fabs fraction 1

Percentage Accurate: 92.1% → 99.8%

Time: 6.1s

Alternatives: 10

Speedup: 1.6×

• 20.6% 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: 92.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-39}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.5e-39)
   (fabs (/ (fma (- 1.0 z) x 4.0) y_m))
   (fabs (fma (- x) (/ z y_m) (/ (+ 4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.5e-39) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y_m));
	} else {
		tmp = fabs(fma(-x, (z / y_m), ((4.0 + x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.5e-39)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y_m));
	else
		tmp = abs(fma(Float64(-x), Float64(z / y_m), Float64(Float64(4.0 + x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.5e-39], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-x) * N[(z / y$95$m), $MachinePrecision] + N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000014e-39

   1. Initial program 85.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 97.9%

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

   if 1.50000000000000014e-39 < y

   1. Initial program 93.6%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -850000:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{y\_m} \cdot x\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -850000.0)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (if (<= x 4.0)
     (fabs (/ (fma (- z) x 4.0) y_m))
     (fabs (* (/ (- 1.0 z) y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -850000.0) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else if (x <= 4.0) {
		tmp = fabs((fma(-z, x, 4.0) / y_m));
	} else {
		tmp = fabs((((1.0 - z) / y_m) * x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -850000.0)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	elseif (x <= 4.0)
		tmp = abs(Float64(fma(Float64(-z), x, 4.0) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(1.0 - z) / y_m) * x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -850000.0], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(N[((-z) * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5e5

   1. Initial program 81.2%

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

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -8.5e5 < x < 4

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.1%

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

   if 4 < x

   1. Initial program 80.5%

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

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -850000:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{y\_m} \cdot x\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.55e-29)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (if (<= x 4.8e-48) (fabs (/ 4.0 y_m)) (fabs (* (/ (- 1.0 z) y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.55e-29) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else if (x <= 4.8e-48) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((((1.0 - z) / y_m) * x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.55d-29)) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else if (x <= 4.8d-48) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((((1.0d0 - z) / y_m) * x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.55e-29) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else if (x <= 4.8e-48) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((((1.0 - z) / y_m) * x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.55e-29:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	elif x <= 4.8e-48:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((((1.0 - z) / y_m) * x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.55e-29)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	elseif (x <= 4.8e-48)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(Float64(Float64(1.0 - z) / y_m) * x));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.55e-29)
		tmp = abs(((x / y_m) * (1.0 - z)));
	elseif (x <= 4.8e-48)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((((1.0 - z) / y_m) * x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.55e-29], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.8e-48], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.54999999999999993e-29

   1. Initial program 83.5%

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

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -2.54999999999999993e-29 < x < 4.8e-48

   1. Initial program 95.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   if 4.8e-48 < x

   1. Initial program 80.2%

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

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 94.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{1 - z}{y\_m} \cdot x\right|\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ (- 1.0 z) y_m) x))))
   (if (<= x -2.55e-29) t_0 (if (<= x 4.8e-48) (fabs (/ 4.0 y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((((1.0 - z) / y_m) * x));
	double tmp;
	if (x <= -2.55e-29) {
		tmp = t_0;
	} else if (x <= 4.8e-48) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((((1.0d0 - z) / y_m) * x))
    if (x <= (-2.55d-29)) then
        tmp = t_0
    else if (x <= 4.8d-48) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((((1.0 - z) / y_m) * x));
	double tmp;
	if (x <= -2.55e-29) {
		tmp = t_0;
	} else if (x <= 4.8e-48) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((((1.0 - z) / y_m) * x))
	tmp = 0
	if x <= -2.55e-29:
		tmp = t_0
	elif x <= 4.8e-48:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(Float64(1.0 - z) / y_m) * x))
	tmp = 0.0
	if (x <= -2.55e-29)
		tmp = t_0;
	elseif (x <= 4.8e-48)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((((1.0 - z) / y_m) * x));
	tmp = 0.0;
	if (x <= -2.55e-29)
		tmp = t_0;
	elseif (x <= 4.8e-48)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.55e-29], t$95$0, If[LessEqual[x, 4.8e-48], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.54999999999999993e-29 or 4.8e-48 < x

   1. Initial program 81.9%

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

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 96.3%

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

   if -2.54999999999999993e-29 < x < 4.8e-48

   1. Initial program 95.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{z \cdot x}{y\_m}\right|\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* z x) y_m))))
   (if (<= z -9.2e+91) t_0 (if (<= z 8.4e+18) (fabs (/ (+ 4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((z * x) / y_m));
	double tmp;
	if (z <= -9.2e+91) {
		tmp = t_0;
	} else if (z <= 8.4e+18) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((z * x) / y_m))
    if (z <= (-9.2d+91)) then
        tmp = t_0
    else if (z <= 8.4d+18) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((z * x) / y_m));
	double tmp;
	if (z <= -9.2e+91) {
		tmp = t_0;
	} else if (z <= 8.4e+18) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((z * x) / y_m))
	tmp = 0
	if z <= -9.2e+91:
		tmp = t_0
	elif z <= 8.4e+18:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(z * x) / y_m))
	tmp = 0.0
	if (z <= -9.2e+91)
		tmp = t_0;
	elseif (z <= 8.4e+18)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((z * x) / y_m));
	tmp = 0.0;
	if (z <= -9.2e+91)
		tmp = t_0;
	elseif (z <= 8.4e+18)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(z * x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -9.2e+91], t$95$0, If[LessEqual[z, 8.4e+18], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.19999999999999965e91 or 8.4e18 < 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

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

   4. Applied rewrites 95.4%

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

   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 80.0

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

   7. Applied rewrites 80.0%

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

   if -9.19999999999999965e91 < z < 8.4e18

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -9.2e+91)
   (fabs (* (/ x y_m) z))
   (if (<= z 8.4e+18) (fabs (/ (+ 4.0 x) y_m)) (fabs (* (/ z y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -9.2e+91) {
		tmp = fabs(((x / y_m) * z));
	} else if (z <= 8.4e+18) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = fabs(((z / y_m) * x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-9.2d+91)) then
        tmp = abs(((x / y_m) * z))
    else if (z <= 8.4d+18) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = abs(((z / y_m) * x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -9.2e+91) {
		tmp = Math.abs(((x / y_m) * z));
	} else if (z <= 8.4e+18) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = Math.abs(((z / y_m) * x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -9.2e+91:
		tmp = math.fabs(((x / y_m) * z))
	elif z <= 8.4e+18:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = math.fabs(((z / y_m) * x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -9.2e+91)
		tmp = abs(Float64(Float64(x / y_m) * z));
	elseif (z <= 8.4e+18)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = abs(Float64(Float64(z / y_m) * x));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -9.2e+91)
		tmp = abs(((x / y_m) * z));
	elseif (z <= 8.4e+18)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = abs(((z / y_m) * x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -9.2e+91], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 8.4e+18], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.19999999999999965e91

   1. Initial program 95.7%

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

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 81.5

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 85.6%

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

   if -9.19999999999999965e91 < z < 8.4e18

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.5%

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

   if 8.4e18 < z

   1. Initial program 75.6%

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

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

Alternative 7: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) z))))
   (if (<= z -9.2e+91) t_0 (if (<= z 8.4e+18) (fabs (/ (+ 4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x / y_m) * z));
	double tmp;
	if (z <= -9.2e+91) {
		tmp = t_0;
	} else if (z <= 8.4e+18) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x / y_m) * z))
    if (z <= (-9.2d+91)) then
        tmp = t_0
    else if (z <= 8.4d+18) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x / y_m) * z));
	double tmp;
	if (z <= -9.2e+91) {
		tmp = t_0;
	} else if (z <= 8.4e+18) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x / y_m) * z))
	tmp = 0
	if z <= -9.2e+91:
		tmp = t_0
	elif z <= 8.4e+18:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (z <= -9.2e+91)
		tmp = t_0;
	elseif (z <= 8.4e+18)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x / y_m) * z));
	tmp = 0.0;
	if (z <= -9.2e+91)
		tmp = t_0;
	elseif (z <= 8.4e+18)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -9.2e+91], t$95$0, If[LessEqual[z, 8.4e+18], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.19999999999999965e91 or 8.4e18 < 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

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 78.1

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 77.7%

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

   if -9.19999999999999965e91 < z < 8.4e18

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.5%

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

Alternative 8: 95.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right| \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ (fma (- 1.0 z) x 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs((fma((1.0 - z), x, 4.0) / y_m));
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	return abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y_m))
end

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. associate-/l* N/A

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

   6. associate--l+ N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. associate-*l* N/A

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

   11. *-rgt-identity N/A

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

   12. associate-*r/ N/A

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

   13. distribute-rgt-out N/A

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

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

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

   15. *-commutative N/A

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

   16. associate-/l* N/A

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

5. Applied rewrites 98.1%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]
6. Add Preprocessing

Alternative 9: 69.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. associate-/l* N/A

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

   6. associate--l+ N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. associate-*l* N/A

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

   11. *-rgt-identity N/A

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

   12. associate-*r/ N/A

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

   13. distribute-rgt-out N/A

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

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

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

   15. *-commutative N/A

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

   16. associate-/l* N/A

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 71.0%

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

Alternative 10: 40.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 41.5

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

5. Applied rewrites 41.5%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))