fabs fraction 1

Percentage Accurate: 91.9% → 99.6%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.5 \cdot 10^{-77}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4.5e-77)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.5e-77) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4.5e-77)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 4.5e-77], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5000000000000001e-77

   1. Initial program 90.5%

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

      1. associate-*l/ 94.0%

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

      2. sub-div 98.2%

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

   4. Applied egg-rr 98.2%

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

   if 4.5000000000000001e-77 < y

   1. Initial program 96.6%

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

      1. fabs-sub 96.6%

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

      2. associate-*l/ 93.6%

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

      3. associate-*r/ 99.8%

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

      4. fma-neg 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. unsub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

Alternative 2: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+16} \lor \neg \left(x \leq 1.72 \cdot 10^{+64}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -1.65e+109)
     t_0
     (if (<= x -1.15e+18)
       t_1
       (if (<= x 2.7e-49)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 3.4e+16) (not (<= x 1.72e+64))) 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((z * (x / y_m)));
	double tmp;
	if (x <= -1.65e+109) {
		tmp = t_0;
	} else if (x <= -1.15e+18) {
		tmp = t_1;
	} else if (x <= 2.7e-49) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 3.4e+16) || !(x <= 1.72e+64)) {
		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((z * (x / y_m)))
    if (x <= (-1.65d+109)) then
        tmp = t_0
    else if (x <= (-1.15d+18)) then
        tmp = t_1
    else if (x <= 2.7d-49) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 3.4d+16) .or. (.not. (x <= 1.72d+64))) 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((z * (x / y_m)));
	double tmp;
	if (x <= -1.65e+109) {
		tmp = t_0;
	} else if (x <= -1.15e+18) {
		tmp = t_1;
	} else if (x <= 2.7e-49) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 3.4e+16) || !(x <= 1.72e+64)) {
		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((z * (x / y_m)))
	tmp = 0
	if x <= -1.65e+109:
		tmp = t_0
	elif x <= -1.15e+18:
		tmp = t_1
	elif x <= 2.7e-49:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 3.4e+16) or not (x <= 1.72e+64):
		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(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -1.65e+109)
		tmp = t_0;
	elseif (x <= -1.15e+18)
		tmp = t_1;
	elseif (x <= 2.7e-49)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 3.4e+16) || !(x <= 1.72e+64))
		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((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -1.65e+109)
		tmp = t_0;
	elseif (x <= -1.15e+18)
		tmp = t_1;
	elseif (x <= 2.7e-49)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 3.4e+16) || ~((x <= 1.72e+64)))
		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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.65e+109], t$95$0, If[LessEqual[x, -1.15e+18], t$95$1, If[LessEqual[x, 2.7e-49], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 3.4e+16], N[Not[LessEqual[x, 1.72e+64]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6499999999999999e109 or 3.4e16 < x < 1.72e64

   1. Initial program 88.6%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in x around inf 95.8%

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

      1. associate-*r/ 95.8%

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

      2. *-commutative 95.8%

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

      3. associate-*r* 95.8%

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

      4. sub-neg 95.8%

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

      5. metadata-eval 95.8%

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

      6. distribute-lft-in 95.8%

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

      7. neg-mul-1 95.8%

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

      8. metadata-eval 95.8%

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

      9. +-commutative 95.8%

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

      10. neg-mul-1 95.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 80.9%

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

   if -1.6499999999999999e109 < x < -1.15e18 or 2.7e-49 < x < 3.4e16 or 1.72e64 < x

   1. Initial program 92.9%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in z around inf 56.5%

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

      1. associate-*r/ 56.5%

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

      2. neg-mul-1 56.5%

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

      3. distribute-rgt-neg-in 56.5%

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

   7. Simplified 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-/l* 69.6%

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

      3. add-sqr-sqrt 36.0%

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

      4. sqrt-unprod 48.5%

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

      5. sqr-neg 48.5%

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

      6. sqrt-unprod 33.4%

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

      7. add-sqr-sqrt 69.6%

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

      8. *-commutative 69.6%

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

   9. Applied egg-rr 69.6%

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

   if -1.15e18 < x < 2.7e-49

   1. Initial program 93.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 75.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+109}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+18}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+16} \lor \neg \left(x \leq 1.72 \cdot 10^{+64}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 18000000000000:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (/ z (/ y_m x)))))
   (if (<= x -1.35e+110)
     t_0
     (if (<= x -1.15e+18)
       t_1
       (if (<= x 5.2e-51)
         (fabs (/ 4.0 y_m))
         (if (<= x 18000000000000.0)
           (fabs (* x (/ z y_m)))
           (if (<= x 2.05e+68) t_0 t_1)))))))

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((z / (y_m / x)));
	double tmp;
	if (x <= -1.35e+110) {
		tmp = t_0;
	} else if (x <= -1.15e+18) {
		tmp = t_1;
	} else if (x <= 5.2e-51) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 18000000000000.0) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 2.05e+68) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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((z / (y_m / x)))
    if (x <= (-1.35d+110)) then
        tmp = t_0
    else if (x <= (-1.15d+18)) then
        tmp = t_1
    else if (x <= 5.2d-51) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 18000000000000.0d0) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 2.05d+68) then
        tmp = t_0
    else
        tmp = t_1
    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((z / (y_m / x)));
	double tmp;
	if (x <= -1.35e+110) {
		tmp = t_0;
	} else if (x <= -1.15e+18) {
		tmp = t_1;
	} else if (x <= 5.2e-51) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 18000000000000.0) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 2.05e+68) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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((z / (y_m / x)))
	tmp = 0
	if x <= -1.35e+110:
		tmp = t_0
	elif x <= -1.15e+18:
		tmp = t_1
	elif x <= 5.2e-51:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 18000000000000.0:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 2.05e+68:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z / Float64(y_m / x)))
	tmp = 0.0
	if (x <= -1.35e+110)
		tmp = t_0;
	elseif (x <= -1.15e+18)
		tmp = t_1;
	elseif (x <= 5.2e-51)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 18000000000000.0)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 2.05e+68)
		tmp = t_0;
	else
		tmp = t_1;
	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((z / (y_m / x)));
	tmp = 0.0;
	if (x <= -1.35e+110)
		tmp = t_0;
	elseif (x <= -1.15e+18)
		tmp = t_1;
	elseif (x <= 5.2e-51)
		tmp = abs((4.0 / y_m));
	elseif (x <= 18000000000000.0)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 2.05e+68)
		tmp = t_0;
	else
		tmp = t_1;
	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[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.35e+110], t$95$0, If[LessEqual[x, -1.15e+18], t$95$1, If[LessEqual[x, 5.2e-51], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 18000000000000.0], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.05e+68], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.35000000000000005e110 or 1.8e13 < x < 2.05e68

   1. Initial program 87.2%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. *-commutative 95.9%

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

      3. associate-*r* 95.9%

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

      4. sub-neg 95.9%

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

      5. metadata-eval 95.9%

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

      6. distribute-lft-in 95.9%

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

      7. neg-mul-1 95.9%

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

      8. metadata-eval 95.9%

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

      9. +-commutative 95.9%

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

      10. neg-mul-1 95.9%

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

      11. associate-/l* 99.8%

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

      12. neg-mul-1 99.8%

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

      13. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 79.9%

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

   if -1.35000000000000005e110 < x < -1.15e18 or 2.05e68 < x

   1. Initial program 92.7%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in z around inf 54.3%

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

      1. associate-*r/ 54.3%

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

      2. neg-mul-1 54.3%

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

      3. distribute-rgt-neg-in 54.3%

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

   7. Simplified 54.3%

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

      1. distribute-rgt-neg-out 54.3%

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

      2. distribute-lft-neg-in 54.3%

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

      3. associate-*r/ 63.6%

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

      4. add-sqr-sqrt 16.5%

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

      5. sqrt-unprod 56.1%

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

      6. sqr-neg 56.1%

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

      7. sqrt-unprod 46.9%

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

      8. add-sqr-sqrt 63.6%

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

      9. *-commutative 63.6%

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

   9. Applied egg-rr 63.6%

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

       1. associate-/r/ 70.5%

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

   11. Applied egg-rr 70.5%

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

   if -1.15e18 < x < 5.2e-51

   1. Initial program 93.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 75.9%

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

   if 5.2e-51 < x < 1.8e13

   1. Initial program 98.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

      3. distribute-rgt-neg-in 68.4%

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

   7. Simplified 68.4%

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

      1. distribute-rgt-neg-out 68.4%

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

      2. distribute-lft-neg-in 68.4%

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

      3. associate-*r/ 68.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 68.3%

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

      6. sqr-neg 68.3%

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

      7. sqrt-unprod 68.0%

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

      8. add-sqr-sqrt 68.3%

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

      9. *-commutative 68.3%

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

   9. Applied egg-rr 68.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 18000000000000:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+68}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 24000000000000:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -1.55e+109)
     t_0
     (if (<= x -1.15e+18)
       t_1
       (if (<= x 2.3e-49)
         (fabs (/ 4.0 y_m))
         (if (<= x 24000000000000.0)
           (fabs (* x (/ z y_m)))
           (if (<= x 1.6e+64) t_0 t_1)))))))

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((z * (x / y_m)));
	double tmp;
	if (x <= -1.55e+109) {
		tmp = t_0;
	} else if (x <= -1.15e+18) {
		tmp = t_1;
	} else if (x <= 2.3e-49) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 24000000000000.0) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 1.6e+64) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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((z * (x / y_m)))
    if (x <= (-1.55d+109)) then
        tmp = t_0
    else if (x <= (-1.15d+18)) then
        tmp = t_1
    else if (x <= 2.3d-49) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 24000000000000.0d0) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 1.6d+64) then
        tmp = t_0
    else
        tmp = t_1
    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((z * (x / y_m)));
	double tmp;
	if (x <= -1.55e+109) {
		tmp = t_0;
	} else if (x <= -1.15e+18) {
		tmp = t_1;
	} else if (x <= 2.3e-49) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 24000000000000.0) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 1.6e+64) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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((z * (x / y_m)))
	tmp = 0
	if x <= -1.55e+109:
		tmp = t_0
	elif x <= -1.15e+18:
		tmp = t_1
	elif x <= 2.3e-49:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 24000000000000.0:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 1.6e+64:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -1.55e+109)
		tmp = t_0;
	elseif (x <= -1.15e+18)
		tmp = t_1;
	elseif (x <= 2.3e-49)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 24000000000000.0)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 1.6e+64)
		tmp = t_0;
	else
		tmp = t_1;
	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((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -1.55e+109)
		tmp = t_0;
	elseif (x <= -1.15e+18)
		tmp = t_1;
	elseif (x <= 2.3e-49)
		tmp = abs((4.0 / y_m));
	elseif (x <= 24000000000000.0)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 1.6e+64)
		tmp = t_0;
	else
		tmp = t_1;
	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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.55e+109], t$95$0, If[LessEqual[x, -1.15e+18], t$95$1, If[LessEqual[x, 2.3e-49], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 24000000000000.0], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.6e+64], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.54999999999999996e109 or 2.4e13 < x < 1.60000000000000009e64

   1. Initial program 87.2%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. *-commutative 95.9%

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

      3. associate-*r* 95.9%

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

      4. sub-neg 95.9%

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

      5. metadata-eval 95.9%

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

      6. distribute-lft-in 95.9%

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

      7. neg-mul-1 95.9%

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

      8. metadata-eval 95.9%

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

      9. +-commutative 95.9%

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

      10. neg-mul-1 95.9%

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

      11. associate-/l* 99.8%

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

      12. neg-mul-1 99.8%

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

      13. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 79.9%

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

   if -1.54999999999999996e109 < x < -1.15e18 or 1.60000000000000009e64 < x

   1. Initial program 92.7%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in z around inf 54.3%

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

      1. associate-*r/ 54.3%

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

      2. neg-mul-1 54.3%

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

      3. distribute-rgt-neg-in 54.3%

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

   7. Simplified 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-/l* 70.4%

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

      3. add-sqr-sqrt 37.1%

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

      4. sqrt-unprod 50.9%

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

      5. sqr-neg 50.9%

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

      6. sqrt-unprod 33.1%

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

      7. add-sqr-sqrt 70.4%

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

      8. *-commutative 70.4%

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

   9. Applied egg-rr 70.4%

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

   if -1.15e18 < x < 2.2999999999999999e-49

   1. Initial program 93.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 75.9%

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

   if 2.2999999999999999e-49 < x < 2.4e13

   1. Initial program 98.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

      3. distribute-rgt-neg-in 68.4%

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

   7. Simplified 68.4%

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

      1. distribute-rgt-neg-out 68.4%

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

      2. distribute-lft-neg-in 68.4%

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

      3. associate-*r/ 68.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 68.3%

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

      6. sqr-neg 68.3%

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

      7. sqrt-unprod 68.0%

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

      8. add-sqr-sqrt 68.3%

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

      9. *-commutative 68.3%

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

   9. Applied egg-rr 68.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+109}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+18}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 24000000000000:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.3e-10)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.3e-10) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	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 <= 1.3d-10) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    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 <= 1.3e-10) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 1.3e-10:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.3e-10)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.3e-10)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.3e-10], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999991e-10

   1. Initial program 91.1%

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

      1. associate-*l/ 94.4%

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

      2. sub-div 98.3%

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

   4. Applied egg-rr 98.3%

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

   if 1.29999999999999991e-10 < y

   1. Initial program 96.2%

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

      1. associate-*l/ 92.6%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.1%

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

      4. un-div-inv 99.2%

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

   4. Applied egg-rr 99.2%

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

Alternative 6: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+104}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -5.3e+104)
   (fabs (/ (* x z) y_m))
   (if (<= z 1.45e+34) (fabs (/ (- -4.0 x) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -5.3e+104) {
		tmp = fabs(((x * z) / y_m));
	} else if (z <= 1.45e+34) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.3d+104)) then
        tmp = abs(((x * z) / y_m))
    else if (z <= 1.45d+34) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -5.3e+104) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (z <= 1.45e+34) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -5.3e+104:
		tmp = math.fabs(((x * z) / y_m))
	elif z <= 1.45e+34:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -5.3e+104)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (z <= 1.45e+34)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -5.3e+104)
		tmp = abs(((x * z) / y_m));
	elseif (z <= 1.45e+34)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -5.3e+104], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.45e+34], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2999999999999999e104

   1. Initial program 86.7%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in z around inf 82.7%

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

      1. associate-*r/ 82.7%

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

      2. neg-mul-1 82.7%

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

      3. distribute-rgt-neg-in 82.7%

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

   7. Simplified 82.7%

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

   if -5.2999999999999999e104 < z < 1.4500000000000001e34

   1. Initial program 95.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in z around 0 91.8%

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

      1. +-commutative 91.8%

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

      2. rem-square-sqrt 43.9%

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

      3. fabs-sqr 43.9%

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

      4. rem-square-sqrt 91.8%

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

      5. fabs-neg 91.8%

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

      6. distribute-neg-frac 91.8%

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

      7. distribute-neg-in 91.8%

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

      8. metadata-eval 91.8%

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

      9. +-commutative 91.8%

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

      10. sub-neg 91.8%

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

      11. rem-square-sqrt 47.3%

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

      12. fabs-sqr 47.3%

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

      13. rem-square-sqrt 91.8%

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

   7. Simplified 91.8%

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

   if 1.4500000000000001e34 < z

   1. Initial program 87.6%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in z around inf 80.5%

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

      1. associate-*r/ 80.5%

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

      2. neg-mul-1 80.5%

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

      3. distribute-rgt-neg-in 80.5%

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

   7. Simplified 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-/l* 86.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 52.5%

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

      5. sqr-neg 52.5%

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

      6. sqrt-unprod 85.9%

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

      7. add-sqr-sqrt 86.2%

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

      8. *-commutative 86.2%

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

   9. Applied egg-rr 86.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+104}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+38}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -3.7e+27)
   (fabs (/ x (/ y_m z)))
   (if (<= z 6.8e+38) (fabs (/ (- -4.0 x) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -3.7e+27) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 6.8e+38) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.7d+27)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 6.8d+38) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -3.7e+27) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 6.8e+38) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -3.7e+27:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 6.8e+38:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -3.7e+27)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 6.8e+38)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -3.7e+27)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 6.8e+38)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -3.7e+27], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 6.8e+38], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.70000000000000002e27

   1. Initial program 89.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in z around inf 72.5%

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

      1. mul-1-neg 72.5%

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

      2. associate-*r/ 73.2%

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

      3. *-commutative 73.2%

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

      4. distribute-rgt-neg-in 73.2%

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

   7. Simplified 73.2%

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

      1. *-commutative 73.2%

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

      2. clear-num 73.3%

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

      3. un-div-inv 74.4%

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

      4. add-sqr-sqrt 28.6%

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

      5. sqrt-unprod 62.1%

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

      6. sqr-neg 62.1%

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

      7. sqrt-unprod 45.5%

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

      8. add-sqr-sqrt 74.4%

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

   9. Applied egg-rr 74.4%

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

   if -3.70000000000000002e27 < z < 6.79999999999999992e38

   1. Initial program 95.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 95.4%

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

      1. +-commutative 95.4%

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

      2. rem-square-sqrt 45.2%

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

      3. fabs-sqr 45.2%

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

      4. rem-square-sqrt 95.4%

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

      5. fabs-neg 95.4%

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

      6. distribute-neg-frac 95.4%

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

      7. distribute-neg-in 95.4%

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

      8. metadata-eval 95.4%

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

      9. +-commutative 95.4%

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

      10. sub-neg 95.4%

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

      11. rem-square-sqrt 49.7%

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

      12. fabs-sqr 49.7%

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

      13. rem-square-sqrt 95.4%

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

   7. Simplified 95.4%

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

   if 6.79999999999999992e38 < z

   1. Initial program 87.6%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in z around inf 80.5%

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

      1. associate-*r/ 80.5%

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

      2. neg-mul-1 80.5%

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

      3. distribute-rgt-neg-in 80.5%

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

   7. Simplified 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-/l* 86.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 52.5%

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

      5. sqr-neg 52.5%

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

      6. sqrt-unprod 85.9%

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

      7. add-sqr-sqrt 86.2%

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

      8. *-commutative 86.2%

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

   9. Applied egg-rr 86.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+38}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 2.7e+16)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (* (- 1.0 z) (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.7d+16) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 2.7e+16:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 2.7e+16)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 2.7e+16)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((1.0 - z) * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 2.7e+16], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7e16

   1. Initial program 92.9%

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

      1. associate-*l/ 97.5%

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

      2. sub-div 99.0%

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

   4. Applied egg-rr 99.0%

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

   if 2.7e16 < x

   1. Initial program 91.6%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in x around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. associate-*r* 88.9%

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

      4. sub-neg 88.9%

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

      5. metadata-eval 88.9%

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

      6. distribute-lft-in 88.9%

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

      7. neg-mul-1 88.9%

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

      8. metadata-eval 88.9%

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

      9. +-commutative 88.9%

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

      10. neg-mul-1 88.9%

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

      11. associate-/l* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

Alternative 9: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.55) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 91.0%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around inf 91.5%

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

      1. associate-*r/ 91.5%

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

      2. *-commutative 91.5%

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

      3. associate-*r* 91.5%

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

      4. sub-neg 91.5%

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

      5. metadata-eval 91.5%

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

      6. distribute-lft-in 91.5%

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

      7. neg-mul-1 91.5%

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

      8. metadata-eval 91.5%

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

      9. +-commutative 91.5%

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

      10. neg-mul-1 91.5%

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

      11. associate-/l* 98.3%

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

      12. neg-mul-1 98.3%

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

      13. unsub-neg 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in z around 0 60.2%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 72.0%

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

4. Final simplification 66.3%

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

Alternative 10: 40.3% 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 92.6%

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

3. Simplified 96.5%

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

5. Taylor expanded in x around 0 39.6%

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

Reproduce

herbie shell --seed 2024089 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))