fabs fraction 1

Percentage Accurate: 91.5% → 99.9%

Time: 7.4s

Alternatives: 9

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.5% 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, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}} - \frac{x + 4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 3e-25)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (- (/ x (/ y_m z)) (/ (+ x 4.0) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3e-25) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(((x / (y_m / z)) - ((x + 4.0) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 3e-25)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(x / Float64(y_m / z)) - 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, 3e-25], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9999999999999998e-25

   1. Initial program 89.5%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 98.0

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

   4. Applied rewrites 98.0%

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

   if 2.9999999999999998e-25 < 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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. associate-/r/ N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-30)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (fma (- x) (/ z y_m) (/ (+ 4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-30) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(fma(-x, (z / y_m), ((4.0 + x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-30)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / 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, 4e-30], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-x) * N[(z / y$95$m), $MachinePrecision] + N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e-30

   1. Initial program 89.4%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 97.9

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

   4. Applied rewrites 97.9%

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

   if 4e-30 < y

   1. Initial program 93.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 3: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+72}:\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.9e+72)
   (fabs (* (/ z y_m) x))
   (if (<= z 4.3e+117)
     (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 tmp;
	if (z <= -1.9e+72) {
		tmp = fabs(((z / y_m) * x));
	} else if (z <= 4.3e+117) {
		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) :: tmp
    if (z <= (-1.9d+72)) then
        tmp = abs(((z / y_m) * x))
    else if (z <= 4.3d+117) 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 tmp;
	if (z <= -1.9e+72) {
		tmp = Math.abs(((z / y_m) * x));
	} else if (z <= 4.3e+117) {
		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):
	tmp = 0
	if z <= -1.9e+72:
		tmp = math.fabs(((z / y_m) * x))
	elif z <= 4.3e+117:
		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)
	tmp = 0.0
	if (z <= -1.9e+72)
		tmp = abs(Float64(Float64(z / y_m) * x));
	elseif (z <= 4.3e+117)
		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)
	tmp = 0.0;
	if (z <= -1.9e+72)
		tmp = abs(((z / y_m) * x));
	elseif (z <= 4.3e+117)
		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_] := If[LessEqual[z, -1.9e+72], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.3e+117], 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 z < -1.90000000000000003e72

   1. Initial program 91.3%

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

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left|\frac{\color{blue}{0 - \left(4 + x\right)}}{y}\right| \]

      3. associate--r+ N/A

         \[\leadsto \left|\frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{4 \cdot 1}\right) - x}{y}\right| \]

      5. lft-mult-inverse N/A

         \[\leadsto \left|\frac{\left(0 - 4 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) - x}{y}\right| \]

      6. associate-*l* N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{\left(4 \cdot \frac{1}{x}\right) \cdot x}\right) - x}{y}\right| \]

      7. neg-sub0 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 35.6

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

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.0

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

   10. Applied rewrites 74.0%

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

   11. Taylor expanded in z around inf

       \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}}\right| \]
   12. 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 78.9

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

   13. Applied rewrites 78.9%

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

   if -1.90000000000000003e72 < z < 4.29999999999999998e117

   1. Initial program 92.1%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left|\frac{\color{blue}{0 - \left(4 + x\right)}}{y}\right| \]

      3. associate--r+ N/A

         \[\leadsto \left|\frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{4 \cdot 1}\right) - x}{y}\right| \]

      5. lft-mult-inverse N/A

         \[\leadsto \left|\frac{\left(0 - 4 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) - x}{y}\right| \]

      6. associate-*l* N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{\left(4 \cdot \frac{1}{x}\right) \cdot x}\right) - x}{y}\right| \]

      7. neg-sub0 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 92.8

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

   7. Applied rewrites 92.8%

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

   if 4.29999999999999998e117 < z

   1. Initial program 80.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. 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 85.7

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

   5. Applied rewrites 85.7%

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

Alternative 4: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+72}:\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.9e+72)
   (fabs (* (/ z y_m) x))
   (if (<= z 4.3e+117) (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 tmp;
	if (z <= -1.9e+72) {
		tmp = fabs(((z / y_m) * x));
	} else if (z <= 4.3e+117) {
		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) :: tmp
    if (z <= (-1.9d+72)) then
        tmp = abs(((z / y_m) * x))
    else if (z <= 4.3d+117) 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 tmp;
	if (z <= -1.9e+72) {
		tmp = Math.abs(((z / y_m) * x));
	} else if (z <= 4.3e+117) {
		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):
	tmp = 0
	if z <= -1.9e+72:
		tmp = math.fabs(((z / y_m) * x))
	elif z <= 4.3e+117:
		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)
	tmp = 0.0
	if (z <= -1.9e+72)
		tmp = abs(Float64(Float64(z / y_m) * x));
	elseif (z <= 4.3e+117)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(-z) * 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 (z <= -1.9e+72)
		tmp = abs(((z / y_m) * x));
	elseif (z <= 4.3e+117)
		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_] := If[LessEqual[z, -1.9e+72], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.3e+117], 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 z < -1.90000000000000003e72

   1. Initial program 91.3%

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

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left|\frac{\color{blue}{0 - \left(4 + x\right)}}{y}\right| \]

      3. associate--r+ N/A

         \[\leadsto \left|\frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{4 \cdot 1}\right) - x}{y}\right| \]

      5. lft-mult-inverse N/A

         \[\leadsto \left|\frac{\left(0 - 4 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) - x}{y}\right| \]

      6. associate-*l* N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{\left(4 \cdot \frac{1}{x}\right) \cdot x}\right) - x}{y}\right| \]

      7. neg-sub0 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 35.6

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

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.0

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

   10. Applied rewrites 74.0%

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

   11. Taylor expanded in z around inf

       \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}}\right| \]
   12. 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 78.9

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

   13. Applied rewrites 78.9%

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

   if -1.90000000000000003e72 < z < 4.29999999999999998e117

   1. Initial program 92.1%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left|\frac{\color{blue}{0 - \left(4 + x\right)}}{y}\right| \]

      3. associate--r+ N/A

         \[\leadsto \left|\frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{4 \cdot 1}\right) - x}{y}\right| \]

      5. lft-mult-inverse N/A

         \[\leadsto \left|\frac{\left(0 - 4 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) - x}{y}\right| \]

      6. associate-*l* N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{\left(4 \cdot \frac{1}{x}\right) \cdot x}\right) - x}{y}\right| \]

      7. neg-sub0 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 92.8

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

   7. Applied rewrites 92.8%

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

   if 4.29999999999999998e117 < z

   1. Initial program 80.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 85.7

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

   5. Applied rewrites 85.7%

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

Alternative 5: 84.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+72} \lor \neg \left(z \leq 4.3 \cdot 10^{+117}\right):\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.9e+72) (not (<= z 4.3e+117)))
   (fabs (* (/ z y_m) x))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.9e+72) || !(z <= 4.3e+117)) {
		tmp = fabs(((z / y_m) * x));
	} 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 <= (-1.9d+72)) .or. (.not. (z <= 4.3d+117))) then
        tmp = abs(((z / y_m) * x))
    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 <= -1.9e+72) || !(z <= 4.3e+117)) {
		tmp = Math.abs(((z / y_m) * x));
	} 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 <= -1.9e+72) or not (z <= 4.3e+117):
		tmp = math.fabs(((z / y_m) * x))
	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 <= -1.9e+72) || !(z <= 4.3e+117))
		tmp = abs(Float64(Float64(z / y_m) * x));
	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 <= -1.9e+72) || ~((z <= 4.3e+117)))
		tmp = abs(((z / y_m) * x));
	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, -1.9e+72], N[Not[LessEqual[z, 4.3e+117]], $MachinePrecision]], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $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 < -1.90000000000000003e72 or 4.29999999999999998e117 < z

   1. Initial program 87.6%

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

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left|\frac{\color{blue}{0 - \left(4 + x\right)}}{y}\right| \]

      3. associate--r+ N/A

         \[\leadsto \left|\frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{4 \cdot 1}\right) - x}{y}\right| \]

      5. lft-mult-inverse N/A

         \[\leadsto \left|\frac{\left(0 - 4 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) - x}{y}\right| \]

      6. associate-*l* N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{\left(4 \cdot \frac{1}{x}\right) \cdot x}\right) - x}{y}\right| \]

      7. neg-sub0 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 35.8

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

   7. Applied rewrites 35.8%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.0

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

   10. Applied rewrites 75.0%

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

   11. Taylor expanded in z around inf

       \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}}\right| \]
   12. 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 80.0

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

   13. Applied rewrites 80.0%

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

   if -1.90000000000000003e72 < z < 4.29999999999999998e117

   1. Initial program 92.1%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left|\frac{\color{blue}{0 - \left(4 + x\right)}}{y}\right| \]

      3. associate--r+ N/A

         \[\leadsto \left|\frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{4 \cdot 1}\right) - x}{y}\right| \]

      5. lft-mult-inverse N/A

         \[\leadsto \left|\frac{\left(0 - 4 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) - x}{y}\right| \]

      6. associate-*l* N/A

         \[\leadsto \left|\frac{\left(0 - \color{blue}{\left(4 \cdot \frac{1}{x}\right) \cdot x}\right) - x}{y}\right| \]

      7. neg-sub0 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 92.8

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

   7. Applied rewrites 92.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+72} \lor \neg \left(z \leq 4.3 \cdot 10^{+117}\right):\\ \;\;\;\;\left|\frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 200.0)
   (fabs (/ (fma z x (- -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 tmp;
	if (x <= 200.0) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 200.0)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 200.0], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $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 2 regimes

2. if x < 200

   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-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 98.1

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

   4. Applied rewrites 98.1%

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

   if 200 < x

   1. Initial program 87.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\left(1 - z\right) \cdot \frac{x}{y}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.2) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.2) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((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 <= (-10.2d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((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 <= -10.2) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -10.2) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -10.2) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(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 <= -10.2) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((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, -10.2], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.199999999999999 or 4 < x

   1. Initial program 82.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 86.9%

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

   9. Applied rewrites 91.9%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 67.1%

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

   if -10.199999999999999 < x < 4

   1. Initial program 96.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 74.4

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

   5. Applied rewrites 74.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% 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 90.5%

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

   1. lift-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. sub-div N/A

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

   10. lower-/.f64 N/A

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. distribute-neg-in N/A

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

   17. unsub-neg N/A

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

   18. lower--.f64 N/A

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

   19. metadata-eval 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \left|\frac{\color{blue}{0 - \left(4 + x\right)}}{y}\right| \]

   3. associate--r+ N/A

      \[\leadsto \left|\frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right| \]

   4. metadata-eval N/A

      \[\leadsto \left|\frac{\left(0 - \color{blue}{4 \cdot 1}\right) - x}{y}\right| \]

   5. lft-mult-inverse N/A

      \[\leadsto \left|\frac{\left(0 - 4 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) - x}{y}\right| \]

   6. associate-*l* N/A

      \[\leadsto \left|\frac{\left(0 - \color{blue}{\left(4 \cdot \frac{1}{x}\right) \cdot x}\right) - x}{y}\right| \]

   7. neg-sub0 N/A

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

   8. lower--.f64 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. mul-1-neg N/A

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

   11. associate-*l* N/A

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

   12. mul-1-neg N/A

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

   13. distribute-rgt-neg-out N/A

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

   14. lft-mult-inverse N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 72.6

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

7. Applied rewrites 72.6%

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

Alternative 9: 34.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 57.1

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

5. Applied rewrites 57.1%

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

7. Applied rewrites 52.0%

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

9. Applied rewrites 55.7%

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

10. Taylor expanded in z around 0

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

12. Applied rewrites 32.4%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))