fabs fraction 1

Percentage Accurate: 91.6% → 99.9%

Time: 7.8s

Alternatives: 11

Speedup: 1.1×

• 20.3% 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: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+20} \lor \neg \left(x \leq 2 \cdot 10^{+19}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.6e+20) (not (<= x 2e+19)))
   (fabs (* (- 1.0 z) (/ x y)))
   (fabs (/ (fma (- 1.0 z) x 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.6e+20) || !(x <= 2e+19)) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.6e+20) || !(x <= 2e+19))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	else
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.6e+20], N[Not[LessEqual[x, 2e+19]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e20 or 2e19 < x

   1. Initial program 89.3%

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

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. 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 -2.6e20 < x < 2e19

   1. Initial program 96.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+20} \lor \neg \left(x \leq 2 \cdot 10^{+19}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 4}{y} - \frac{x}{y} \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-164} \lor \neg \left(t\_0 \leq 10^{+207}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y) (* (/ x y) z))))
   (if (or (<= t_0 -2e-164) (not (<= t_0 1e+207)))
     (fabs (* (- 1.0 z) (/ x y)))
     (fabs (/ (- x -4.0) y)))))

C

double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - ((x / y) * z);
	double tmp;
	if ((t_0 <= -2e-164) || !(t_0 <= 1e+207)) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else {
		tmp = fabs(((x - -4.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y) - ((x / y) * z)
    if ((t_0 <= (-2d-164)) .or. (.not. (t_0 <= 1d+207))) then
        tmp = abs(((1.0d0 - z) * (x / y)))
    else
        tmp = abs(((x - (-4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - ((x / y) * z);
	double tmp;
	if ((t_0 <= -2e-164) || !(t_0 <= 1e+207)) {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x + 4.0) / y) - ((x / y) * z)
	tmp = 0
	if (t_0 <= -2e-164) or not (t_0 <= 1e+207):
		tmp = math.fabs(((1.0 - z) * (x / y)))
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z))
	tmp = 0.0
	if ((t_0 <= -2e-164) || !(t_0 <= 1e+207))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x + 4.0) / y) - ((x / y) * z);
	tmp = 0.0;
	if ((t_0 <= -2e-164) || ~((t_0 <= 1e+207)))
		tmp = abs(((1.0 - z) * (x / y)));
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-164], N[Not[LessEqual[t$95$0, 1e+207]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -1.99999999999999992e-164 or 1e207 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 90.3%

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

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -1.99999999999999992e-164 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1e207

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 79.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -2 \cdot 10^{-164} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{+207}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y}, \frac{4 + x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+57)
   (fabs (/ (fma (- 1.0 z) x 4.0) y))
   (fabs (fma (- x) (/ z y) (/ (+ 4.0 x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+57) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y));
	} else {
		tmp = fabs(fma(-x, (z / y), ((4.0 + x) / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+57)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y));
	else
		tmp = abs(fma(Float64(-x), Float64(z / y), Float64(Float64(4.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+57], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-x) * N[(z / y), $MachinePrecision] + N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000019e57

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 97.0%

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

   if 4.00000000000000019e57 < y

   1. Initial program 92.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 4: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+125}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{1 - z}{y}, \frac{4}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+125)
   (fabs (/ (fma (- 1.0 z) x 4.0) y))
   (fabs (fma x (/ (- 1.0 z) y) (/ 4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+125) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y));
	} else {
		tmp = fabs(fma(x, ((1.0 - z) / y), (4.0 / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+125)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y));
	else
		tmp = abs(fma(x, Float64(Float64(1.0 - z) / y), Float64(4.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+125], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999999e125

   1. Initial program 93.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 97.3%

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

   if 6.4999999999999999e125 < y

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 99.7%

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

Alternative 5: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+15} \lor \neg \left(x \leq 9 \cdot 10^{+18}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x - 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.8e+15) (not (<= x 9e+18)))
   (fabs (* (- 1.0 z) (/ x y)))
   (fabs (/ (- (* z x) 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e+15) || !(x <= 9e+18)) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else {
		tmp = fabs((((z * x) - 4.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.8d+15)) .or. (.not. (x <= 9d+18))) then
        tmp = abs(((1.0d0 - z) * (x / y)))
    else
        tmp = abs((((z * x) - 4.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e+15) || !(x <= 9e+18)) {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	} else {
		tmp = Math.abs((((z * x) - 4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.8e+15) or not (x <= 9e+18):
		tmp = math.fabs(((1.0 - z) * (x / y)))
	else:
		tmp = math.fabs((((z * x) - 4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.8e+15) || !(x <= 9e+18))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	else
		tmp = abs(Float64(Float64(Float64(z * x) - 4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.8e+15) || ~((x <= 9e+18)))
		tmp = abs(((1.0 - z) * (x / y)));
	else
		tmp = abs((((z * x) - 4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.8e+15], N[Not[LessEqual[x, 9e+18]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(z * x), $MachinePrecision] - 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8e15 or 9e18 < x

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. 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 -8.8e15 < x < 9e18

   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

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

   5. Applied rewrites 95.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+15} \lor \neg \left(x \leq 9 \cdot 10^{+18}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x - 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+77} \lor \neg \left(z \leq 1.7 \cdot 10^{+90}\right):\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.2e+77) (not (<= z 1.7e+90)))
   (fabs (* (- z) (/ x y)))
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2e+77) || !(z <= 1.7e+90)) {
		tmp = fabs((-z * (x / y)));
	} else {
		tmp = fabs(((x - -4.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.2d+77)) .or. (.not. (z <= 1.7d+90))) then
        tmp = abs((-z * (x / y)))
    else
        tmp = abs(((x - (-4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2e+77) || !(z <= 1.7e+90)) {
		tmp = Math.abs((-z * (x / y)));
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.2e+77) or not (z <= 1.7e+90):
		tmp = math.fabs((-z * (x / y)))
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.2e+77) || !(z <= 1.7e+90))
		tmp = abs(Float64(Float64(-z) * Float64(x / y)));
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.2e+77) || ~((z <= 1.7e+90)))
		tmp = abs((-z * (x / y)));
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.2e+77], N[Not[LessEqual[z, 1.7e+90]], $MachinePrecision]], N[Abs[N[((-z) * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000002e77 or 1.70000000000000009e90 < z

   1. Initial program 88.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -3.2000000000000002e77 < z < 1.70000000000000009e90

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 91.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+77} \lor \neg \left(z \leq 1.7 \cdot 10^{+90}\right):\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+145} \lor \neg \left(z \leq 7 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.4e+145) (not (<= z 7e+216)))
   (* (/ x y) (- z))
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.4e+145) || !(z <= 7e+216)) {
		tmp = (x / y) * -z;
	} else {
		tmp = fabs(((x - -4.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.4d+145)) .or. (.not. (z <= 7d+216))) then
        tmp = (x / y) * -z
    else
        tmp = abs(((x - (-4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.4e+145) || !(z <= 7e+216)) {
		tmp = (x / y) * -z;
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.4e+145) or not (z <= 7e+216):
		tmp = (x / y) * -z
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.4e+145) || !(z <= 7e+216))
		tmp = Float64(Float64(x / y) * Float64(-z));
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.4e+145) || ~((z <= 7e+216)))
		tmp = (x / y) * -z;
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.4e+145], N[Not[LessEqual[z, 7e+216]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * (-z)), $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e145 or 6.99999999999999984e216 < z

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 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. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 83.3

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

   5. Applied rewrites 83.3%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 29.4

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

   7. Applied rewrites 29.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 29.4%

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

   if -1.3999999999999999e145 < z < 6.99999999999999984e216

   1. Initial program 93.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 82.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+145} \lor \neg \left(z \leq 7 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.52) (fabs (/ x y)) (if (<= x 4.0) (fabs (/ 4.0 y)) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.52) {
		tmp = fabs((x / y));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.52d0)) then
        tmp = abs((x / y))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.52) {
		tmp = Math.abs((x / y));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.52:
		tmp = math.fabs((x / y))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.52)
		tmp = abs(Float64(x / y));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.52)
		tmp = abs((x / y));
	elseif (x <= 4.0)
		tmp = abs((4.0 / y));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.52], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   1. Initial program 90.2%

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

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 65.4%

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

   if -1.52 < x < 4

   1. Initial program 96.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   if 4 < x

   1. Initial program 89.6%

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

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 70.4%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 47.9

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

   10. Applied rewrites 47.9%

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

Alternative 9: 70.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. associate-/l* N/A

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

   6. associate--l+ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. div-add N/A

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

   10. div-sub N/A

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

   11. lower-/.f64 N/A

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

5. Applied rewrites 95.5%

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

6. Taylor expanded in z around 0

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. div-add N/A

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

   4. lower-/.f64 N/A

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

8. Applied rewrites 69.0%

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

Alternative 10: 34.2% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fabs (/ x y)))

C

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

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 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 93.0%

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

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

   6. fp-cancel-sub-sign-inv N/A

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

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

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt1-in N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. fp-cancel-sign-sub-inv N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. lower--.f64 N/A

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

   19. lower-/.f64 65.0

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

5. Applied rewrites 65.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 36.9%

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

Alternative 11: 17.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ x y))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return x / y;
}

Python

def code(x, y, z):
	return x / y

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 93.0%

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

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

   6. fp-cancel-sub-sign-inv N/A

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

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

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt1-in N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. fp-cancel-sign-sub-inv N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. lower--.f64 N/A

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

   19. lower-/.f64 65.0

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

5. Applied rewrites 65.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 36.9%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 21.2

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

10. Applied rewrites 21.2%

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

Reproduce

herbie shell --seed 2024333 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))