fabs fraction 1

Percentage Accurate: 92.2% → 99.9%

Time: 11.7s

Alternatives: 10

Speedup: 1.0×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.2% 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

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 100000:\\ \;\;\;\;\left|\frac{4 - x \cdot \left(z + -1\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m} - x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 100000.0)
   (fabs (/ (- 4.0 (* x (+ z -1.0))) y_m))
   (fabs (- (/ (+ 4.0 x) y_m) (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 100000.0) {
		tmp = fabs(((4.0 - (x * (z + -1.0))) / y_m));
	} else {
		tmp = fabs((((4.0 + x) / y_m) - (x * (z / y_m))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 100000.0d0) then
        tmp = abs(((4.0d0 - (x * (z + (-1.0d0)))) / y_m))
    else
        tmp = abs((((4.0d0 + x) / y_m) - (x * (z / y_m))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 100000.0) {
		tmp = Math.abs(((4.0 - (x * (z + -1.0))) / y_m));
	} else {
		tmp = Math.abs((((4.0 + x) / y_m) - (x * (z / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 100000.0:
		tmp = math.fabs(((4.0 - (x * (z + -1.0))) / y_m))
	else:
		tmp = math.fabs((((4.0 + x) / y_m) - (x * (z / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 100000.0)
		tmp = abs(Float64(Float64(4.0 - Float64(x * Float64(z + -1.0))) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / y_m) - Float64(x * Float64(z / y_m))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 100000.0)
		tmp = abs(((4.0 - (x * (z + -1.0))) / y_m));
	else
		tmp = abs((((4.0 + x) / y_m) - (x * (z / y_m))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 100000.0], N[Abs[N[(N[(4.0 - N[(x * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e5

   1. Initial program 90.7%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(x + 4\right) - x \cdot z\right), y\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(x + 4\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 + x\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 - \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 - \left(\left(\mathsf{neg}\left(x\right)\right) - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(\left(\mathsf{neg}\left(x\right)\right) - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(-1 \cdot x - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot -1 - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot -1 - x \cdot \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot \left(-1 - \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), y\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 - \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), y\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 + z\right)\right)\right), y\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(z + -1\right)\right)\right), y\right)\right) \]

      19. +-lowering-+.f64 97.1%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(z, -1\right)\right)\right), y\right)\right) \]

   3. Simplified 97.1%

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

   if 1e5 < y

   1. Initial program 97.2%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 4\right), y\right), \left(\frac{x \cdot z}{y}\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 4\right), y\right), \left(x \cdot \frac{z}{y}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 4\right), y\right), \mathsf{*.f64}\left(x, \left(\frac{z}{y}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 4\right), y\right), \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, y\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 100000:\\ \;\;\;\;\left|\frac{4 - x \cdot \left(z + -1\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+99}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.55e+140)
     t_0
     (if (<= x -7.8e-112)
       (fabs (* x (/ z y_m)))
       (if (<= x 2e-47)
         (fabs (/ 4.0 y_m))
         (if (<= x 6e+99) (fabs (/ x (/ y_m z))) 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 <= -1.55e+140) {
		tmp = t_0;
	} else if (x <= -7.8e-112) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 2e-47) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 6e+99) {
		tmp = fabs((x / (y_m / z)));
	} 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 <= (-1.55d+140)) then
        tmp = t_0
    else if (x <= (-7.8d-112)) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 2d-47) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 6d+99) then
        tmp = abs((x / (y_m / z)))
    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 <= -1.55e+140) {
		tmp = t_0;
	} else if (x <= -7.8e-112) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 2e-47) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 6e+99) {
		tmp = Math.abs((x / (y_m / z)));
	} 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 <= -1.55e+140:
		tmp = t_0
	elif x <= -7.8e-112:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 2e-47:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 6e+99:
		tmp = math.fabs((x / (y_m / z)))
	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 <= -1.55e+140)
		tmp = t_0;
	elseif (x <= -7.8e-112)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 2e-47)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 6e+99)
		tmp = abs(Float64(x / Float64(y_m / z)));
	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 <= -1.55e+140)
		tmp = t_0;
	elseif (x <= -7.8e-112)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 2e-47)
		tmp = abs((4.0 / y_m));
	elseif (x <= 6e+99)
		tmp = abs((x / (y_m / z)));
	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, -1.55e+140], t$95$0, If[LessEqual[x, -7.8e-112], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2e-47], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6e+99], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.55e140 or 6.00000000000000029e99 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{1 - z}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 - z\right)}{y}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)}{y}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + -1 \cdot z\right)}{y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + -1 \cdot z\right)\right), y\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), y\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      9. --lowering--.f64 89.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   5. Simplified 89.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 73.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, y\right)\right) \]

   8. Simplified 73.4%

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

   if -1.55e140 < x < -7.8000000000000002e-112

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 87.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]

   5. Simplified 87.8%

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

      1. fabs-sub N/A

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

      2. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y} - \frac{x}{y}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{1}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{1}{\frac{y}{x}}\right)\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z - 1}{\frac{y}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(z - 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      10. /-lowering-/.f64 87.7%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{/.f64}\left(y, x\right)\right)\right) \]

   7. Applied egg-rr 87.7%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x \cdot z}{y}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{z}{y}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{z}{y}\right)\right)\right) \]

      3. /-lowering-/.f64 68.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, y\right)\right)\right) \]

   10. Simplified 68.5%

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

   if -7.8000000000000002e-112 < x < 1.9999999999999999e-47

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 84.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

   5. Simplified 84.5%

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

   if 1.9999999999999999e-47 < x < 6.00000000000000029e99

   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 inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 93.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]

   5. Simplified 93.4%

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

      1. fabs-sub N/A

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

      2. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y} - \frac{x}{y}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{1}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{1}{\frac{y}{x}}\right)\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z - 1}{\frac{y}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(z - 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      10. /-lowering-/.f64 96.7%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{/.f64}\left(y, x\right)\right)\right) \]

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x \cdot z}{y}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{z}{y}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{z}{y}\right)\right)\right) \]

      3. /-lowering-/.f64 78.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, y\right)\right)\right) \]

   10. Simplified 78.7%

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

       1. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{z}{y}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right) \]

       3. un-div-inv N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{\frac{y}{z}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{z}\right)\right)\right) \]

       5. /-lowering-/.f64 78.7%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   12. Applied egg-rr 78.7%

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

Alternative 3: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* x (/ z y_m)))))
   (if (<= x -6.6e+140)
     t_0
     (if (<= x -7.8e-112)
       t_1
       (if (<= x 2.5e-46) (fabs (/ 4.0 y_m)) (if (<= x 2.25e+98) t_1 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((x * (z / y_m)));
	double tmp;
	if (x <= -6.6e+140) {
		tmp = t_0;
	} else if (x <= -7.8e-112) {
		tmp = t_1;
	} else if (x <= 2.5e-46) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 2.25e+98) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((x * (z / y_m)))
    if (x <= (-6.6d+140)) then
        tmp = t_0
    else if (x <= (-7.8d-112)) then
        tmp = t_1
    else if (x <= 2.5d-46) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 2.25d+98) then
        tmp = t_1
    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 t_1 = Math.abs((x * (z / y_m)));
	double tmp;
	if (x <= -6.6e+140) {
		tmp = t_0;
	} else if (x <= -7.8e-112) {
		tmp = t_1;
	} else if (x <= 2.5e-46) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 2.25e+98) {
		tmp = t_1;
	} 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))
	t_1 = math.fabs((x * (z / y_m)))
	tmp = 0
	if x <= -6.6e+140:
		tmp = t_0
	elif x <= -7.8e-112:
		tmp = t_1
	elif x <= 2.5e-46:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 2.25e+98:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(x * Float64(z / y_m)))
	tmp = 0.0
	if (x <= -6.6e+140)
		tmp = t_0;
	elseif (x <= -7.8e-112)
		tmp = t_1;
	elseif (x <= 2.5e-46)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 2.25e+98)
		tmp = t_1;
	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));
	t_1 = abs((x * (z / y_m)));
	tmp = 0.0;
	if (x <= -6.6e+140)
		tmp = t_0;
	elseif (x <= -7.8e-112)
		tmp = t_1;
	elseif (x <= 2.5e-46)
		tmp = abs((4.0 / y_m));
	elseif (x <= 2.25e+98)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.6e+140], t$95$0, If[LessEqual[x, -7.8e-112], t$95$1, If[LessEqual[x, 2.5e-46], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.25e+98], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6000000000000003e140 or 2.2500000000000001e98 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{1 - z}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 - z\right)}{y}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)}{y}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + -1 \cdot z\right)}{y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + -1 \cdot z\right)\right), y\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), y\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      9. --lowering--.f64 89.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   5. Simplified 89.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 73.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, y\right)\right) \]

   8. Simplified 73.4%

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

   if -6.6000000000000003e140 < x < -7.8000000000000002e-112 or 2.49999999999999996e-46 < x < 2.2500000000000001e98

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 89.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]

   5. Simplified 89.9%

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

      1. fabs-sub N/A

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

      2. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y} - \frac{x}{y}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{1}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{1}{\frac{y}{x}}\right)\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z - 1}{\frac{y}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(z - 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      10. /-lowering-/.f64 91.2%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{/.f64}\left(y, x\right)\right)\right) \]

   7. Applied egg-rr 91.2%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x \cdot z}{y}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{z}{y}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{z}{y}\right)\right)\right) \]

      3. /-lowering-/.f64 72.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, y\right)\right)\right) \]

   10. Simplified 72.4%

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

   if -7.8000000000000002e-112 < x < 2.49999999999999996e-46

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 84.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

   5. Simplified 84.5%

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

Alternative 4: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -27.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 46000000:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -27.5)
     t_0
     (if (<= x 46000000.0) (fabs (/ (- 4.0 (* x z)) 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 <= -27.5) {
		tmp = t_0;
	} else if (x <= 46000000.0) {
		tmp = fabs(((4.0 - (x * z)) / 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 <= (-27.5d0)) then
        tmp = t_0
    else if (x <= 46000000.0d0) then
        tmp = abs(((4.0d0 - (x * z)) / 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 <= -27.5) {
		tmp = t_0;
	} else if (x <= 46000000.0) {
		tmp = Math.abs(((4.0 - (x * z)) / 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 <= -27.5:
		tmp = t_0
	elif x <= 46000000.0:
		tmp = math.fabs(((4.0 - (x * z)) / 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 <= -27.5)
		tmp = t_0;
	elseif (x <= 46000000.0)
		tmp = abs(Float64(Float64(4.0 - Float64(x * z)) / 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 <= -27.5)
		tmp = t_0;
	elseif (x <= 46000000.0)
		tmp = abs(((4.0 - (x * z)) / 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, -27.5], t$95$0, If[LessEqual[x, 46000000.0], N[Abs[N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -27.5 or 4.6e7 < x

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{1 - z}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 - z\right)}{y}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)}{y}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + -1 \cdot z\right)}{y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + -1 \cdot z\right)\right), y\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), y\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      9. --lowering--.f64 90.1%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   5. Simplified 90.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(1 - z\right) \cdot x}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(1 - z\right) \cdot \frac{x}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(1 - z\right), \left(\frac{x}{y}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \left(\frac{x}{y}\right)\right)\right) \]

      5. /-lowering-/.f64 99.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   7. Applied egg-rr 99.7%

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

   if -27.5 < x < 4.6e7

   1. Initial program 98.0%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(x + 4\right) - x \cdot z\right), y\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(x + 4\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 + x\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 - \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 - \left(\left(\mathsf{neg}\left(x\right)\right) - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(\left(\mathsf{neg}\left(x\right)\right) - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(-1 \cdot x - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot -1 - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot -1 - x \cdot \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot \left(-1 - \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), y\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 - \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), y\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 + z\right)\right)\right), y\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(z + -1\right)\right)\right), y\right)\right) \]

      19. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(z, -1\right)\right)\right), y\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \color{blue}{\left(x \cdot z\right)}\right), y\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, z\right)\right), y\right)\right) \]

   7. Simplified 99.9%

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

Alternative 5: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (- 1.0 z) (/ x y_m)))))
   (if (<= x -7.8e-112)
     t_0
     (if (<= x 1.45e-46) (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(((1.0 - z) * (x / y_m)));
	double tmp;
	if (x <= -7.8e-112) {
		tmp = t_0;
	} else if (x <= 1.45e-46) {
		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(((1.0d0 - z) * (x / y_m)))
    if (x <= (-7.8d-112)) then
        tmp = t_0
    else if (x <= 1.45d-46) 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(((1.0 - z) * (x / y_m)));
	double tmp;
	if (x <= -7.8e-112) {
		tmp = t_0;
	} else if (x <= 1.45e-46) {
		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(((1.0 - z) * (x / y_m)))
	tmp = 0
	if x <= -7.8e-112:
		tmp = t_0
	elif x <= 1.45e-46:
		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(1.0 - z) * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -7.8e-112)
		tmp = t_0;
	elseif (x <= 1.45e-46)
		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(((1.0 - z) * (x / y_m)));
	tmp = 0.0;
	if (x <= -7.8e-112)
		tmp = t_0;
	elseif (x <= 1.45e-46)
		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[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7.8e-112], t$95$0, If[LessEqual[x, 1.45e-46], 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 x < -7.8000000000000002e-112 or 1.45000000000000002e-46 < x

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{1 - z}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 - z\right)}{y}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)}{y}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + -1 \cdot z\right)}{y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + -1 \cdot z\right)\right), y\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), y\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      9. --lowering--.f64 88.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   5. Simplified 88.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(1 - z\right) \cdot x}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(1 - z\right) \cdot \frac{x}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(1 - z\right), \left(\frac{x}{y}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \left(\frac{x}{y}\right)\right)\right) \]

      5. /-lowering-/.f64 95.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   7. Applied egg-rr 95.7%

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

   if -7.8000000000000002e-112 < x < 1.45000000000000002e-46

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)}\right) \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} - -1 \cdot \frac{x}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 \cdot 1}{y} - -1 \cdot \frac{x}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} - -1 \cdot \frac{x}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} - \frac{-1 \cdot x}{y}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 - -1 \cdot x}{y}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}{y}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + x}{y}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 + x\right), y\right)\right) \]

      12. +-lowering-+.f64 84.5%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(4, x\right), y\right)\right) \]

   5. Simplified 84.5%

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

Alternative 6: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y_m x)))))
   (if (<= z -8.2e+76) t_0 (if (<= z 4.7e+27) (fabs (/ (+ 4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z / (y_m / x)));
	double tmp;
	if (z <= -8.2e+76) {
		tmp = t_0;
	} else if (z <= 4.7e+27) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((z / (y_m / x)))
    if (z <= (-8.2d+76)) then
        tmp = t_0
    else if (z <= 4.7d+27) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z / (y_m / x)));
	double tmp;
	if (z <= -8.2e+76) {
		tmp = t_0;
	} else if (z <= 4.7e+27) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z / (y_m / x)))
	tmp = 0
	if z <= -8.2e+76:
		tmp = t_0
	elif z <= 4.7e+27:
		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(z / Float64(y_m / x)))
	tmp = 0.0
	if (z <= -8.2e+76)
		tmp = t_0;
	elseif (z <= 4.7e+27)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z / (y_m / x)));
	tmp = 0.0;
	if (z <= -8.2e+76)
		tmp = t_0;
	elseif (z <= 4.7e+27)
		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[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -8.2e+76], t$95$0, If[LessEqual[z, 4.7e+27], 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 < -8.1999999999999997e76 or 4.69999999999999976e27 < z

   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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 75.0%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]

   5. Simplified 75.0%

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

      1. fabs-sub N/A

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

      2. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y} - \frac{x}{y}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{1}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{1}{\frac{y}{x}}\right)\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z - 1}{\frac{y}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(z - 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      10. /-lowering-/.f64 80.5%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{/.f64}\left(y, x\right)\right)\right) \]

   7. Applied egg-rr 80.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\color{blue}{z}, \mathsf{/.f64}\left(y, x\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 80.5%

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

   if -8.1999999999999997e76 < z < 4.69999999999999976e27

   1. Initial program 92.5%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)}\right) \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} - -1 \cdot \frac{x}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 \cdot 1}{y} - -1 \cdot \frac{x}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} - -1 \cdot \frac{x}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} - \frac{-1 \cdot x}{y}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 - -1 \cdot x}{y}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}{y}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + x}{y}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 + x\right), y\right)\right) \]

      12. +-lowering-+.f64 95.2%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(4, x\right), y\right)\right) \]

   5. Simplified 95.2%

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

Alternative 7: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y_m x)))))
   (if (<= x -7.8e-112) t_0 (if (<= x 1.2e-46) (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((z / (y_m / x)));
	double tmp;
	if (x <= -7.8e-112) {
		tmp = t_0;
	} else if (x <= 1.2e-46) {
		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((z / (y_m / x)))
    if (x <= (-7.8d-112)) then
        tmp = t_0
    else if (x <= 1.2d-46) 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((z / (y_m / x)));
	double tmp;
	if (x <= -7.8e-112) {
		tmp = t_0;
	} else if (x <= 1.2e-46) {
		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((z / (y_m / x)))
	tmp = 0
	if x <= -7.8e-112:
		tmp = t_0
	elif x <= 1.2e-46:
		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(z / Float64(y_m / x)))
	tmp = 0.0
	if (x <= -7.8e-112)
		tmp = t_0;
	elseif (x <= 1.2e-46)
		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((z / (y_m / x)));
	tmp = 0.0;
	if (x <= -7.8e-112)
		tmp = t_0;
	elseif (x <= 1.2e-46)
		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[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7.8e-112], t$95$0, If[LessEqual[x, 1.2e-46], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.8000000000000002e-112 or 1.20000000000000007e-46 < x

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 85.1%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]

   5. Simplified 85.1%

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

      1. fabs-sub N/A

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

      2. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y} - \frac{x}{y}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{1}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{x}{y}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{\frac{y}{x}} - \frac{1}{\frac{y}{x}}\right)\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z - 1}{\frac{y}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(z - 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\frac{y}{x}\right)\right)\right) \]

      10. /-lowering-/.f64 95.6%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{/.f64}\left(y, x\right)\right)\right) \]

   7. Applied egg-rr 95.6%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\color{blue}{z}, \mathsf{/.f64}\left(y, x\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 70.0%

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

   if -7.8000000000000002e-112 < x < 1.20000000000000007e-46

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 84.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

   5. Simplified 84.5%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - x \cdot \left(z + -1\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2e+20)
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (- 4.0 (* x (+ z -1.0))) y_m))))

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2d+20)) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs(((4.0d0 - (x * (z + (-1.0d0)))) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2e+20) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs(((4.0 - (x * (z + -1.0))) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2e+20:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = math.fabs(((4.0 - (x * (z + -1.0))) / y_m))
	return tmp

Julia

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

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2e+20)
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = abs(((4.0 - (x * (z + -1.0))) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2e+20], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 - N[(x * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e20

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{1 - z}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 - z\right)}{y}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)}{y}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + -1 \cdot z\right)}{y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + -1 \cdot z\right)\right), y\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), y\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      9. --lowering--.f64 87.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   5. Simplified 87.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(1 - z\right) \cdot x}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(1 - z\right) \cdot \frac{x}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(1 - z\right), \left(\frac{x}{y}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \left(\frac{x}{y}\right)\right)\right) \]

      5. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   7. Applied egg-rr 99.9%

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

   if -2e20 < x

   1. Initial program 94.8%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(x + 4\right) - x \cdot z\right), y\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(x + 4\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 + x\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\left(4 - \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\mathsf{neg}\left(x \cdot z\right)\right)\right), y\right)\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 - \left(\left(\mathsf{neg}\left(x\right)\right) - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(\left(\mathsf{neg}\left(x\right)\right) - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(-1 \cdot x - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot -1 - \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot -1 - x \cdot \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \left(x \cdot \left(-1 - \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), y\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 - \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), y\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(-1 + z\right)\right)\right), y\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(z + -1\right)\right)\right), y\right)\right) \]

      19. +-lowering-+.f64 98.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(z, -1\right)\right)\right), y\right)\right) \]

   3. Simplified 98.0%

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

Alternative 9: 69.7% accurate, 1.0× 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 86.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{1 - z}{y}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 - z\right)}{y}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)}{y}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot \left(1 + -1 \cdot z\right)}{y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + -1 \cdot z\right)\right), y\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), y\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      9. --lowering--.f64 90.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   5. Simplified 90.3%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 61.1%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, y\right)\right) \]

   8. Simplified 61.1%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 69.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

   5. Simplified 69.5%

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

Alternative 10: 40.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 37.4%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

5. Simplified 37.4%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))