fabs fraction 1

Percentage Accurate: 91.9% → 99.6%

Time: 7.4s

Alternatives: 10

Speedup: 1.1×

• 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: 91.9% 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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-61}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-61)
   (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-61) {
		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-61)
		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-61], 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 < 4.0000000000000002e-61

   1. Initial program 92.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.1%

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

   if 4.0000000000000002e-61 < y

   1. Initial program 97.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot z - \frac{4 + x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e+84)
   (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 <= 2e+84) {
		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 <= 2e+84)
		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, 2e+84], 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 < 2.00000000000000012e84

   1. Initial program 93.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.7%

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

   if 2.00000000000000012e84 < y

   1. Initial program 94.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z - \frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{+42}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1.75e+16)
     t_0
     (if (<= x 1e+42) (fabs (/ (fma z x (- -4.0 x)) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((1.0 - z) * (x / y_m)));
	double tmp;
	if (x <= -1.75e+16) {
		tmp = t_0;
	} else if (x <= 1e+42) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -1.75e+16)
		tmp = t_0;
	elseif (x <= 1e+42)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / 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, -1.75e+16], t$95$0, If[LessEqual[x, 1e+42], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e16 or 1.00000000000000004e42 < x

   1. Initial program 90.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.75e16 < x < 1.00000000000000004e42

   1. Initial program 97.0%

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

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

Alternative 4: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\left|\frac{x \cdot z - 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (- 1.0 z) (/ x y_m)))))
   (if (<= x -1.55) t_0 (if (<= x 400.0) (fabs (/ (- (* x z) 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 <= -1.55) {
		tmp = t_0;
	} else if (x <= 400.0) {
		tmp = fabs((((x * z) - 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((1.0d0 - z) * (x / y_m)))
    if (x <= (-1.55d0)) then
        tmp = t_0
    else if (x <= 400.0d0) then
        tmp = abs((((x * z) - 4.0d0) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((1.0 - z) * (x / y_m)));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 400.0) {
		tmp = Math.abs((((x * z) - 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((1.0 - z) * (x / y_m)))
	tmp = 0
	if x <= -1.55:
		tmp = t_0
	elif x <= 400.0:
		tmp = math.fabs((((x * z) - 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 <= -1.55)
		tmp = t_0;
	elseif (x <= 400.0)
		tmp = abs(Float64(Float64(Float64(x * z) - 4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((1.0 - z) * (x / y_m)));
	tmp = 0.0;
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 400.0)
		tmp = abs((((x * z) - 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[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.55], t$95$0, If[LessEqual[x, 400.0], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 400 < x

   1. Initial program 91.3%

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

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

   5. Applied rewrites 98.4%

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

   if -1.55000000000000004 < x < 400

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\left|\frac{x \cdot z - 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (- 1.0 z) (/ x y_m)))))
   (if (<= x -2.5e-15) t_0 (if (<= x 1.5e-33) (fabs (/ 4.0 y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((1.0 - z) * (x / y_m)));
	double tmp;
	if (x <= -2.5e-15) {
		tmp = t_0;
	} else if (x <= 1.5e-33) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((1.0d0 - z) * (x / y_m)))
    if (x <= (-2.5d-15)) then
        tmp = t_0
    else if (x <= 1.5d-33) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((1.0 - z) * (x / y_m)));
	double tmp;
	if (x <= -2.5e-15) {
		tmp = t_0;
	} else if (x <= 1.5e-33) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((1.0 - z) * (x / y_m)))
	tmp = 0
	if x <= -2.5e-15:
		tmp = t_0
	elif x <= 1.5e-33:
		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(1.0 - z) * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -2.5e-15)
		tmp = t_0;
	elseif (x <= 1.5e-33)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((1.0 - z) * (x / y_m)));
	tmp = 0.0;
	if (x <= -2.5e-15)
		tmp = t_0;
	elseif (x <= 1.5e-33)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.5e-15], t$95$0, If[LessEqual[x, 1.5e-33], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e-15 or 1.5000000000000001e-33 < x

   1. Initial program 92.0%

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

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

   5. Applied rewrites 96.6%

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

   if -2.5e-15 < x < 1.5000000000000001e-33

   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 79.2

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

   5. Applied rewrites 79.2%

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

Alternative 6: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+71}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.02e+71)
   (fabs (* (/ x y_m) z))
   (if (<= z 2.5e+15) (fabs (/ (- -4.0 x) y_m)) (fabs (* (/ z y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.02e+71) {
		tmp = fabs(((x / y_m) * z));
	} else if (z <= 2.5e+15) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((z / y_m) * x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.02d+71)) then
        tmp = abs(((x / y_m) * z))
    else if (z <= 2.5d+15) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((z / y_m) * x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.02e+71) {
		tmp = Math.abs(((x / y_m) * z));
	} else if (z <= 2.5e+15) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((z / y_m) * x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.02e+71:
		tmp = math.fabs(((x / y_m) * z))
	elif z <= 2.5e+15:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((z / y_m) * x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.02e+71)
		tmp = abs(Float64(Float64(x / y_m) * z));
	elseif (z <= 2.5e+15)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(z / y_m) * x));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.02e+71)
		tmp = abs(((x / y_m) * z));
	elseif (z <= 2.5e+15)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((z / y_m) * x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.02e+71], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e+15], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02000000000000003e71

   1. Initial program 97.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 92.8%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.8

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

   7. Applied rewrites 74.8%

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

   9. Applied rewrites 84.1%

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

   if -1.02000000000000003e71 < z < 2.5e15

   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 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. 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 94.4

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

   7. Applied rewrites 94.4%

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

   if 2.5e15 < z

   1. Initial program 89.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 79.6%

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

   5. Taylor expanded in z around inf

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

      1. 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 72.9

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

   7. Applied rewrites 72.9%

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

Alternative 7: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) z))))
   (if (<= z -1.02e+71)
     t_0
     (if (<= z 2.5e+15) (fabs (/ (- -4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x / y_m) * z));
	double tmp;
	if (z <= -1.02e+71) {
		tmp = t_0;
	} else if (z <= 2.5e+15) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x / y_m) * z))
    if (z <= (-1.02d+71)) then
        tmp = t_0
    else if (z <= 2.5d+15) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x / y_m) * z));
	double tmp;
	if (z <= -1.02e+71) {
		tmp = t_0;
	} else if (z <= 2.5e+15) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x / y_m) * z))
	tmp = 0
	if z <= -1.02e+71:
		tmp = t_0
	elif z <= 2.5e+15:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (z <= -1.02e+71)
		tmp = t_0;
	elseif (z <= 2.5e+15)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x / y_m) * z));
	tmp = 0.0;
	if (z <= -1.02e+71)
		tmp = t_0;
	elseif (z <= 2.5e+15)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.02e+71], t$95$0, If[LessEqual[z, 2.5e+15], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000003e71 or 2.5e15 < z

   1. Initial program 93.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 85.4%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   9. Applied rewrites 75.9%

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

   if -1.02000000000000003e71 < z < 2.5e15

   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 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. 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 94.4

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

   7. Applied rewrites 94.4%

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

Alternative 8: 69.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.55) 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 <= -1.55) {
		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 <= (-1.55d0)) 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 <= -1.55) {
		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 <= -1.55:
		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(x / y_m))
	tmp = 0.0
	if (x <= -1.55)
		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 <= -1.55)
		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, -1.55], 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 < -1.55000000000000004 or 4 < x

   1. Initial program 91.4%

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

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 64.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 64.0%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 9: 70.9% 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 94.0%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 94.7%

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

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 70.2

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

7. Applied rewrites 70.2%

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

Alternative 10: 34.4% 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 94.0%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 62.1

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

5. Applied rewrites 62.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 34.8%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 34.8%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))