fabs fraction 1

Percentage Accurate: 92.0% → 95.7%

Time: 7.8s

Alternatives: 6

Speedup: 1.6×

• 20.1% 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: 92.0% 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: 95.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.6%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.00035:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (fma x z -4.0) y))))
   (if (<= z -4.1e+40) t_0 (if (<= z 0.00035) (fabs (/ (+ x 4.0) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((fma(x, z, -4.0) / y));
	double tmp;
	if (z <= -4.1e+40) {
		tmp = t_0;
	} else if (z <= 0.00035) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = abs(Float64(fma(x, z, -4.0) / y))
	tmp = 0.0
	if (z <= -4.1e+40)
		tmp = t_0;
	elseif (z <= 0.00035)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -4.1e+40], t$95$0, If[LessEqual[z, 0.00035], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1000000000000002e40 or 3.49999999999999996e-4 < z

   1. Initial program 84.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.6

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

   4. Applied rewrites 96.6%

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

   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 95.8%

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

   if -4.1000000000000002e40 < z < 3.49999999999999996e-4

   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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 0.00035:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x \cdot z}{y}\right|\\ \mathbf{if}\;z \leq -8 \cdot 10^{+108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+83}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* x z) y))))
   (if (<= z -8e+108) t_0 (if (<= z 2.6e+83) (fabs (/ (+ x 4.0) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x * z) / y));
	double tmp;
	if (z <= -8e+108) {
		tmp = t_0;
	} else if (z <= 2.6e+83) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x * z) / y))
    if (z <= (-8d+108)) then
        tmp = t_0
    else if (z <= 2.6d+83) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((x * z) / y));
	double tmp;
	if (z <= -8e+108) {
		tmp = t_0;
	} else if (z <= 2.6e+83) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x * z) / y))
	tmp = 0
	if z <= -8e+108:
		tmp = t_0
	elif z <= 2.6e+83:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x * z) / y))
	tmp = 0.0
	if (z <= -8e+108)
		tmp = t_0;
	elseif (z <= 2.6e+83)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x * z) / y));
	tmp = 0.0;
	if (z <= -8e+108)
		tmp = t_0;
	elseif (z <= 2.6e+83)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -8e+108], t$95$0, If[LessEqual[z, 2.6e+83], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.0000000000000003e108 or 2.6000000000000001e83 < z

   1. Initial program 85.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

   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 71.0

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

   7. Applied rewrites 71.0%

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

   if -8.0000000000000003e108 < z < 2.6000000000000001e83

   1. Initial program 91.8%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-frac-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+83}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- x) y))))
   (if (<= x -1.5) t_0 (if (<= x 4.0) (fabs (/ 4.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((-x / y));
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((-x / y))
    if (x <= (-1.5d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((-x / y));
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((-x / y))
	tmp = 0
	if x <= -1.5:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(-x) / y))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((-x / y));
	tmp = 0.0;
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[((-x) / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 4 < x

   1. Initial program 81.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.6

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

   4. Applied rewrites 96.6%

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

   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 69.0

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

   7. Applied rewrites 69.0%

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

   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 67.6%

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

   if -1.5 < x < 4

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 80.4

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

   5. Applied rewrites 80.4%

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

Alternative 5: 70.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in z around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. associate-*r/ N/A

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

   8. neg-mul-1 N/A

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

   9. mul-1-neg N/A

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

   10. distribute-frac-neg N/A

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

   11. remove-double-neg N/A

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

   12. lower-/.f64 N/A

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

   13. lower-+.f64 75.7

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

5. Applied rewrites 75.7%

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

6. Final simplification 75.7%

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

Alternative 6: 40.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 89.6%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 47.0

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

5. Applied rewrites 47.0%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))