+21], 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 < 2e21

   1. Initial program 91.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. sub-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 98.4

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

   4. Applied rewrites 98.4%

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

   if 2e21 < y

   1. Initial program 98.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+193}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(-z\right)\right|\\ \end{array} \end{array} \]

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 1.1e-6)
       (fabs (/ (+ x 4.0) y_m))
       (if (<= z 3.8e+193) t_0 (fabs (* (/ x y_m) (- z))))))))

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 <= 1.1e-6) {
		tmp = fabs(((x + 4.0) / y_m));
	} else if (z <= 3.8e+193) {
		tmp = t_0;
	} else {
		tmp = fabs(((x / y_m) * -z));
	}
	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 <= 1.1e-6)
		tmp = abs(Float64(Float64(x + 4.0) / y_m));
	elseif (z <= 3.8e+193)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(-z)));
	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, 1.1e-6], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.8e+193], t$95$0, N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * (-z)), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.1000000000000001e-6 < z < 3.79999999999999972e193

   1. Initial program 94.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. sub-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 95.6

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

   4. Applied rewrites 95.6%

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

   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. Applied rewrites 95.6%

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

   if -1 < z < 1.1000000000000001e-6

   1. Initial program 94.2%

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

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

      13. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 3.79999999999999972e193 < z

   1. Initial program 83.3%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 96.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+193}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m} - z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z 6e+52)
   (fabs (/ (fma x z (- -4.0 x)) y_m))
   (fabs (- (/ 4.0 y_m) (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= 6e+52) {
		tmp = fabs((fma(x, z, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(((4.0 / y_m) - (z * (x / y_m))));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= 6e+52)
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(4.0 / y_m) - Float64(z * Float64(x / y_m))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, 6e+52], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6e52

   1. Initial program 94.5%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. sub-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

   if 6e52 < z

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 1.1e-6) (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 <= 1.1e-6) {
		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 <= 1.1e-6)
		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, 1.1e-6], 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 1.1000000000000001e-6 < z

   1. Initial program 91.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. sub-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 91.8

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

   4. Applied rewrites 91.8%

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

   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. Applied rewrites 91.8%

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

   if -1 < z < 1.1000000000000001e-6

   1. Initial program 94.2%

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

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

      13. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* x z) y_m))))
   (if (<= z -1.65e+108)
     t_0
     (if (<= z 6.6e+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(((x * z) / y_m));
	double tmp;
	if (z <= -1.65e+108) {
		tmp = t_0;
	} else if (z <= 6.6e+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(((x * z) / y_m))
    if (z <= (-1.65d+108)) then
        tmp = t_0
    else if (z <= 6.6d+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(((x * z) / y_m));
	double tmp;
	if (z <= -1.65e+108) {
		tmp = t_0;
	} else if (z <= 6.6e+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(((x * z) / y_m))
	tmp = 0
	if z <= -1.65e+108:
		tmp = t_0
	elif z <= 6.6e+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(Float64(x * z) / y_m))
	tmp = 0.0
	if (z <= -1.65e+108)
		tmp = t_0;
	elseif (z <= 6.6e+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(((x * z) / y_m));
	tmp = 0.0;
	if (z <= -1.65e+108)
		tmp = t_0;
	elseif (z <= 6.6e+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[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.65e+108], t$95$0, If[LessEqual[z, 6.6e+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 < -1.6500000000000001e108 or 6.6000000000000001e45 < 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. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. sub-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -1.6500000000000001e108 < z < 6.6000000000000001e45

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-frac-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+193}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(-z\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z 3.8e+193)
   (fabs (/ (fma x z (- -4.0 x)) y_m))
   (fabs (* (/ x y_m) (- z)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= 3.8e+193) {
		tmp = fabs((fma(x, z, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(((x / y_m) * -z));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= 3.8e+193)
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(-z)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, 3.8e+193], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * (-z)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.79999999999999972e193

   1. Initial program 94.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. sub-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 98.3

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

   4. Applied rewrites 98.3%

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

   if 3.79999999999999972e193 < z

   1. Initial program 83.3%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 96.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+193}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{-x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- x) y_m))))
   (if (<= x -1.55) 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 <= -1.55) {
		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 <= (-1.55d0)) 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 <= -1.55) {
		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 <= -1.55:
		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 <= -1.55)
		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 <= -1.55)
		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, -1.55], 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 < -1.55000000000000004 or 4 < x

   1. Initial program 89.0%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. sub-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 93.1

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

   4. Applied rewrites 93.1%

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

   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 64.9

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

   7. Applied rewrites 64.9%

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

   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 -1.55000000000000004 < x < 4

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 8: 70.2% 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 93.0%

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

3. Taylor expanded in z around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. associate-*r/ N/A

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

   8. neg-mul-1 N/A

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

   9. mul-1-neg N/A

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

   10. distribute-frac-neg N/A

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

   11. remove-double-neg N/A

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

   12. lower-/.f64 N/A

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

   13. lower-+.f64 69.5

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

5. Applied rewrites 69.5%

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

6. Final simplification 69.5%

   \[\leadsto \left|\frac{x + 4}{y}\right| \]
7. 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 93.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 36.9

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

5. Applied rewrites 36.9%

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

Reproduce

herbie shell --seed 2024231 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))