fabs fraction 1

Percentage Accurate: 92.4% → 99.9%

Time: 8.2s

Alternatives: 11

Speedup: 1.2×

• 20.8% 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: 92.4% 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

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-14}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m} - \frac{x + 4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1e-14)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (- (* 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 <= 1e-14) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(((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 <= 1e-14)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(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, 1e-14], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999999e-15

   1. Initial program 89.0%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 96.9%

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

   if 9.99999999999999999e-15 < y

   1. Initial program 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-14}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y} - \frac{x + 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\left(z - 1\right) \cdot \frac{x}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 -5e-49)
     (* (- z 1.0) (/ x y_m))
     (if (<= t_0 5e+260)
       (fabs (/ (+ 4.0 x) y_m))
       (fabs (* (- 1.0 z) (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= -5e-49) {
		tmp = (z - 1.0) * (x / y_m);
	} else if (t_0 <= 5e+260) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y_m) - ((x / y_m) * z)
    if (t_0 <= (-5d-49)) then
        tmp = (z - 1.0d0) * (x / y_m)
    else if (t_0 <= 5d+260) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= -5e-49) {
		tmp = (z - 1.0) * (x / y_m);
	} else if (t_0 <= 5e+260) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z)
	tmp = 0
	if t_0 <= -5e-49:
		tmp = (z - 1.0) * (x / y_m)
	elif t_0 <= 5e+260:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= -5e-49)
		tmp = Float64(Float64(z - 1.0) * Float64(x / y_m));
	elseif (t_0 <= 5e+260)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	tmp = 0.0;
	if (t_0 <= -5e-49)
		tmp = (z - 1.0) * (x / y_m);
	elseif (t_0 <= 5e+260)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = abs(((1.0 - z) * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-49], N[(N[(z - 1.0), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+260], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -4.9999999999999999e-49

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.1%

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

   if -4.9999999999999999e-49 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.9999999999999996e260

   1. Initial program 95.3%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if 4.9999999999999996e260 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 60.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\left(z - 1\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\left(z - 1\right) \cdot \frac{x}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 -5e-49)
     (* (- z 1.0) (/ x y_m))
     (if (<= t_0 5e+297) (fabs (/ (+ 4.0 x) y_m)) (fabs (* z (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= -5e-49) {
		tmp = (z - 1.0) * (x / y_m);
	} else if (t_0 <= 5e+297) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y_m) - ((x / y_m) * z)
    if (t_0 <= (-5d-49)) then
        tmp = (z - 1.0d0) * (x / y_m)
    else if (t_0 <= 5d+297) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= -5e-49) {
		tmp = (z - 1.0) * (x / y_m);
	} else if (t_0 <= 5e+297) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z)
	tmp = 0
	if t_0 <= -5e-49:
		tmp = (z - 1.0) * (x / y_m)
	elif t_0 <= 5e+297:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= -5e-49)
		tmp = Float64(Float64(z - 1.0) * Float64(x / y_m));
	elseif (t_0 <= 5e+297)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	tmp = 0.0;
	if (t_0 <= -5e-49)
		tmp = (z - 1.0) * (x / y_m);
	elseif (t_0 <= 5e+297)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-49], N[(N[(z - 1.0), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+297], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -4.9999999999999999e-49

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.1%

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

   if -4.9999999999999999e-49 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.9999999999999998e297

   1. Initial program 95.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 75.5

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

   7. Applied rewrites 75.5%

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

   if 4.9999999999999998e297 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 52.7%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\left(z - 1\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\frac{z \cdot x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= (- (/ (+ x 4.0) y_m) (* (/ x y_m) z)) -2e+145)
   (/ (* z x) y_m)
   (fabs (/ (+ 4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((((x + 4.0) / y_m) - ((x / y_m) * z)) <= -2e+145) {
		tmp = (z * x) / y_m;
	} else {
		tmp = fabs(((4.0 + x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((x + 4.0d0) / y_m) - ((x / y_m) * z)) <= (-2d+145)) then
        tmp = (z * x) / y_m
    else
        tmp = abs(((4.0d0 + x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((((x + 4.0) / y_m) - ((x / y_m) * z)) <= -2e+145) {
		tmp = (z * x) / y_m;
	} else {
		tmp = Math.abs(((4.0 + x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (((x + 4.0) / y_m) - ((x / y_m) * z)) <= -2e+145:
		tmp = (z * x) / y_m
	else:
		tmp = math.fabs(((4.0 + x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z)) <= -2e+145)
		tmp = Float64(Float64(z * x) / y_m);
	else
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((((x + 4.0) / y_m) - ((x / y_m) * z)) <= -2e+145)
		tmp = (z * x) / y_m;
	else
		tmp = abs(((4.0 + x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], -2e+145], N[(N[(z * x), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -2e145

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 59.0%

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

   if -2e145 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 88.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 5: 86.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+37} \lor \neg \left(z \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.55e+37) (not (<= z 7.5e+76)))
   (fabs (* z (/ x y_m)))
   (fabs (/ (+ 4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.55e+37) || !(z <= 7.5e+76)) {
		tmp = fabs((z * (x / y_m)));
	} else {
		tmp = fabs(((4.0 + x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.55d+37)) .or. (.not. (z <= 7.5d+76))) then
        tmp = abs((z * (x / y_m)))
    else
        tmp = abs(((4.0d0 + x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.55e+37) || !(z <= 7.5e+76)) {
		tmp = Math.abs((z * (x / y_m)));
	} else {
		tmp = Math.abs(((4.0 + x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.55e+37) or not (z <= 7.5e+76):
		tmp = math.fabs((z * (x / y_m)))
	else:
		tmp = math.fabs(((4.0 + x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.55e+37) || !(z <= 7.5e+76))
		tmp = abs(Float64(z * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.55e+37) || ~((z <= 7.5e+76)))
		tmp = abs((z * (x / y_m)));
	else
		tmp = abs(((4.0 + x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -1.55e+37], N[Not[LessEqual[z, 7.5e+76]], $MachinePrecision]], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5500000000000001e37 or 7.4999999999999995e76 < z

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.4%

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

   if -1.5500000000000001e37 < z < 7.4999999999999995e76

   1. Initial program 91.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 97.7

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

   7. Applied rewrites 97.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+37} \lor \neg \left(z \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+37} \lor \neg \left(z \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.55e+37) (not (<= z 7.5e+76)))
   (fabs (* x (/ z y_m)))
   (fabs (/ (+ 4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.55e+37) || !(z <= 7.5e+76)) {
		tmp = fabs((x * (z / y_m)));
	} else {
		tmp = fabs(((4.0 + x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.55d+37)) .or. (.not. (z <= 7.5d+76))) then
        tmp = abs((x * (z / y_m)))
    else
        tmp = abs(((4.0d0 + x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.55e+37) || !(z <= 7.5e+76)) {
		tmp = Math.abs((x * (z / y_m)));
	} else {
		tmp = Math.abs(((4.0 + x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.55e+37) or not (z <= 7.5e+76):
		tmp = math.fabs((x * (z / y_m)))
	else:
		tmp = math.fabs(((4.0 + x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.55e+37) || !(z <= 7.5e+76))
		tmp = abs(Float64(x * Float64(z / y_m)));
	else
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.55e+37) || ~((z <= 7.5e+76)))
		tmp = abs((x * (z / y_m)));
	else
		tmp = abs(((4.0 + x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -1.55e+37], N[Not[LessEqual[z, 7.5e+76]], $MachinePrecision]], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5500000000000001e37 or 7.4999999999999995e76 < z

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.4%

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

   10. Applied rewrites 70.4%

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

   if -1.5500000000000001e37 < z < 7.4999999999999995e76

   1. Initial program 91.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 97.7

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

   7. Applied rewrites 97.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+37} \lor \neg \left(z \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+127}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -7e+127)
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (fma z x (- -4.0 x)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -7e+127) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -7e+127)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -7e+127], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999956e127

   1. Initial program 89.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -6.99999999999999956e127 < x

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+127}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y_m)) (/ 4.0 y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = 4.0 / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = 4.0d0 / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = 4.0 / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = 4.0 / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = Float64(4.0 / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -10.5) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = 4.0 / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[(4.0 / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 65.0%

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

   if -10.5 < x < 4

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

         \[\leadsto \color{blue}{\sqrt{\frac{4}{y} \cdot \frac{4}{y}}} \]

      3. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{4}{y}} \cdot \sqrt{\frac{4}{y}}} \]

      4. rem-square-sqrt 37.5

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

   7. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{4}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.6%

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

   10. Applied rewrites 23.6%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 31.4%

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

   if -4 < x < 4

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

         \[\leadsto \color{blue}{\sqrt{\frac{4}{y} \cdot \frac{4}{y}}} \]

      3. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{4}{y}} \cdot \sqrt{\frac{4}{y}}} \]

      4. rem-square-sqrt 37.5

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

   7. Applied rewrites 37.5%

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

   if 4 < x

   1. Initial program 85.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 64.3%

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

4. Final simplification 43.1%

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

Alternative 10: 54.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 93.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 49.7

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

   5. Applied rewrites 49.7%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

         \[\leadsto \color{blue}{\sqrt{\frac{4}{y} \cdot \frac{4}{y}}} \]

      3. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{4}{y}} \cdot \sqrt{\frac{4}{y}}} \]

      4. rem-square-sqrt 26.6

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

   7. Applied rewrites 26.6%

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

   if 4 < x

   1. Initial program 85.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.4%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 34.8

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

   9. Applied rewrites 34.8%

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

   10. Taylor expanded in z around 0

       \[\leadsto \frac{x}{\color{blue}{y}} \]
   11. Step-by-step derivation

   12. Applied rewrites 23.4%

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

Alternative 11: 18.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x / y_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. div-sub N/A

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

   3. unsub-neg N/A

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

   4. mul-1-neg N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 63.9

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

5. Applied rewrites 63.9%

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

7. Applied rewrites 63.8%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 29.7

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

9. Applied rewrites 28.7%

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

10. Taylor expanded in z around 0

    \[\leadsto \frac{x}{\color{blue}{y}} \]
11. Step-by-step derivation

12. Applied rewrites 16.4%

    \[\leadsto \frac{x}{\color{blue}{y}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))