fabs fraction 1

Percentage Accurate: 91.5% → 99.8%

Time: 7.5s

Alternatives: 9

Speedup: 1.2×

• 21.3% 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: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.4e+47)
   (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 <= 1.4e+47) {
		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 <= 1.4e+47)
		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, 1.4e+47], 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 < 1.39999999999999994e47

   1. Initial program 94.4%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 96.7%

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

   if 1.39999999999999994e47 < y

   1. Initial program 99.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot z - \frac{4 + x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 8e+102)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (- (* (/ x y_m) z) (/ (+ 4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8e+102) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs((((x / y_m) * z) - ((4.0 + x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 8e+102)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x / y_m) * z) - 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, 8e+102], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision] - N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999982e102

   1. Initial program 94.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.9%

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

   if 7.99999999999999982e102 < y

   1. Initial program 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z - \frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+154}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (fma z x -4.0) y_m))))
   (if (<= z -1.85e+154)
     (fabs (* (/ x y_m) z))
     (if (<= z -1.05e+19)
       t_0
       (if (<= z 6.5e-10) (fabs (/ (- x -4.0) y_m)) t_0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((fma(z, x, -4.0) / y_m));
	double tmp;
	if (z <= -1.85e+154) {
		tmp = fabs(((x / y_m) * z));
	} else if (z <= -1.05e+19) {
		tmp = t_0;
	} else if (z <= 6.5e-10) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(fma(z, x, -4.0) / y_m))
	tmp = 0.0
	if (z <= -1.85e+154)
		tmp = abs(Float64(Float64(x / y_m) * z));
	elseif (z <= -1.05e+19)
		tmp = t_0;
	elseif (z <= 6.5e-10)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.85e+154], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -1.05e+19], t$95$0, If[LessEqual[z, 6.5e-10], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999997e154

   1. Initial program 97.2%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if -1.84999999999999997e154 < z < -1.05e19 or 6.5000000000000003e-10 < z

   1. Initial program 95.0%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 93.5%

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

   if -1.05e19 < z < 6.5000000000000003e-10

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-frac-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+154}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -430:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (- 1.0 z) (/ x y_m)))))
   (if (<= x -430.0)
     t_0
     (if (<= x 0.0032) (fabs (/ (fma z x -4.0) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((1.0 - z) * (x / y_m)));
	double tmp;
	if (x <= -430.0) {
		tmp = t_0;
	} else if (x <= 0.0032) {
		tmp = fabs((fma(z, x, -4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -430.0)
		tmp = t_0;
	elseif (x <= 0.0032)
		tmp = abs(Float64(fma(z, x, -4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -430.0], t$95$0, If[LessEqual[x, 0.0032], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -430 or 0.00320000000000000015 < x

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   if -430 < x < 0.00320000000000000015

   1. Initial program 98.3%

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

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.7%

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

Alternative 5: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+74}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) z))))
   (if (<= z -5.2e+137)
     t_0
     (if (<= z 2.25e+74) (fabs (/ (- x -4.0) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x / y_m) * z));
	double tmp;
	if (z <= -5.2e+137) {
		tmp = t_0;
	} else if (z <= 2.25e+74) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x / y_m) * z))
    if (z <= (-5.2d+137)) then
        tmp = t_0
    else if (z <= 2.25d+74) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x / y_m) * z));
	double tmp;
	if (z <= -5.2e+137) {
		tmp = t_0;
	} else if (z <= 2.25e+74) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x / y_m) * z))
	tmp = 0
	if z <= -5.2e+137:
		tmp = t_0
	elif z <= 2.25e+74:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (z <= -5.2e+137)
		tmp = t_0;
	elseif (z <= 2.25e+74)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x / y_m) * z));
	tmp = 0.0;
	if (z <= -5.2e+137)
		tmp = t_0;
	elseif (z <= 2.25e+74)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -5.2e+137], t$95$0, If[LessEqual[z, 2.25e+74], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999998e137 or 2.25e74 < z

   1. Initial program 94.5%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 87.5

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

   7. Applied rewrites 87.5%

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

   if -5.1999999999999998e137 < z < 2.25e74

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-frac-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 90.3

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

   5. Applied rewrites 90.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+137}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+74}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5e+25)
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (fma z x (- -4.0 x)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5e+25) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5e+25)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -5e+25], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000024e25

   1. Initial program 89.7%

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

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

   if -5.00000000000000024e25 < x

   1. Initial program 96.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.1%

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

Alternative 7: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{-x}{y\_m}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- x) y_m))))
   (if (<= x -10.5) t_0 (if (<= x 4.0) (fabs (/ 4.0 y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((-x / y_m));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((-x / y_m))
    if (x <= (-10.5d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((-x / y_m));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((-x / y_m))
	tmp = 0
	if x <= -10.5:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(-x) / y_m))
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((-x / y_m));
	tmp = 0.0;
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[((-x) / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 92.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 90.4%

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

   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. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 57.4%

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

   if -10.5 < x < 4

   1. Initial program 98.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

Alternative 8: 69.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in z around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. associate-*r/ N/A

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

   8. neg-mul-1 N/A

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

   9. mul-1-neg N/A

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

   10. distribute-frac-neg N/A

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

   11. remove-double-neg N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval 68.6

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

5. Applied rewrites 68.6%

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

Alternative 9: 39.3% 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 95.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 44.1

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

5. Applied rewrites 44.1%

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

Reproduce

herbie shell --seed 2024257 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))