fabs fraction 1

Percentage Accurate: 91.8% → 99.8%

Time: 9.6s

Alternatives: 16

Speedup: 1.0×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.65 \cdot 10^{-44}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 3.65e-44)
   (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 <= 3.65e-44) {
		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 <= 3.65e-44)
		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, 3.65e-44], 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 < 3.64999999999999993e-44

   1. Initial program 88.4%

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

      1. associate-*l/ 90.9%

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

      2. sub-div 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 3.64999999999999993e-44 < y

   1. Initial program 96.9%

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

      1. fabs-sub 96.9%

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

      2. associate-*l/ 92.2%

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

      3. associate-*r/ 99.9%

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

      4. fma-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+73} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+292}\right):\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m))))))
   (if (or (<= t_0 2e+73) (not (<= t_0 4e+292)))
     (fabs (/ (- (+ x 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((((x + 4.0) / y_m) - (z * (x / y_m))));
	double tmp;
	if ((t_0 <= 2e+73) || !(t_0 <= 4e+292)) {
		tmp = fabs((((x + 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((((x + 4.0d0) / y_m) - (z * (x / y_m))))
    if ((t_0 <= 2d+73) .or. (.not. (t_0 <= 4d+292))) then
        tmp = abs((((x + 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((((x + 4.0) / y_m) - (z * (x / y_m))));
	double tmp;
	if ((t_0 <= 2e+73) || !(t_0 <= 4e+292)) {
		tmp = Math.abs((((x + 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((((x + 4.0) / y_m) - (z * (x / y_m))))
	tmp = 0
	if (t_0 <= 2e+73) or not (t_0 <= 4e+292):
		tmp = math.fabs((((x + 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(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m))))
	tmp = 0.0
	if ((t_0 <= 2e+73) || !(t_0 <= 4e+292))
		tmp = abs(Float64(Float64(Float64(x + 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((((x + 4.0) / y_m) - (z * (x / y_m))));
	tmp = 0.0;
	if ((t_0 <= 2e+73) || ~((t_0 <= 4e+292)))
		tmp = abs((((x + 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[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e+73], N[Not[LessEqual[t$95$0, 4e+292]], $MachinePrecision]], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1.99999999999999997e73 or 4.0000000000000001e292 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 84.9%

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

      1. associate-*l/ 93.8%

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

      2. sub-div 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 1.99999999999999997e73 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 4.0000000000000001e292

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+73} \lor \neg \left(\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 4 \cdot 10^{+292}\right):\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+14} \lor \neg \left(z \leq 7.5 \cdot 10^{-13}\right):\\ \;\;\;\;\left|\frac{4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -3.7e+14) (not (<= z 7.5e-13)))
   (fabs (- (/ 4.0 y_m) (/ x (/ y_m z))))
   (fabs (/ (- x -4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -3.7e+14) || !(z <= 7.5e-13)) {
		tmp = fabs(((4.0 / y_m) - (x / (y_m / z))));
	} else {
		tmp = fabs(((x - -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 ((z <= (-3.7d+14)) .or. (.not. (z <= 7.5d-13))) then
        tmp = abs(((4.0d0 / y_m) - (x / (y_m / z))))
    else
        tmp = abs(((x - (-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 ((z <= -3.7e+14) || !(z <= 7.5e-13)) {
		tmp = Math.abs(((4.0 / y_m) - (x / (y_m / z))));
	} else {
		tmp = Math.abs(((x - -4.0) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -3.7e+14) or not (z <= 7.5e-13):
		tmp = math.fabs(((4.0 / y_m) - (x / (y_m / z))))
	else:
		tmp = math.fabs(((x - -4.0) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -3.7e+14) || !(z <= 7.5e-13))
		tmp = abs(Float64(Float64(4.0 / y_m) - Float64(x / Float64(y_m / z))));
	else
		tmp = abs(Float64(Float64(x - -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 ((z <= -3.7e+14) || ~((z <= 7.5e-13)))
		tmp = abs(((4.0 / y_m) - (x / (y_m / z))));
	else
		tmp = abs(((x - -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[z, -3.7e+14], N[Not[LessEqual[z, 7.5e-13]], $MachinePrecision]], N[Abs[N[(N[(4.0 / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7e14 or 7.5000000000000004e-13 < z

   1. Initial program 85.4%

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

      1. associate-*l/ 81.2%

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

      2. associate-*r/ 80.5%

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

      3. clear-num 80.4%

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

      4. un-div-inv 80.6%

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

   4. Applied egg-rr 80.6%

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

   5. Taylor expanded in x around 0 87.9%

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

   if -3.7e14 < z < 7.5000000000000004e-13

   1. Initial program 94.4%

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

      1. fabs-sub 94.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-*r/ 99.3%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. sub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+14} \lor \neg \left(z \leq 7.5 \cdot 10^{-13}\right):\\ \;\;\;\;\left|\frac{4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e-12)
   (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 <= 5e-12) {
		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 <= 5d-12) 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 <= 5e-12) {
		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 <= 5e-12:
		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 <= 5e-12)
		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 <= 5e-12)
		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, 5e-12], 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 < 4.9999999999999997e-12

   1. Initial program 88.6%

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

      1. associate-*l/ 91.1%

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

      2. sub-div 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 4.9999999999999997e-12 < y

   1. Initial program 96.7%

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

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

Alternative 5: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+47} \lor \neg \left(z \leq 2.6 \cdot 10^{+31}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.45e+47) (not (<= z 2.6e+31)))
   (fabs (* z (/ x y_m)))
   (fabs (/ (- x -4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.45e+47) || !(z <= 2.6e+31)) {
		tmp = fabs((z * (x / y_m)));
	} else {
		tmp = fabs(((x - -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 ((z <= (-1.45d+47)) .or. (.not. (z <= 2.6d+31))) then
        tmp = abs((z * (x / y_m)))
    else
        tmp = abs(((x - (-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 ((z <= -1.45e+47) || !(z <= 2.6e+31)) {
		tmp = Math.abs((z * (x / y_m)));
	} else {
		tmp = Math.abs(((x - -4.0) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.45e+47) or not (z <= 2.6e+31):
		tmp = math.fabs((z * (x / y_m)))
	else:
		tmp = math.fabs(((x - -4.0) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.45e+47) || !(z <= 2.6e+31))
		tmp = abs(Float64(z * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(x - -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 ((z <= -1.45e+47) || ~((z <= 2.6e+31)))
		tmp = abs((z * (x / y_m)));
	else
		tmp = abs(((x - -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[z, -1.45e+47], N[Not[LessEqual[z, 2.6e+31]], $MachinePrecision]], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4499999999999999e47 or 2.6e31 < z

   1. Initial program 85.2%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in z around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. distribute-frac-neg2 69.1%

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

      3. associate-*l/ 74.9%

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

      4. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if -1.4499999999999999e47 < z < 2.6e31

   1. Initial program 94.0%

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

      1. fabs-sub 94.0%

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

      2. associate-*l/ 98.7%

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

      3. associate-*r/ 98.7%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 98.2%

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

      1. associate-*r/ 98.2%

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

      2. distribute-lft-in 98.2%

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

      3. metadata-eval 98.2%

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

      4. neg-mul-1 98.2%

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

      5. sub-neg 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+47} \lor \neg \left(z \leq 2.6 \cdot 10^{+31}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. associate-*l/ 91.2%

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

   2. sub-div 95.1%

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

4. Applied egg-rr 95.1%

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

Alternative 7: 79.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ \mathbf{if}\;x \leq -4:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{x}{\frac{y\_m}{z}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)))
   (if (<= x -4.0)
     (- (* x (/ z y_m)) t_0)
     (if (<= x 2.6e-272)
       (/ (- (+ x 4.0) (* x z)) y_m)
       (+ t_0 (/ x (/ y_m z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -4.0) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= 2.6e-272) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    if (x <= (-4.0d0)) then
        tmp = (x * (z / y_m)) - t_0
    else if (x <= 2.6d-272) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else
        tmp = t_0 + (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 t_0 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -4.0) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= 2.6e-272) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = t_0 + (x / (y_m / z));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	tmp = 0
	if x <= -4.0:
		tmp = (x * (z / y_m)) - t_0
	elif x <= 2.6e-272:
		tmp = ((x + 4.0) - (x * z)) / y_m
	else:
		tmp = t_0 + (x / (y_m / z))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (x <= 2.6e-272)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	else
		tmp = Float64(t_0 + Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	tmp = 0.0;
	if (x <= -4.0)
		tmp = (x * (z / y_m)) - t_0;
	elseif (x <= 2.6e-272)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	else
		tmp = t_0 + (x / (y_m / z));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[x, -4.0], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 2.6e-272], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(t$95$0 + N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 84.6%

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

      1. fabs-sub 84.6%

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

      2. div-inv 84.5%

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

      3. cancel-sign-sub-inv 84.5%

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

      4. +-commutative 84.5%

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

      5. distribute-neg-in 84.5%

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

      6. metadata-eval 84.5%

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

      7. sub-neg 84.5%

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

      8. div-inv 84.6%

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

      9. associate-*l/ 79.9%

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

      10. associate-*r/ 91.6%

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

      11. fma-undefine 94.4%

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

      12. add-sqr-sqrt 51.0%

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

      13. fabs-sqr 51.0%

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

      14. add-sqr-sqrt 51.6%

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

      15. fma-undefine 51.6%

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

      16. associate-*r/ 43.8%

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

      17. associate-*l/ 46.1%

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

      18. div-inv 46.0%

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

   4. Applied egg-rr 51.6%

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

   if -4 < x < 2.59999999999999992e-272

   1. Initial program 92.8%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 40.2%

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

      4. associate-*l/ 41.3%

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

      5. associate-*r/ 34.4%

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

   4. Applied egg-rr 34.4%

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

      1. clear-num 34.4%

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

      2. un-div-inv 34.6%

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

   6. Applied egg-rr 34.6%

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

   7. Taylor expanded in y around 0 41.3%

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

   if 2.59999999999999992e-272 < x

   1. Initial program 92.5%

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

      1. add-sqr-sqrt 42.7%

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

      2. fabs-sqr 42.7%

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

      3. add-sqr-sqrt 43.6%

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

      4. associate-*l/ 43.6%

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

      5. associate-*r/ 45.4%

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

   4. Applied egg-rr 45.4%

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

      1. add-sqr-sqrt 21.7%

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

      2. sqrt-unprod 35.3%

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

      3. sqr-neg 35.3%

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

      4. sqrt-unprod 20.8%

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

      5. add-sqr-sqrt 42.7%

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

      6. distribute-frac-neg 42.7%

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

      7. distribute-rgt-neg-in 42.7%

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

      8. distribute-lft-neg-in 42.7%

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

      9. clear-num 42.7%

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

      10. un-div-inv 42.7%

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

   6. Applied egg-rr 42.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{x}{\frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x \cdot z + \left(-4 - x\right)}{y\_m}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.0)
   (/ (+ (* x z) (- -4.0 x)) y_m)
   (if (<= x 6.5e+23) (/ (- 4.0 (* x z)) y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = ((x * z) + (-4.0 - x)) / y_m;
	} else if (x <= 6.5e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = 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 <= (-4.0d0)) then
        tmp = ((x * z) + ((-4.0d0) - x)) / y_m
    else if (x <= 6.5d+23) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = 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 <= -4.0) {
		tmp = ((x * z) + (-4.0 - x)) / y_m;
	} else if (x <= 6.5e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.0:
		tmp = ((x * z) + (-4.0 - x)) / y_m
	elif x <= 6.5e+23:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(Float64(x * z) + Float64(-4.0 - x)) / y_m);
	elseif (x <= 6.5e+23)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = 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 <= -4.0)
		tmp = ((x * z) + (-4.0 - x)) / y_m;
	elseif (x <= 6.5e+23)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -4.0], N[(N[(N[(x * z), $MachinePrecision] + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 6.5e+23], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 84.6%

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

      1. associate-*l/ 79.9%

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

      2. sub-div 88.2%

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

   4. Applied egg-rr 88.2%

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

   6. Applied egg-rr 49.4%

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

   if -4 < x < 6.4999999999999996e23

   1. Initial program 94.4%

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

      1. add-sqr-sqrt 39.4%

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

      2. fabs-sqr 39.4%

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

      3. add-sqr-sqrt 40.6%

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

      4. sub-neg 40.6%

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

      5. +-commutative 40.6%

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

      6. associate-*l/ 42.7%

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

      7. associate-*r/ 38.7%

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

      8. frac-2neg 38.7%

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

      9. +-commutative 38.7%

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

      10. distribute-neg-in 38.7%

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

      11. metadata-eval 38.7%

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

      12. sub-neg 38.7%

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

      13. distribute-frac-neg2 38.7%

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

      14. distribute-neg-in 38.7%

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

      15. fma-undefine 38.7%

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

      16. neg-sub0 38.7%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.7%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 42.7%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 40.6%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 42.7%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in x around 0 42.1%

      \[\leadsto 0 - \frac{x \cdot z - \color{blue}{4}}{y} \]

   if 6.4999999999999996e23 < x

   1. Initial program 88.0%

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

      1. fabs-sub 88.0%

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

      2. associate-*l/ 84.4%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.9%

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

      5. distribute-neg-frac 95.9%

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

      6. +-commutative 95.9%

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

      7. distribute-neg-in 95.9%

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

      8. unsub-neg 95.9%

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

      9. metadata-eval 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in z around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. distribute-lft-in 75.7%

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

      3. metadata-eval 75.7%

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

      4. neg-mul-1 75.7%

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

      5. sub-neg 75.7%

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

   7. Simplified 75.7%

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

      1. add-sqr-sqrt 75.3%

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

      2. sqrt-unprod 64.7%

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

      3. neg-fabs 64.7%

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

      4. neg-fabs 64.7%

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

      5. sqr-abs 64.7%

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

      6. distribute-frac-neg2 64.7%

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

      7. sub-neg 64.7%

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

      8. metadata-eval 64.7%

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

      9. distribute-neg-in 64.7%

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

      10. +-commutative 64.7%

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

      11. frac-2neg 64.7%

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

      12. distribute-frac-neg2 64.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 64.7%

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

      14. metadata-eval 64.7%

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

      15. distribute-neg-in 64.7%

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

      16. +-commutative 64.7%

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

      17. frac-2neg 64.7%

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

      18. sqrt-unprod 38.4%

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

      19. add-sqr-sqrt 39.1%

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

      20. clear-num 39.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 39.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 39.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x \cdot z + \left(-4 - x\right)}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+51}:\\ \;\;\;\;\frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.22e+51)
   (/ (- x) y_m)
   (if (<= x 6.5e+23) (/ (- 4.0 (* x z)) y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.22e+51) {
		tmp = -x / y_m;
	} else if (x <= 6.5e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = 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 <= (-1.22d+51)) then
        tmp = -x / y_m
    else if (x <= 6.5d+23) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = 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 <= -1.22e+51) {
		tmp = -x / y_m;
	} else if (x <= 6.5e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.22e+51:
		tmp = -x / y_m
	elif x <= 6.5e+23:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.22e+51)
		tmp = Float64(Float64(-x) / y_m);
	elseif (x <= 6.5e+23)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = 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 <= -1.22e+51)
		tmp = -x / y_m;
	elseif (x <= 6.5e+23)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.22e+51], N[((-x) / y$95$m), $MachinePrecision], If[LessEqual[x, 6.5e+23], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.22000000000000005e51

   1. Initial program 82.9%

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

      1. fabs-sub 82.9%

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

      2. associate-*l/ 77.2%

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

      3. associate-*r/ 91.5%

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

      4. fma-neg 94.9%

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

      5. distribute-neg-frac 94.9%

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

      6. +-commutative 94.9%

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

      7. distribute-neg-in 94.9%

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

      8. unsub-neg 94.9%

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

      9. metadata-eval 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in z around 0 69.7%

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

      1. associate-*r/ 69.7%

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

      2. distribute-lft-in 69.7%

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

      3. metadata-eval 69.7%

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

      4. neg-mul-1 69.7%

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

      5. sub-neg 69.7%

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

   7. Simplified 69.7%

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

      1. add-sqr-sqrt 36.2%

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

      2. fabs-sqr 36.2%

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

      3. add-sqr-sqrt 36.8%

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

      4. div-inv 36.7%

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

   9. Applied egg-rr 36.7%

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

   10. Taylor expanded in x around inf 36.8%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   11. Step-by-step derivation

       1. mul-1-neg 36.8%

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

       2. distribute-frac-neg2 36.8%

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

   12. Simplified 36.8%

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

   if -1.22000000000000005e51 < x < 6.4999999999999996e23

   1. Initial program 94.2%

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

      1. add-sqr-sqrt 40.6%

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

      2. fabs-sqr 40.6%

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

      3. add-sqr-sqrt 41.8%

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

      4. sub-neg 41.8%

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

      5. +-commutative 41.8%

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

      6. associate-*l/ 43.7%

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

      7. associate-*r/ 40.0%

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

      8. frac-2neg 40.0%

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

      9. +-commutative 40.0%

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

      10. distribute-neg-in 40.0%

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

      11. metadata-eval 40.0%

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

      12. sub-neg 40.0%

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

      13. distribute-frac-neg2 40.0%

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

      14. distribute-neg-in 40.0%

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

      15. fma-undefine 40.0%

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

      16. neg-sub0 40.0%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 40.0%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 43.7%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 41.8%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in x around 0 42.7%

      \[\leadsto 0 - \frac{x \cdot z - \color{blue}{4}}{y} \]

   if 6.4999999999999996e23 < x

   1. Initial program 88.0%

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

      1. fabs-sub 88.0%

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

      2. associate-*l/ 84.4%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.9%

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

      5. distribute-neg-frac 95.9%

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

      6. +-commutative 95.9%

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

      7. distribute-neg-in 95.9%

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

      8. unsub-neg 95.9%

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

      9. metadata-eval 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in z around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. distribute-lft-in 75.7%

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

      3. metadata-eval 75.7%

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

      4. neg-mul-1 75.7%

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

      5. sub-neg 75.7%

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

   7. Simplified 75.7%

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

      1. add-sqr-sqrt 75.3%

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

      2. sqrt-unprod 64.7%

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

      3. neg-fabs 64.7%

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

      4. neg-fabs 64.7%

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

      5. sqr-abs 64.7%

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

      6. distribute-frac-neg2 64.7%

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

      7. sub-neg 64.7%

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

      8. metadata-eval 64.7%

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

      9. distribute-neg-in 64.7%

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

      10. +-commutative 64.7%

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

      11. frac-2neg 64.7%

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

      12. distribute-frac-neg2 64.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 64.7%

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

      14. metadata-eval 64.7%

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

      15. distribute-neg-in 64.7%

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

      16. +-commutative 64.7%

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

      17. frac-2neg 64.7%

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

      18. sqrt-unprod 38.4%

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

      19. add-sqr-sqrt 39.1%

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

      20. clear-num 39.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 39.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 39.1%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+51}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (x * (z / y_m)) - ((x + 4.0) / y_m);
	} else {
		tmp = ((x + 4.0) - (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 (x <= (-4.0d0)) then
        tmp = (x * (z / y_m)) - ((x + 4.0d0) / y_m)
    else
        tmp = ((x + 4.0d0) - (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 (x <= -4.0) {
		tmp = (x * (z / y_m)) - ((x + 4.0) / y_m);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 84.6%

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

      1. fabs-sub 84.6%

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

      2. div-inv 84.5%

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

      3. cancel-sign-sub-inv 84.5%

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

      4. +-commutative 84.5%

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

      5. distribute-neg-in 84.5%

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

      6. metadata-eval 84.5%

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

      7. sub-neg 84.5%

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

      8. div-inv 84.6%

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

      9. associate-*l/ 79.9%

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

      10. associate-*r/ 91.6%

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

      11. fma-undefine 94.4%

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

      12. add-sqr-sqrt 51.0%

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

      13. fabs-sqr 51.0%

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

      14. add-sqr-sqrt 51.6%

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

      15. fma-undefine 51.6%

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

      16. associate-*r/ 43.8%

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

      17. associate-*l/ 46.1%

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

      18. div-inv 46.0%

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

   4. Applied egg-rr 51.6%

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

   if -4 < x

   1. Initial program 92.7%

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

      1. add-sqr-sqrt 41.1%

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

      2. fabs-sqr 41.1%

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

      3. add-sqr-sqrt 42.2%

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

      4. associate-*l/ 42.7%

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

      5. associate-*r/ 40.8%

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

   4. Applied egg-rr 40.8%

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

      1. clear-num 40.8%

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

      2. un-div-inv 40.8%

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

   6. Applied egg-rr 40.8%

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

   7. Taylor expanded in y around 0 43.8%

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

4. Final simplification 46.0%

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

Alternative 11: 80.1% accurate, 7.9× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = ((x * z) + (-4.0 - x)) / y_m;
	} else {
		tmp = ((x + 4.0) - (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 (x <= (-4.0d0)) then
        tmp = ((x * z) + ((-4.0d0) - x)) / y_m
    else
        tmp = ((x + 4.0d0) - (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 (x <= -4.0) {
		tmp = ((x * z) + (-4.0 - x)) / y_m;
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 84.6%

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

      1. associate-*l/ 79.9%

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

      2. sub-div 88.2%

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

   4. Applied egg-rr 88.2%

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

   6. Applied egg-rr 49.4%

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

   if -4 < x

   1. Initial program 92.7%

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

      1. add-sqr-sqrt 41.1%

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

      2. fabs-sqr 41.1%

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

      3. add-sqr-sqrt 42.2%

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

      4. associate-*l/ 42.7%

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

      5. associate-*r/ 40.8%

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

   4. Applied egg-rr 40.8%

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

      1. clear-num 40.8%

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

      2. un-div-inv 40.8%

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

   6. Applied egg-rr 40.8%

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

   7. Taylor expanded in y around 0 43.8%

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

4. Final simplification 45.3%

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

Alternative 12: 70.0% accurate, 8.5× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -10.5) {
		tmp = -x / y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = 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 <= (-10.5d0)) then
        tmp = -x / y_m
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = 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 <= -10.5) {
		tmp = -x / y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -10.5)
		tmp = Float64(Float64(-x) / y_m);
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = 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 <= -10.5)
		tmp = -x / y_m;
	elseif (x <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -10.5

   1. Initial program 84.4%

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

      1. fabs-sub 84.4%

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

      2. associate-*l/ 79.6%

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

      3. associate-*r/ 91.5%

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

      4. fma-neg 94.3%

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

      5. distribute-neg-frac 94.3%

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

      6. +-commutative 94.3%

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

      7. distribute-neg-in 94.3%

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

      8. unsub-neg 94.3%

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

      9. metadata-eval 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in z around 0 65.6%

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

      1. associate-*r/ 65.6%

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

      2. distribute-lft-in 65.6%

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

      3. metadata-eval 65.6%

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

      4. neg-mul-1 65.6%

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

      5. sub-neg 65.6%

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

   7. Simplified 65.6%

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

      1. add-sqr-sqrt 35.9%

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

      2. fabs-sqr 35.9%

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

      3. add-sqr-sqrt 36.5%

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

      4. div-inv 36.4%

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

   9. Applied egg-rr 36.4%

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

   10. Taylor expanded in x around inf 36.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   11. Step-by-step derivation

       1. mul-1-neg 36.5%

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

       2. distribute-frac-neg2 36.5%

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

   12. Simplified 36.5%

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

   if -10.5 < x < 4

   1. Initial program 94.3%

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

      1. add-sqr-sqrt 39.5%

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

      2. fabs-sqr 39.5%

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

      3. add-sqr-sqrt 40.8%

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

      4. associate-*l/ 42.9%

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

      5. associate-*r/ 38.8%

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

   4. Applied egg-rr 38.8%

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

      1. clear-num 38.8%

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

      2. un-div-inv 38.9%

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

   6. Applied egg-rr 38.9%

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

   7. Taylor expanded in x around 0 31.1%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 88.8%

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

      1. fabs-sub 88.8%

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

      2. associate-*l/ 85.5%

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

      3. associate-*r/ 92.5%

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

      4. fma-neg 96.2%

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

      5. distribute-neg-frac 96.2%

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

      6. +-commutative 96.2%

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

      7. distribute-neg-in 96.2%

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

      8. unsub-neg 96.2%

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

      9. metadata-eval 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in z around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. distribute-lft-in 72.2%

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

      3. metadata-eval 72.2%

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

      4. neg-mul-1 72.2%

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

      5. sub-neg 72.2%

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

   7. Simplified 72.2%

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

      1. add-sqr-sqrt 71.8%

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

      2. sqrt-unprod 61.9%

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

      3. neg-fabs 61.9%

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

      4. neg-fabs 61.9%

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

      5. sqr-abs 61.9%

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

      6. distribute-frac-neg2 61.9%

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

      7. sub-neg 61.9%

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

      8. metadata-eval 61.9%

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

      9. distribute-neg-in 61.9%

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

      10. +-commutative 61.9%

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

      11. frac-2neg 61.9%

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

      12. distribute-frac-neg2 61.9%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 61.9%

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

      14. metadata-eval 61.9%

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

      15. distribute-neg-in 61.9%

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

      16. +-commutative 61.9%

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

      17. frac-2neg 61.9%

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

      18. sqrt-unprod 35.8%

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

      19. add-sqr-sqrt 36.4%

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

      20. clear-num 36.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 36.4%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.0% accurate, 11.1× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-4.0 - x) / y_m;
	} else {
		tmp = (x + 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 <= (-4.0d0)) then
        tmp = ((-4.0d0) - x) / y_m
    else
        tmp = (x + 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 <= -4.0) {
		tmp = (-4.0 - x) / y_m;
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(-4.0 - x) / y_m);
	else
		tmp = Float64(Float64(x + 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 <= -4.0)
		tmp = (-4.0 - x) / y_m;
	else
		tmp = (x + 4.0) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 84.6%

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

      1. fabs-sub 84.6%

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

      2. associate-*l/ 79.9%

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

      3. associate-*r/ 91.6%

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

      4. fma-neg 94.4%

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

      5. distribute-neg-frac 94.4%

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

      6. +-commutative 94.4%

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

      7. distribute-neg-in 94.4%

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

      8. unsub-neg 94.4%

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

      9. metadata-eval 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in z around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. distribute-lft-in 66.1%

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

      3. metadata-eval 66.1%

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

      4. neg-mul-1 66.1%

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

      5. sub-neg 66.1%

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

   7. Simplified 66.1%

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

      1. add-sqr-sqrt 35.4%

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

      2. fabs-sqr 35.4%

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

      3. add-sqr-sqrt 36.0%

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

      4. div-inv 35.9%

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

   9. Applied egg-rr 35.9%

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

   10. Taylor expanded in x around 0 36.0%

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

       1. mul-1-neg 36.0%

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

       2. neg-sub0 36.0%

          \[\leadsto \color{blue}{\left(0 - \frac{x}{y}\right)} - 4 \cdot \frac{1}{y} \]

       3. associate-*r/ 36.0%

          \[\leadsto \left(0 - \frac{x}{y}\right) - \color{blue}{\frac{4 \cdot 1}{y}} \]

       4. metadata-eval 36.0%

          \[\leadsto \left(0 - \frac{x}{y}\right) - \frac{\color{blue}{4}}{y} \]

       5. associate--r+ 36.0%

          \[\leadsto \color{blue}{0 - \left(\frac{x}{y} + \frac{4}{y}\right)} \]

       6. +-commutative 36.0%

          \[\leadsto 0 - \color{blue}{\left(\frac{4}{y} + \frac{x}{y}\right)} \]

       7. associate--r+ 36.0%

          \[\leadsto \color{blue}{\left(0 - \frac{4}{y}\right) - \frac{x}{y}} \]

       8. neg-sub0 36.0%

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

       9. distribute-neg-frac 36.0%

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

       10. metadata-eval 36.0%

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

       11. div-sub 36.0%

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

   12. Simplified 36.0%

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

   if -4 < x

   1. Initial program 92.7%

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

      1. fabs-sub 92.7%

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

      2. associate-*l/ 95.7%

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

      3. associate-*r/ 90.7%

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

      4. fma-neg 91.8%

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

      5. distribute-neg-frac 91.8%

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

      6. +-commutative 91.8%

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

      7. distribute-neg-in 91.8%

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

      8. unsub-neg 91.8%

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

      9. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in z around 0 74.2%

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

      1. associate-*r/ 74.2%

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

      2. distribute-lft-in 74.2%

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

      3. metadata-eval 74.2%

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

      4. neg-mul-1 74.2%

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

      5. sub-neg 74.2%

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

   7. Simplified 74.2%

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

      1. add-sqr-sqrt 73.7%

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

      2. sqrt-unprod 47.6%

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

      3. neg-fabs 47.6%

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

      4. neg-fabs 47.6%

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

      5. sqr-abs 47.6%

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

      6. distribute-frac-neg2 47.6%

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

      7. sub-neg 47.6%

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

      8. metadata-eval 47.6%

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

      9. distribute-neg-in 47.6%

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

      10. +-commutative 47.6%

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

      11. frac-2neg 47.6%

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

      12. distribute-frac-neg2 47.6%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 47.6%

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

      14. metadata-eval 47.6%

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

      15. distribute-neg-in 47.6%

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

      16. +-commutative 47.6%

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

      17. frac-2neg 47.6%

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

      18. sqrt-unprod 32.1%

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

      19. add-sqr-sqrt 33.2%

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

   9. Applied egg-rr 33.2%

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

Alternative 14: 70.8% accurate, 11.1× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = -x / y_m;
	} else {
		tmp = (x + 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 <= (-4.0d0)) then
        tmp = -x / y_m
    else
        tmp = (x + 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 <= -4.0) {
		tmp = -x / y_m;
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(-x) / y_m);
	else
		tmp = Float64(Float64(x + 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 <= -4.0)
		tmp = -x / y_m;
	else
		tmp = (x + 4.0) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 84.6%

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

      1. fabs-sub 84.6%

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

      2. associate-*l/ 79.9%

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

      3. associate-*r/ 91.6%

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

      4. fma-neg 94.4%

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

      5. distribute-neg-frac 94.4%

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

      6. +-commutative 94.4%

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

      7. distribute-neg-in 94.4%

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

      8. unsub-neg 94.4%

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

      9. metadata-eval 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in z around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. distribute-lft-in 66.1%

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

      3. metadata-eval 66.1%

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

      4. neg-mul-1 66.1%

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

      5. sub-neg 66.1%

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

   7. Simplified 66.1%

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

      1. add-sqr-sqrt 35.4%

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

      2. fabs-sqr 35.4%

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

      3. add-sqr-sqrt 36.0%

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

      4. div-inv 35.9%

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

   9. Applied egg-rr 35.9%

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

   10. Taylor expanded in x around inf 36.0%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   11. Step-by-step derivation

       1. mul-1-neg 36.0%

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

       2. distribute-frac-neg2 36.0%

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

   12. Simplified 36.0%

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

   if -4 < x

   1. Initial program 92.7%

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

      1. fabs-sub 92.7%

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

      2. associate-*l/ 95.7%

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

      3. associate-*r/ 90.7%

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

      4. fma-neg 91.8%

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

      5. distribute-neg-frac 91.8%

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

      6. +-commutative 91.8%

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

      7. distribute-neg-in 91.8%

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

      8. unsub-neg 91.8%

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

      9. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in z around 0 74.2%

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

      1. associate-*r/ 74.2%

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

      2. distribute-lft-in 74.2%

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

      3. metadata-eval 74.2%

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

      4. neg-mul-1 74.2%

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

      5. sub-neg 74.2%

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

   7. Simplified 74.2%

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

      1. add-sqr-sqrt 73.7%

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

      2. sqrt-unprod 47.6%

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

      3. neg-fabs 47.6%

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

      4. neg-fabs 47.6%

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

      5. sqr-abs 47.6%

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

      6. distribute-frac-neg2 47.6%

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

      7. sub-neg 47.6%

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

      8. metadata-eval 47.6%

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

      9. distribute-neg-in 47.6%

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

      10. +-commutative 47.6%

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

      11. frac-2neg 47.6%

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

      12. distribute-frac-neg2 47.6%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 47.6%

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

      14. metadata-eval 47.6%

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

      15. distribute-neg-in 47.6%

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

      16. +-commutative 47.6%

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

      17. frac-2neg 47.6%

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

      18. sqrt-unprod 32.1%

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

      19. add-sqr-sqrt 33.2%

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

   9. Applied egg-rr 33.2%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.5% accurate, 13.8× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = 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 <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = 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 <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = 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 <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 90.8%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 39.9%

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

      4. associate-*l/ 40.4%

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

      5. associate-*r/ 39.1%

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

   4. Applied egg-rr 39.1%

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

      1. clear-num 39.1%

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

      2. un-div-inv 39.2%

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

   6. Applied egg-rr 39.2%

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

   7. Taylor expanded in x around 0 21.3%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 88.8%

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

      1. fabs-sub 88.8%

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

      2. associate-*l/ 85.5%

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

      3. associate-*r/ 92.5%

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

      4. fma-neg 96.2%

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

      5. distribute-neg-frac 96.2%

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

      6. +-commutative 96.2%

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

      7. distribute-neg-in 96.2%

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

      8. unsub-neg 96.2%

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

      9. metadata-eval 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in z around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. distribute-lft-in 72.2%

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

      3. metadata-eval 72.2%

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

      4. neg-mul-1 72.2%

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

      5. sub-neg 72.2%

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

   7. Simplified 72.2%

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

      1. add-sqr-sqrt 71.8%

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

      2. sqrt-unprod 61.9%

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

      3. neg-fabs 61.9%

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

      4. neg-fabs 61.9%

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

      5. sqr-abs 61.9%

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

      6. distribute-frac-neg2 61.9%

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

      7. sub-neg 61.9%

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

      8. metadata-eval 61.9%

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

      9. distribute-neg-in 61.9%

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

      10. +-commutative 61.9%

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

      11. frac-2neg 61.9%

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

      12. distribute-frac-neg2 61.9%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 61.9%

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

      14. metadata-eval 61.9%

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

      15. distribute-neg-in 61.9%

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

      16. +-commutative 61.9%

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

      17. frac-2neg 61.9%

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

      18. sqrt-unprod 35.8%

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

      19. add-sqr-sqrt 36.4%

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

      20. clear-num 36.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 36.4%

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

Alternative 16: 41.5% accurate, 37.0× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return 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 = 4.0d0 / y_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. add-sqr-sqrt 40.4%

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

   2. fabs-sqr 40.4%

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

   3. add-sqr-sqrt 41.3%

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

   4. associate-*l/ 41.0%

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

   5. associate-*r/ 40.7%

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

4. Applied egg-rr 40.7%

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

   1. clear-num 40.7%

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

   2. un-div-inv 40.8%

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

6. Applied egg-rr 40.8%

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

7. Taylor expanded in x around 0 17.5%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))