fabs fraction 1

Percentage Accurate: 91.5% → 99.4%

Time: 37.2s

Alternatives: 17

Speedup: 1.0×

• 21.0% 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: 91.5% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.05e-78)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.05e-78) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.05d-78) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.05e-78) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 1.05e-78:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.05e-78)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.05e-78)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.05e-78], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e-78

   1. Initial program 89.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. sub-div 97.1%

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

   4. Applied egg-rr 97.1%

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

   if 1.05e-78 < y

   1. Initial program 97.6%

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

      1. associate-*l/ 95.7%

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

      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 2: 91.9% accurate, 0.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -4.0) (not (<= x 4e-15)))
   (fabs (* (/ x y_m) (- 1.0 z)))
   (/ 1.0 (/ y_m (+ x (+ 4.0 (* x z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -4.0) || !(x <= 4e-15)) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = 1.0 / (y_m / (x + (4.0 + (x * 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 ((x <= (-4.0d0)) .or. (.not. (x <= 4d-15))) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = 1.0d0 / (y_m / (x + (4.0d0 + (x * 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 ((x <= -4.0) || !(x <= 4e-15)) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = 1.0 / (y_m / (x + (4.0 + (x * z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -4.0) or not (x <= 4e-15):
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = 1.0 / (y_m / (x + (4.0 + (x * z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -4.0) || !(x <= 4e-15))
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = Float64(1.0 / Float64(y_m / Float64(x + Float64(4.0 + Float64(x * z)))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -4.0) || ~((x <= 4e-15)))
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = 1.0 / (y_m / (x + (4.0 + (x * z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -4.0], N[Not[LessEqual[x, 4e-15]], $MachinePrecision]], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(y$95$m / N[(x + N[(4.0 + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4 or 4.0000000000000003e-15 < x

   1. Initial program 88.2%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in x around inf 90.7%

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

      1. mul-1-neg 90.7%

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

      2. associate-/l* 97.6%

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

      3. distribute-rgt-neg-in 97.6%

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

      4. mul-1-neg 97.6%

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

      5. associate-*r/ 97.6%

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

      6. sub-neg 97.6%

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

      7. metadata-eval 97.6%

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

      8. distribute-lft-in 97.6%

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

      9. neg-mul-1 97.6%

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

      10. metadata-eval 97.6%

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

      11. +-commutative 97.6%

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

      12. neg-mul-1 97.6%

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

      13. associate-/l* 90.7%

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

      14. *-commutative 90.7%

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

      15. associate-/l* 97.8%

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

      16. neg-mul-1 97.8%

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

      17. unsub-neg 97.8%

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

   7. Simplified 97.8%

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

   if -4 < x < 4.0000000000000003e-15

   1. Initial program 95.7%

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

      1. associate-*l/ 99.9%

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

      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%

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

      1. add-sqr-sqrt 49.9%

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

      2. fabs-sqr 49.9%

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

      3. add-sqr-sqrt 51.3%

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

      4. clear-num 51.3%

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

      5. *-commutative 51.3%

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

      6. cancel-sign-sub-inv 51.3%

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

      7. *-commutative 51.3%

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

      8. add-sqr-sqrt 30.4%

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

      9. sqrt-unprod 43.1%

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

      10. sqr-neg 43.1%

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

      11. sqrt-unprod 21.1%

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

      12. add-sqr-sqrt 54.1%

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

      13. associate-+l+ 54.1%

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

   6. Applied egg-rr 54.1%

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

4. Final simplification 74.4%

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

Alternative 3: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+97} \lor \neg \left(z \leq 3.9 \cdot 10^{+15}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.65e+97) (not (<= z 3.9e+15)))
   (fabs (* z (/ x y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.65e+97) || !(z <= 3.9e+15)) {
		tmp = fabs((z * (x / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.65d+97)) .or. (.not. (z <= 3.9d+15))) then
        tmp = abs((z * (x / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.65e+97) || !(z <= 3.9e+15)) {
		tmp = Math.abs((z * (x / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.65e+97) or not (z <= 3.9e+15):
		tmp = math.fabs((z * (x / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.65e+97) || !(z <= 3.9e+15))
		tmp = abs(Float64(z * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.65e+97) || ~((z <= 3.9e+15)))
		tmp = abs((z * (x / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -1.65e+97], N[Not[LessEqual[z, 3.9e+15]], $MachinePrecision]], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6500000000000001e97 or 3.9e15 < z

   1. Initial program 88.6%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in z around inf 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-frac-neg2 74.4%

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

      3. associate-*l/ 78.1%

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

      4. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   if -1.6500000000000001e97 < z < 3.9e15

   1. Initial program 94.3%

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

      1. fabs-sub 94.3%

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

      2. associate-*l/ 97.5%

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

      3. associate-*r/ 97.5%

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

      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 96.5%

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

      1. associate-*r/ 96.5%

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

      2. distribute-lft-in 96.5%

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

      3. metadata-eval 96.5%

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

      4. neg-mul-1 96.5%

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

      5. sub-neg 96.5%

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

   7. Simplified 96.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+97} \lor \neg \left(z \leq 3.9 \cdot 10^{+15}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+97}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.65e+97)
   (fabs (* z (/ x y_m)))
   (if (<= z 1.9e+15) (fabs (/ (- -4.0 x) y_m)) (fabs (/ (* x z) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.65e+97) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 1.9e+15) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((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 (z <= (-1.65d+97)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 1.9d+15) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((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 (z <= -1.65e+97) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 1.9e+15) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((x * z) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.65e+97:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 1.9e+15:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((x * z) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.65e+97)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 1.9e+15)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(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 (z <= -1.65e+97)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 1.9e+15)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((x * z) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.65e+97], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.9e+15], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6500000000000001e97

   1. Initial program 92.6%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in z around inf 70.0%

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

      1. mul-1-neg 70.0%

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

      2. distribute-frac-neg2 70.0%

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

      3. associate-*l/ 77.7%

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

      4. *-commutative 77.7%

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

   7. Simplified 77.7%

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

   if -1.6500000000000001e97 < z < 1.9e15

   1. Initial program 94.3%

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

      1. fabs-sub 94.3%

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

      2. associate-*l/ 97.5%

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

      3. associate-*r/ 97.5%

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

      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 96.5%

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

      1. associate-*r/ 96.5%

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

      2. distribute-lft-in 96.5%

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

      3. metadata-eval 96.5%

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

      4. neg-mul-1 96.5%

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

      5. sub-neg 96.5%

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

   7. Simplified 96.5%

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

   if 1.9e15 < z

   1. Initial program 84.7%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around inf 78.7%

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

      1. associate-*r/ 78.7%

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

      2. neg-mul-1 78.7%

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

      3. distribute-rgt-neg-in 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+97}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.55e+45)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (fabs (/ (- (+ 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 <= -1.55e+45) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = fabs((((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 <= (-1.55d+45)) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = abs((((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 <= -1.55e+45) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = Math.abs((((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 <= -1.55e+45:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = math.fabs((((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 <= -1.55e+45)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = abs(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 <= -1.55e+45)
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = abs((((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, -1.55e+45], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.54999999999999994e45

   1. Initial program 80.7%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf 88.4%

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

      1. mul-1-neg 88.4%

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

      2. associate-/l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. mul-1-neg 99.8%

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

      5. associate-*r/ 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. neg-mul-1 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

      12. neg-mul-1 99.8%

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

      13. associate-/l* 88.4%

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

      14. *-commutative 88.4%

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

      15. associate-/l* 100.0%

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

      16. neg-mul-1 100.0%

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

      17. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.54999999999999994e45 < x

   1. Initial program 95.5%

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

      1. associate-*l/ 98.0%

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

      2. sub-div 99.0%

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

   4. Applied egg-rr 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z + -1}}\\ \mathbf{elif}\;x \leq 10.2:\\ \;\;\;\;\frac{\left(--4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.2e-8)
   (/ x (/ y_m (+ z -1.0)))
   (if (<= x 10.2) (/ (- (- -4.0) (* x z)) y_m) (/ x (/ y_m (+ z 1.0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.2e-8) {
		tmp = x / (y_m / (z + -1.0));
	} else if (x <= 10.2) {
		tmp = (-(-4.0) - (x * z)) / y_m;
	} else {
		tmp = x / (y_m / (z + 1.0));
	}
	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) :: tmp
    if (x <= (-8.2d-8)) then
        tmp = x / (y_m / (z + (-1.0d0)))
    else if (x <= 10.2d0) then
        tmp = (-(-4.0d0) - (x * z)) / y_m
    else
        tmp = x / (y_m / (z + 1.0d0))
    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 <= -8.2e-8) {
		tmp = x / (y_m / (z + -1.0));
	} else if (x <= 10.2) {
		tmp = (-(-4.0) - (x * z)) / y_m;
	} else {
		tmp = x / (y_m / (z + 1.0));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -8.2e-8:
		tmp = x / (y_m / (z + -1.0))
	elif x <= 10.2:
		tmp = (-(-4.0) - (x * z)) / y_m
	else:
		tmp = x / (y_m / (z + 1.0))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -8.2e-8)
		tmp = Float64(x / Float64(y_m / Float64(z + -1.0)));
	elseif (x <= 10.2)
		tmp = Float64(Float64(Float64(-(-4.0)) - Float64(x * z)) / y_m);
	else
		tmp = Float64(x / Float64(y_m / Float64(z + 1.0)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -8.2e-8)
		tmp = x / (y_m / (z + -1.0));
	elseif (x <= 10.2)
		tmp = (-(-4.0) - (x * z)) / y_m;
	else
		tmp = x / (y_m / (z + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -8.2e-8], N[(x / N[(y$95$m / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 10.2], N[(N[((--4.0) - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x / N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.20000000000000063e-8

   1. Initial program 84.0%

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

      1. add-sqr-sqrt 46.1%

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

      2. fabs-sqr 46.1%

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

      3. add-sqr-sqrt 46.7%

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

      4. sub-neg 46.7%

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

      5. +-commutative 46.7%

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

      6. associate-*l/ 48.4%

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

      7. associate-*r/ 52.5%

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

      8. frac-2neg 52.5%

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

      9. +-commutative 52.5%

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

      10. distribute-neg-in 52.5%

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

      11. metadata-eval 52.5%

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

      12. sub-neg 52.5%

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

      13. distribute-frac-neg2 52.5%

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

      14. distribute-neg-in 52.5%

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

      15. fma-undefine 59.7%

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

      16. neg-sub0 59.7%

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

      17. fma-undefine 52.5%

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

      18. associate-*r/ 48.4%

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

      19. associate-*l/ 46.7%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in x around inf 51.7%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 55.7%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 55.7%

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

      3. metadata-eval 55.7%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 55.7%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto 0 - \color{blue}{\sqrt{x \cdot \frac{z + -1}{y}} \cdot \sqrt{x \cdot \frac{z + -1}{y}}} \]

      2. sqrt-unprod 0.9%

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

      3. sqr-neg 0.9%

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

      4. sub0-neg 0.9%

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

      5. sub0-neg 0.9%

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

      6. sqrt-unprod 0.4%

         \[\leadsto 0 - \color{blue}{\sqrt{0 - x \cdot \frac{z + -1}{y}} \cdot \sqrt{0 - x \cdot \frac{z + -1}{y}}} \]

      7. add-sqr-sqrt 40.4%

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

      8. cancel-sign-sub-inv 40.4%

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

      9. +-lft-identity 40.4%

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

      10. clear-num 40.4%

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

      11. un-div-inv 40.4%

          \[\leadsto 0 - \color{blue}{\frac{-x}{\frac{y}{z + -1}}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto 0 - \color{blue}{\frac{-x}{\frac{y}{z + -1}}} \]

   if -8.20000000000000063e-8 < x < 10.199999999999999

   1. Initial program 95.6%

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

      1. associate-*l/ 99.9%

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

      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%

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

      1. add-sqr-sqrt 51.1%

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

      2. fabs-sqr 51.1%

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

      3. add-sqr-sqrt 52.4%

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

      4. frac-2neg 52.4%

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

      5. sub-neg 52.4%

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

      6. distribute-rgt-neg-out 52.4%

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

      7. distribute-neg-in 52.4%

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

      8. +-commutative 52.4%

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

      9. distribute-neg-in 52.4%

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

      10. metadata-eval 52.4%

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

      11. sub-neg 52.4%

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

      12. distribute-rgt-neg-out 52.4%

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

      13. remove-double-neg 52.4%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in x around 0 52.1%

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

   if 10.199999999999999 < x

   1. Initial program 94.0%

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

      1. associate-*l/ 92.3%

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

      2. sub-div 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. add-sqr-sqrt 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. frac-2neg 55.4%

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

      5. sub-neg 55.4%

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

      6. distribute-rgt-neg-out 55.4%

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

      7. distribute-neg-in 55.4%

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

      8. +-commutative 55.4%

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

      9. distribute-neg-in 55.4%

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

      10. metadata-eval 55.4%

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

      11. sub-neg 55.4%

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

      12. distribute-rgt-neg-out 55.4%

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

      13. remove-double-neg 55.4%

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

   6. Applied egg-rr 55.4%

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

   7. Taylor expanded in x around inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. sub-neg 55.4%

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

      3. metadata-eval 55.4%

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

      4. neg-mul-1 55.4%

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

      5. distribute-rgt-neg-in 55.4%

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

      6. +-commutative 55.4%

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

      7. distribute-neg-in 55.4%

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

      8. metadata-eval 55.4%

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

      9. neg-mul-1 55.4%

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

      10. associate-/l* 59.0%

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

      11. neg-mul-1 59.0%

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

      12. sub-neg 59.0%

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

   9. Simplified 59.0%

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

       1. clear-num 59.0%

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

       2. un-div-inv 59.1%

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

       3. *-un-lft-identity 59.1%

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

       4. *-un-lft-identity 59.1%

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

       5. sub-neg 59.1%

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

       6. +-commutative 59.1%

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

       7. add-sqr-sqrt 29.3%

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

       8. sqrt-unprod 53.4%

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

       9. sqr-neg 53.4%

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

       10. sqrt-unprod 31.4%

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

       11. add-sqr-sqrt 49.5%

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

   11. Applied egg-rr 49.5%

       \[\leadsto \color{blue}{\frac{x}{\frac{y}{z + 1}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{\frac{y}{z + -1}}\\ \mathbf{elif}\;x \leq 10.2:\\ \;\;\;\;\frac{\left(--4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{z + -1}{y\_m}\\ \mathbf{elif}\;x \leq 23000:\\ \;\;\;\;\frac{\left(--4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.2e-8)
   (* x (/ (+ z -1.0) y_m))
   (if (<= x 23000.0) (/ (- (- -4.0) (* x z)) y_m) (/ x (/ y_m (+ z 1.0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.2e-8) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 23000.0) {
		tmp = (-(-4.0) - (x * z)) / y_m;
	} else {
		tmp = x / (y_m / (z + 1.0));
	}
	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) :: tmp
    if (x <= (-8.2d-8)) then
        tmp = x * ((z + (-1.0d0)) / y_m)
    else if (x <= 23000.0d0) then
        tmp = (-(-4.0d0) - (x * z)) / y_m
    else
        tmp = x / (y_m / (z + 1.0d0))
    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 <= -8.2e-8) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 23000.0) {
		tmp = (-(-4.0) - (x * z)) / y_m;
	} else {
		tmp = x / (y_m / (z + 1.0));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -8.2e-8:
		tmp = x * ((z + -1.0) / y_m)
	elif x <= 23000.0:
		tmp = (-(-4.0) - (x * z)) / y_m
	else:
		tmp = x / (y_m / (z + 1.0))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -8.2e-8)
		tmp = Float64(x * Float64(Float64(z + -1.0) / y_m));
	elseif (x <= 23000.0)
		tmp = Float64(Float64(Float64(-(-4.0)) - Float64(x * z)) / y_m);
	else
		tmp = Float64(x / Float64(y_m / Float64(z + 1.0)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -8.2e-8)
		tmp = x * ((z + -1.0) / y_m);
	elseif (x <= 23000.0)
		tmp = (-(-4.0) - (x * z)) / y_m;
	else
		tmp = x / (y_m / (z + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -8.2e-8], N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 23000.0], N[(N[((--4.0) - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x / N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.20000000000000063e-8

   1. Initial program 84.0%

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

      1. add-sqr-sqrt 46.1%

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

      2. fabs-sqr 46.1%

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

      3. add-sqr-sqrt 46.7%

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

      4. sub-neg 46.7%

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

      5. +-commutative 46.7%

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

      6. associate-*l/ 48.4%

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

      7. associate-*r/ 52.5%

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

      8. frac-2neg 52.5%

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

      9. +-commutative 52.5%

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

      10. distribute-neg-in 52.5%

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

      11. metadata-eval 52.5%

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

      12. sub-neg 52.5%

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

      13. distribute-frac-neg2 52.5%

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

      14. distribute-neg-in 52.5%

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

      15. fma-undefine 59.7%

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

      16. neg-sub0 59.7%

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

      17. fma-undefine 52.5%

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

      18. associate-*r/ 48.4%

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

      19. associate-*l/ 46.7%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in x around inf 51.7%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 55.7%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 55.7%

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

      3. metadata-eval 55.7%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 55.7%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 55.2%

         \[\leadsto \color{blue}{\sqrt{0 - x \cdot \frac{z + -1}{y}} \cdot \sqrt{0 - x \cdot \frac{z + -1}{y}}} \]

      2. sqrt-unprod 69.0%

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

      3. sub0-neg 69.0%

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

      4. sub0-neg 69.0%

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

      5. sqr-neg 69.0%

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

      6. sqrt-unprod 39.8%

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

      7. add-sqr-sqrt 40.4%

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

      8. *-commutative 40.4%

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

   9. Applied egg-rr 40.4%

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

   if -8.20000000000000063e-8 < x < 23000

   1. Initial program 95.6%

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

      1. associate-*l/ 99.9%

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

      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%

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

      1. add-sqr-sqrt 51.1%

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

      2. fabs-sqr 51.1%

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

      3. add-sqr-sqrt 52.4%

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

      4. frac-2neg 52.4%

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

      5. sub-neg 52.4%

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

      6. distribute-rgt-neg-out 52.4%

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

      7. distribute-neg-in 52.4%

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

      8. +-commutative 52.4%

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

      9. distribute-neg-in 52.4%

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

      10. metadata-eval 52.4%

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

      11. sub-neg 52.4%

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

      12. distribute-rgt-neg-out 52.4%

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

      13. remove-double-neg 52.4%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in x around 0 52.1%

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

   if 23000 < x

   1. Initial program 94.0%

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

      1. associate-*l/ 92.3%

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

      2. sub-div 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. add-sqr-sqrt 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. frac-2neg 55.4%

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

      5. sub-neg 55.4%

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

      6. distribute-rgt-neg-out 55.4%

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

      7. distribute-neg-in 55.4%

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

      8. +-commutative 55.4%

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

      9. distribute-neg-in 55.4%

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

      10. metadata-eval 55.4%

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

      11. sub-neg 55.4%

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

      12. distribute-rgt-neg-out 55.4%

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

      13. remove-double-neg 55.4%

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

   6. Applied egg-rr 55.4%

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

   7. Taylor expanded in x around inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. sub-neg 55.4%

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

      3. metadata-eval 55.4%

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

      4. neg-mul-1 55.4%

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

      5. distribute-rgt-neg-in 55.4%

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

      6. +-commutative 55.4%

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

      7. distribute-neg-in 55.4%

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

      8. metadata-eval 55.4%

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

      9. neg-mul-1 55.4%

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

      10. associate-/l* 59.0%

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

      11. neg-mul-1 59.0%

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

      12. sub-neg 59.0%

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

   9. Simplified 59.0%

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

       1. clear-num 59.0%

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

       2. un-div-inv 59.1%

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

       3. *-un-lft-identity 59.1%

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

       4. *-un-lft-identity 59.1%

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

       5. sub-neg 59.1%

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

       6. +-commutative 59.1%

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

       7. add-sqr-sqrt 29.3%

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

       8. sqrt-unprod 53.4%

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

       9. sqr-neg 53.4%

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

       10. sqrt-unprod 31.4%

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

       11. add-sqr-sqrt 49.5%

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

   11. Applied egg-rr 49.5%

       \[\leadsto \color{blue}{\frac{x}{\frac{y}{z + 1}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{z + -1}{y}\\ \mathbf{elif}\;x \leq 23000:\\ \;\;\;\;\frac{\left(--4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot \frac{z + -1}{y\_m}\\ \mathbf{elif}\;x \leq 620000000000:\\ \;\;\;\;\frac{-1}{\frac{y\_m}{-4 - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.0)
   (* x (/ (+ z -1.0) y_m))
   (if (<= x 620000000000.0)
     (/ -1.0 (/ y_m (- -4.0 x)))
     (/ x (/ y_m (+ z 1.0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.0) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 620000000000.0) {
		tmp = -1.0 / (y_m / (-4.0 - x));
	} else {
		tmp = x / (y_m / (z + 1.0));
	}
	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) :: tmp
    if (x <= (-2.0d0)) then
        tmp = x * ((z + (-1.0d0)) / y_m)
    else if (x <= 620000000000.0d0) then
        tmp = (-1.0d0) / (y_m / ((-4.0d0) - x))
    else
        tmp = x / (y_m / (z + 1.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.0) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 620000000000.0) {
		tmp = -1.0 / (y_m / (-4.0 - x));
	} else {
		tmp = x / (y_m / (z + 1.0));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.0:
		tmp = x * ((z + -1.0) / y_m)
	elif x <= 620000000000.0:
		tmp = -1.0 / (y_m / (-4.0 - x))
	else:
		tmp = x / (y_m / (z + 1.0))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(x * Float64(Float64(z + -1.0) / y_m));
	elseif (x <= 620000000000.0)
		tmp = Float64(-1.0 / Float64(y_m / Float64(-4.0 - x)));
	else
		tmp = Float64(x / Float64(y_m / Float64(z + 1.0)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = x * ((z + -1.0) / y_m);
	elseif (x <= 620000000000.0)
		tmp = -1.0 / (y_m / (-4.0 - x));
	else
		tmp = x / (y_m / (z + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.0], N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 620000000000.0], N[(-1.0 / N[(y$95$m / N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 45.9%

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

      2. fabs-sqr 45.9%

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

      3. add-sqr-sqrt 46.4%

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

      4. sub-neg 46.4%

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

      5. +-commutative 46.4%

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

      6. associate-*l/ 48.2%

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

      7. associate-*r/ 52.5%

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

      8. frac-2neg 52.5%

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

      9. +-commutative 52.5%

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

      10. distribute-neg-in 52.5%

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

      11. metadata-eval 52.5%

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

      12. sub-neg 52.5%

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

      13. distribute-frac-neg2 52.5%

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

      14. distribute-neg-in 52.5%

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

      15. fma-undefine 60.2%

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

      16. neg-sub0 60.2%

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

      17. fma-undefine 52.5%

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

      18. associate-*r/ 48.2%

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

      19. associate-*l/ 46.4%

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

   4. Applied egg-rr 55.9%

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

   5. Taylor expanded in x around inf 54.7%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 59.0%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 59.0%

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

      3. metadata-eval 59.0%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 59.0%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 58.5%

         \[\leadsto \color{blue}{\sqrt{0 - x \cdot \frac{z + -1}{y}} \cdot \sqrt{0 - x \cdot \frac{z + -1}{y}}} \]

      2. sqrt-unprod 72.6%

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

      3. sub0-neg 72.6%

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

      4. sub0-neg 72.6%

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

      5. sqr-neg 72.6%

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

      6. sqrt-unprod 39.7%

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

      7. add-sqr-sqrt 40.3%

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

      8. *-commutative 40.3%

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

   9. Applied egg-rr 40.3%

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

   if -2 < x < 6.2e11

   1. Initial program 95.8%

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

      1. fabs-sub 95.8%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.3%

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

      4. fma-neg 95.3%

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

      5. distribute-neg-frac 95.3%

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

      6. +-commutative 95.3%

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

      7. distribute-neg-in 95.3%

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

      8. unsub-neg 95.3%

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

      9. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in z around 0 76.7%

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

      1. associate-*r/ 76.7%

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

      2. distribute-lft-in 76.7%

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

      3. metadata-eval 76.7%

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

      4. neg-mul-1 76.7%

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

      5. sub-neg 76.7%

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

   7. Simplified 76.7%

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

      1. add-sqr-sqrt 76.1%

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

      2. sqrt-unprod 49.6%

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

      3. neg-fabs 49.6%

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

      4. neg-fabs 49.6%

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

      5. sqr-abs 49.6%

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

      6. distribute-frac-neg2 49.6%

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

      7. sub-neg 49.6%

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

      8. metadata-eval 49.6%

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

      9. distribute-neg-in 49.6%

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

      10. +-commutative 49.6%

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

      11. frac-2neg 49.6%

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

      12. distribute-frac-neg2 49.6%

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

      13. sub-neg 49.6%

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

      14. metadata-eval 49.6%

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

      15. distribute-neg-in 49.6%

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

      16. +-commutative 49.6%

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

      17. frac-2neg 49.6%

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

      18. sqrt-unprod 40.3%

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

   9. Applied egg-rr 41.7%

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

   if 6.2e11 < x

   1. Initial program 94.0%

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

      1. associate-*l/ 92.3%

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

      2. sub-div 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. add-sqr-sqrt 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. frac-2neg 55.4%

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

      5. sub-neg 55.4%

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

      6. distribute-rgt-neg-out 55.4%

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

      7. distribute-neg-in 55.4%

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

      8. +-commutative 55.4%

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

      9. distribute-neg-in 55.4%

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

      10. metadata-eval 55.4%

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

      11. sub-neg 55.4%

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

      12. distribute-rgt-neg-out 55.4%

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

      13. remove-double-neg 55.4%

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

   6. Applied egg-rr 55.4%

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

   7. Taylor expanded in x around inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. sub-neg 55.4%

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

      3. metadata-eval 55.4%

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

      4. neg-mul-1 55.4%

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

      5. distribute-rgt-neg-in 55.4%

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

      6. +-commutative 55.4%

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

      7. distribute-neg-in 55.4%

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

      8. metadata-eval 55.4%

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

      9. neg-mul-1 55.4%

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

      10. associate-/l* 59.0%

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

      11. neg-mul-1 59.0%

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

      12. sub-neg 59.0%

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

   9. Simplified 59.0%

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

       1. clear-num 59.0%

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

       2. un-div-inv 59.1%

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

       3. *-un-lft-identity 59.1%

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

       4. *-un-lft-identity 59.1%

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

       5. sub-neg 59.1%

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

       6. +-commutative 59.1%

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

       7. add-sqr-sqrt 29.3%

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

       8. sqrt-unprod 53.4%

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

       9. sqr-neg 53.4%

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

       10. sqrt-unprod 31.4%

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

       11. add-sqr-sqrt 49.5%

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

   11. Applied egg-rr 49.5%

       \[\leadsto \color{blue}{\frac{x}{\frac{y}{z + 1}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot \frac{z + -1}{y}\\ \mathbf{elif}\;x \leq 620000000000:\\ \;\;\;\;\frac{-1}{\frac{y}{-4 - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;x \cdot \frac{z + -1}{y\_m}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.8)
   (* x (/ (+ z -1.0) y_m))
   (if (<= x 4e-15) (/ (+ x 4.0) y_m) (* x (/ (- 1.0 z) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.8) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 4e-15) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - 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 <= (-2.8d0)) then
        tmp = x * ((z + (-1.0d0)) / y_m)
    else if (x <= 4d-15) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = x * ((1.0d0 - 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 <= -2.8) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 4e-15) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - z) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.8:
		tmp = x * ((z + -1.0) / y_m)
	elif x <= 4e-15:
		tmp = (x + 4.0) / y_m
	else:
		tmp = x * ((1.0 - z) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.8)
		tmp = Float64(x * Float64(Float64(z + -1.0) / y_m));
	elseif (x <= 4e-15)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(x * Float64(Float64(1.0 - 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 <= -2.8)
		tmp = x * ((z + -1.0) / y_m);
	elseif (x <= 4e-15)
		tmp = (x + 4.0) / y_m;
	else
		tmp = x * ((1.0 - z) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.8], N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-15], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999998

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 45.9%

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

      2. fabs-sqr 45.9%

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

      3. add-sqr-sqrt 46.4%

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

      4. sub-neg 46.4%

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

      5. +-commutative 46.4%

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

      6. associate-*l/ 48.2%

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

      7. associate-*r/ 52.5%

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

      8. frac-2neg 52.5%

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

      9. +-commutative 52.5%

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

      10. distribute-neg-in 52.5%

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

      11. metadata-eval 52.5%

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

      12. sub-neg 52.5%

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

      13. distribute-frac-neg2 52.5%

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

      14. distribute-neg-in 52.5%

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

      15. fma-undefine 60.2%

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

      16. neg-sub0 60.2%

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

      17. fma-undefine 52.5%

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

      18. associate-*r/ 48.2%

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

      19. associate-*l/ 46.4%

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

   4. Applied egg-rr 55.9%

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

   5. Taylor expanded in x around inf 54.7%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 59.0%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 59.0%

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

      3. metadata-eval 59.0%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 59.0%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 58.5%

         \[\leadsto \color{blue}{\sqrt{0 - x \cdot \frac{z + -1}{y}} \cdot \sqrt{0 - x \cdot \frac{z + -1}{y}}} \]

      2. sqrt-unprod 72.6%

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

      3. sub0-neg 72.6%

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

      4. sub0-neg 72.6%

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

      5. sqr-neg 72.6%

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

      6. sqrt-unprod 39.7%

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

      7. add-sqr-sqrt 40.3%

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

      8. *-commutative 40.3%

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

   9. Applied egg-rr 40.3%

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

   if -2.7999999999999998 < x < 4.0000000000000003e-15

   1. Initial program 95.6%

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

      1. fabs-sub 95.6%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.1%

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

      4. fma-neg 95.1%

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

      5. distribute-neg-frac 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. unsub-neg 95.1%

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

      9. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in z around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. distribute-lft-in 77.4%

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

      3. metadata-eval 77.4%

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

      4. neg-mul-1 77.4%

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

      5. sub-neg 77.4%

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

   7. Simplified 77.4%

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

      1. add-sqr-sqrt 76.9%

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

      2. sqrt-unprod 49.5%

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

      3. neg-fabs 49.5%

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

      4. neg-fabs 49.5%

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

      5. sqr-abs 49.5%

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

      6. distribute-frac-neg2 49.5%

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

      7. sub-neg 49.5%

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

      8. metadata-eval 49.5%

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

      9. distribute-neg-in 49.5%

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

      10. +-commutative 49.5%

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

      11. frac-2neg 49.5%

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

      12. distribute-frac-neg2 49.5%

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

      13. sub-neg 49.5%

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

      14. metadata-eval 49.5%

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

      15. distribute-neg-in 49.5%

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

      16. +-commutative 49.5%

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

      17. frac-2neg 49.5%

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

      18. sqrt-unprod 40.7%

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

      19. add-sqr-sqrt 42.1%

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

   9. Applied egg-rr 42.1%

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

   if 4.0000000000000003e-15 < x

   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. associate-*l/ 92.8%

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

      2. sub-div 96.4%

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

   4. Applied egg-rr 96.4%

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

      1. add-sqr-sqrt 56.3%

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

      2. fabs-sqr 56.3%

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

      3. add-sqr-sqrt 56.8%

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

      4. frac-2neg 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-rgt-neg-out 56.8%

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

      7. distribute-neg-in 56.8%

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

      8. +-commutative 56.8%

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

      9. distribute-neg-in 56.8%

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

      10. metadata-eval 56.8%

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

      11. sub-neg 56.8%

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

      12. distribute-rgt-neg-out 56.8%

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

      13. remove-double-neg 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in x around inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. sub-neg 55.2%

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

      3. metadata-eval 55.2%

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

      4. neg-mul-1 55.2%

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

      5. distribute-rgt-neg-in 55.2%

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

      6. +-commutative 55.2%

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

      7. distribute-neg-in 55.2%

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

      8. metadata-eval 55.2%

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

      9. neg-mul-1 55.2%

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

      10. associate-/l* 58.6%

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

      11. neg-mul-1 58.6%

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

      12. sub-neg 58.6%

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

   9. Simplified 58.6%

      \[\leadsto \color{blue}{x \cdot \frac{1 - z}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;x \cdot \frac{z + -1}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y\_m}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.0)
   (/ (- -4.0 x) y_m)
   (if (<= x 4e-15) (/ (+ x 4.0) y_m) (* x (/ (- 1.0 z) 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 if (x <= 4e-15) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - 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 = ((-4.0d0) - x) / y_m
    else if (x <= 4d-15) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = x * ((1.0d0 - 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 = (-4.0 - x) / y_m;
	} else if (x <= 4e-15) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - z) / 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
	elif x <= 4e-15:
		tmp = (x + 4.0) / y_m
	else:
		tmp = x * ((1.0 - 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(-4.0 - x) / y_m);
	elseif (x <= 4e-15)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(x * Float64(Float64(1.0 - 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 = (-4.0 - x) / y_m;
	elseif (x <= 4e-15)
		tmp = (x + 4.0) / y_m;
	else
		tmp = x * ((1.0 - 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[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 4e-15], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 82.8%

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

      1. fabs-sub 82.8%

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

      2. div-inv 82.7%

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

      3. cancel-sign-sub-inv 82.7%

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

      4. associate-*l/ 78.7%

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

      5. associate-*r/ 88.9%

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

      6. +-commutative 88.9%

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

      7. distribute-neg-in 88.9%

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

      8. metadata-eval 88.9%

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

      9. sub-neg 88.9%

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

      10. div-inv 89.0%

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

      11. fma-undefine 98.4%

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

      12. add-sqr-sqrt 37.3%

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

      13. fabs-sqr 37.3%

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

      14. add-sqr-sqrt 37.9%

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

      15. fma-undefine 36.4%

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

      16. associate-*r/ 30.5%

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

      17. associate-*l/ 36.4%

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

      18. div-inv 36.3%

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

   4. Applied egg-rr 36.4%

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

   5. Taylor expanded in z around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. distribute-neg-in 21.8%

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

      3. distribute-lft-neg-in 21.8%

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

      4. metadata-eval 21.8%

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

      5. associate-*r/ 21.8%

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

      6. metadata-eval 21.8%

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

      7. sub-neg 21.8%

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

      8. div-sub 21.8%

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

   7. Simplified 21.8%

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

   if -4 < x < 4.0000000000000003e-15

   1. Initial program 95.7%

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

      1. fabs-sub 95.7%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.2%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in z around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. distribute-lft-in 76.9%

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

      3. metadata-eval 76.9%

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

      4. neg-mul-1 76.9%

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

      5. sub-neg 76.9%

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

   7. Simplified 76.9%

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

      1. add-sqr-sqrt 76.3%

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

      2. sqrt-unprod 49.1%

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

      3. neg-fabs 49.1%

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

      4. neg-fabs 49.1%

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

      5. sqr-abs 49.1%

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

      6. distribute-frac-neg2 49.1%

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

      7. sub-neg 49.1%

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

      8. metadata-eval 49.1%

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

      9. distribute-neg-in 49.1%

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

      10. +-commutative 49.1%

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

      11. frac-2neg 49.1%

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

      12. distribute-frac-neg2 49.1%

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

      13. sub-neg 49.1%

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

      14. metadata-eval 49.1%

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

      15. distribute-neg-in 49.1%

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

      16. +-commutative 49.1%

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

      17. frac-2neg 49.1%

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

      18. sqrt-unprod 40.5%

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

      19. add-sqr-sqrt 41.8%

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

   9. Applied egg-rr 41.8%

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

   if 4.0000000000000003e-15 < x

   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. associate-*l/ 92.8%

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

      2. sub-div 96.4%

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

   4. Applied egg-rr 96.4%

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

      1. add-sqr-sqrt 56.3%

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

      2. fabs-sqr 56.3%

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

      3. add-sqr-sqrt 56.8%

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

      4. frac-2neg 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-rgt-neg-out 56.8%

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

      7. distribute-neg-in 56.8%

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

      8. +-commutative 56.8%

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

      9. distribute-neg-in 56.8%

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

      10. metadata-eval 56.8%

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

      11. sub-neg 56.8%

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

      12. distribute-rgt-neg-out 56.8%

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

      13. remove-double-neg 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in x around inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. sub-neg 55.2%

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

      3. metadata-eval 55.2%

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

      4. neg-mul-1 55.2%

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

      5. distribute-rgt-neg-in 55.2%

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

      6. +-commutative 55.2%

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

      7. distribute-neg-in 55.2%

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

      8. metadata-eval 55.2%

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

      9. neg-mul-1 55.2%

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

      10. associate-/l* 58.6%

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

      11. neg-mul-1 58.6%

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

      12. sub-neg 58.6%

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

   9. Simplified 58.6%

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

Alternative 11: 77.5% accurate, 6.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ y_m (+ x (+ 4.0 (* x z))))))
   (if (<= x -4.0) (/ -1.0 t_0) (/ 1.0 t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = y_m / (x + (4.0 + (x * z)));
	double tmp;
	if (x <= -4.0) {
		tmp = -1.0 / t_0;
	} else {
		tmp = 1.0 / 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 = y_m / (x + (4.0d0 + (x * z)))
    if (x <= (-4.0d0)) then
        tmp = (-1.0d0) / t_0
    else
        tmp = 1.0d0 / 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 = y_m / (x + (4.0 + (x * z)));
	double tmp;
	if (x <= -4.0) {
		tmp = -1.0 / t_0;
	} else {
		tmp = 1.0 / t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = y_m / (x + (4.0 + (x * z)))
	tmp = 0
	if x <= -4.0:
		tmp = -1.0 / t_0
	else:
		tmp = 1.0 / t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(y_m / Float64(x + Float64(4.0 + Float64(x * z))))
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(-1.0 / t_0);
	else
		tmp = Float64(1.0 / t_0);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 82.8%

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

      1. associate-*l/ 78.8%

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

      2. sub-div 89.7%

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

   4. Applied egg-rr 89.7%

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

   6. Applied egg-rr 36.5%

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

   if -4 < x

   1. Initial program 95.3%

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

      1. associate-*l/ 97.9%

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

      2. sub-div 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. add-sqr-sqrt 51.8%

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

      2. fabs-sqr 51.8%

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

      3. add-sqr-sqrt 52.9%

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

      4. clear-num 52.8%

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

      5. *-commutative 52.8%

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

      6. cancel-sign-sub-inv 52.8%

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

      7. *-commutative 52.8%

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

      8. add-sqr-sqrt 30.0%

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

      9. sqrt-unprod 46.5%

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

      10. sqr-neg 46.5%

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

      11. sqrt-unprod 24.0%

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

      12. add-sqr-sqrt 52.3%

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

      13. associate-+l+ 52.3%

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

   6. Applied egg-rr 52.3%

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

Alternative 12: 80.5% accurate, 7.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.7) (/ x (/ y_m (+ z -1.0))) (/ (- (- 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 <= -2.7) {
		tmp = x / (y_m / (z + -1.0));
	} 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 <= (-2.7d0)) then
        tmp = x / (y_m / (z + (-1.0d0)))
    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 <= -2.7) {
		tmp = x / (y_m / (z + -1.0));
	} 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 <= -2.7:
		tmp = x / (y_m / (z + -1.0))
	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 <= -2.7)
		tmp = Float64(x / Float64(y_m / Float64(z + -1.0)));
	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 <= -2.7)
		tmp = x / (y_m / (z + -1.0));
	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, -2.7], N[(x / N[(y$95$m / N[(z + -1.0), $MachinePrecision]), $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 < -2.7000000000000002

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 45.9%

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

      2. fabs-sqr 45.9%

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

      3. add-sqr-sqrt 46.4%

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

      4. sub-neg 46.4%

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

      5. +-commutative 46.4%

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

      6. associate-*l/ 48.2%

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

      7. associate-*r/ 52.5%

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

      8. frac-2neg 52.5%

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

      9. +-commutative 52.5%

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

      10. distribute-neg-in 52.5%

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

      11. metadata-eval 52.5%

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

      12. sub-neg 52.5%

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

      13. distribute-frac-neg2 52.5%

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

      14. distribute-neg-in 52.5%

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

      15. fma-undefine 60.2%

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

      16. neg-sub0 60.2%

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

      17. fma-undefine 52.5%

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

      18. associate-*r/ 48.2%

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

      19. associate-*l/ 46.4%

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

   4. Applied egg-rr 55.9%

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

   5. Taylor expanded in x around inf 54.7%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 59.0%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 59.0%

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

      3. metadata-eval 59.0%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 59.0%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.3%

         \[\leadsto 0 - \color{blue}{\sqrt{x \cdot \frac{z + -1}{y}} \cdot \sqrt{x \cdot \frac{z + -1}{y}}} \]

      2. sqrt-unprod 0.8%

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

      3. sqr-neg 0.8%

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

      4. sub0-neg 0.8%

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

      5. sub0-neg 0.8%

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

      6. sqrt-unprod 0.4%

         \[\leadsto 0 - \color{blue}{\sqrt{0 - x \cdot \frac{z + -1}{y}} \cdot \sqrt{0 - x \cdot \frac{z + -1}{y}}} \]

      7. add-sqr-sqrt 40.3%

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

      8. cancel-sign-sub-inv 40.3%

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

      9. +-lft-identity 40.3%

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

      10. clear-num 40.3%

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

      11. un-div-inv 40.3%

          \[\leadsto 0 - \color{blue}{\frac{-x}{\frac{y}{z + -1}}} \]

   9. Applied egg-rr 40.3%

      \[\leadsto 0 - \color{blue}{\frac{-x}{\frac{y}{z + -1}}} \]

   if -2.7000000000000002 < x

   1. Initial program 95.3%

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

      1. associate-*l/ 97.9%

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

      2. sub-div 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. add-sqr-sqrt 52.0%

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

      2. fabs-sqr 52.0%

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

      3. add-sqr-sqrt 53.2%

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

      4. frac-2neg 53.2%

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

      5. sub-neg 53.2%

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

      6. distribute-rgt-neg-out 53.2%

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

      7. distribute-neg-in 53.2%

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

      8. +-commutative 53.2%

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

      9. distribute-neg-in 53.2%

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

      10. metadata-eval 53.2%

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

      11. sub-neg 53.2%

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

      12. distribute-rgt-neg-out 53.2%

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

      13. remove-double-neg 53.2%

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

   6. Applied egg-rr 53.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;\frac{x}{\frac{y}{z + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - -4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8e-8) (* z (/ 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 <= -8e-8) {
		tmp = z * (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 <= (-8d-8)) then
        tmp = z * (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 <= -8e-8) {
		tmp = z * (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 <= -8e-8:
		tmp = z * (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 <= -8e-8)
		tmp = Float64(z * 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 <= -8e-8)
		tmp = z * (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, -8e-8], N[(z * N[(x / y$95$m), $MachinePrecision]), $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 < -8.0000000000000002e-8

   1. Initial program 84.0%

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

      1. add-sqr-sqrt 46.1%

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

      2. fabs-sqr 46.1%

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

      3. add-sqr-sqrt 46.7%

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

      4. sub-neg 46.7%

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

      5. +-commutative 46.7%

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

      6. associate-*l/ 48.4%

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

      7. associate-*r/ 52.5%

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

      8. frac-2neg 52.5%

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

      9. +-commutative 52.5%

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

      10. distribute-neg-in 52.5%

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

      11. metadata-eval 52.5%

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

      12. sub-neg 52.5%

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

      13. distribute-frac-neg2 52.5%

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

      14. distribute-neg-in 52.5%

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

      15. fma-undefine 59.7%

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

      16. neg-sub0 59.7%

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

      17. fma-undefine 52.5%

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

      18. associate-*r/ 48.4%

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

      19. associate-*l/ 46.7%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in z around inf 20.7%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 20.7%

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

      2. associate-*r/ 27.5%

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

   7. Simplified 27.5%

      \[\leadsto 0 - \color{blue}{z \cdot \frac{x}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 27.5%

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

      2. *-commutative 27.5%

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

      3. distribute-lft-neg-in 27.5%

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

      4. add-sqr-sqrt 10.7%

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

      5. sqrt-unprod 30.5%

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

      6. sqr-neg 30.5%

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

      7. sqrt-unprod 24.0%

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

      8. add-sqr-sqrt 39.8%

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

      9. distribute-frac-neg 39.8%

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

      10. frac-2neg 39.8%

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

   9. Applied egg-rr 39.8%

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

   if -8.0000000000000002e-8 < x < 4

   1. Initial program 95.6%

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

      1. associate-*l/ 99.9%

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

      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%

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

      1. add-sqr-sqrt 51.1%

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

      2. fabs-sqr 51.1%

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

      3. add-sqr-sqrt 52.4%

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

      4. frac-2neg 52.4%

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

      5. sub-neg 52.4%

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

      6. distribute-rgt-neg-out 52.4%

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

      7. distribute-neg-in 52.4%

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

      8. +-commutative 52.4%

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

      9. distribute-neg-in 52.4%

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

      10. metadata-eval 52.4%

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

      11. sub-neg 52.4%

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

      12. distribute-rgt-neg-out 52.4%

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

      13. remove-double-neg 52.4%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in x around 0 41.0%

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

   if 4 < x

   1. Initial program 94.0%

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

      1. add-sqr-sqrt 54.7%

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

      2. fabs-sqr 54.7%

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

      3. add-sqr-sqrt 55.2%

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

      4. sub-neg 55.2%

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

      5. +-commutative 55.2%

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

      6. associate-*l/ 51.4%

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

      7. associate-*r/ 55.1%

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

      8. frac-2neg 55.1%

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

      9. +-commutative 55.1%

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

      10. distribute-neg-in 55.1%

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

      11. metadata-eval 55.1%

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

      12. sub-neg 55.1%

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

      13. distribute-frac-neg2 55.1%

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

      14. distribute-neg-in 55.1%

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

      15. fma-undefine 57.1%

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

      16. neg-sub0 57.1%

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

      17. fma-undefine 55.1%

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

      18. associate-*r/ 51.4%

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

      19. associate-*l/ 55.2%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around inf 55.4%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 59.0%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 59.0%

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

      3. metadata-eval 59.0%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 59.0%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]

   8. Taylor expanded in z around 0 37.1%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.7% 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 82.8%

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

      1. fabs-sub 82.8%

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

      2. div-inv 82.7%

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

      3. cancel-sign-sub-inv 82.7%

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

      4. associate-*l/ 78.7%

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

      5. associate-*r/ 88.9%

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

      6. +-commutative 88.9%

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

      7. distribute-neg-in 88.9%

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

      8. metadata-eval 88.9%

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

      9. sub-neg 88.9%

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

      10. div-inv 89.0%

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

      11. fma-undefine 98.4%

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

      12. add-sqr-sqrt 37.3%

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

      13. fabs-sqr 37.3%

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

      14. add-sqr-sqrt 37.9%

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

      15. fma-undefine 36.4%

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

      16. associate-*r/ 30.5%

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

      17. associate-*l/ 36.4%

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

      18. div-inv 36.3%

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

   4. Applied egg-rr 36.4%

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

   5. Taylor expanded in z around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. distribute-neg-in 21.8%

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

      3. distribute-lft-neg-in 21.8%

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

      4. metadata-eval 21.8%

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

      5. associate-*r/ 21.8%

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

      6. metadata-eval 21.8%

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

      7. sub-neg 21.8%

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

      8. div-sub 21.8%

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

   7. Simplified 21.8%

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

   if -4 < x

   1. Initial program 95.3%

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

      1. fabs-sub 95.3%

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

      2. associate-*l/ 97.9%

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

      3. associate-*r/ 95.5%

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

      4. fma-neg 96.0%

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

      5. distribute-neg-frac 96.0%

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

      6. +-commutative 96.0%

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

      7. distribute-neg-in 96.0%

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

      8. unsub-neg 96.0%

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

      9. metadata-eval 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around 0 73.8%

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

      1. associate-*r/ 73.8%

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

      2. distribute-lft-in 73.8%

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

      3. metadata-eval 73.8%

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

      4. neg-mul-1 73.8%

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

      5. sub-neg 73.8%

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

   7. Simplified 73.8%

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

      1. add-sqr-sqrt 73.2%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right|} \cdot \sqrt{\left|\frac{-4 - x}{y}\right|}} \]

      2. sqrt-unprod 50.0%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right| \cdot \left|\frac{-4 - x}{y}\right|}} \]

      3. neg-fabs 50.0%

         \[\leadsto \sqrt{\color{blue}{\left|-\frac{-4 - x}{y}\right|} \cdot \left|\frac{-4 - x}{y}\right|} \]

      4. neg-fabs 50.0%

         \[\leadsto \sqrt{\left|-\frac{-4 - x}{y}\right| \cdot \color{blue}{\left|-\frac{-4 - x}{y}\right|}} \]

      5. sqr-abs 50.0%

         \[\leadsto \sqrt{\color{blue}{\left(-\frac{-4 - x}{y}\right) \cdot \left(-\frac{-4 - x}{y}\right)}} \]

      6. distribute-frac-neg2 50.0%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 - x}{-y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      7. sub-neg 50.0%

         \[\leadsto \sqrt{\frac{\color{blue}{-4 + \left(-x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      8. metadata-eval 50.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      9. distribute-neg-in 50.0%

         \[\leadsto \sqrt{\frac{\color{blue}{-\left(4 + x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      10. +-commutative 50.0%

          \[\leadsto \sqrt{\frac{-\color{blue}{\left(x + 4\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      11. frac-2neg 50.0%

          \[\leadsto \sqrt{\color{blue}{\frac{x + 4}{y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      12. distribute-frac-neg2 50.0%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 50.0%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-4 + \left(-x\right)}}{-y}} \]

      14. metadata-eval 50.0%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y}} \]

      15. distribute-neg-in 50.0%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-\left(4 + x\right)}}{-y}} \]

      16. +-commutative 50.0%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{-\color{blue}{\left(x + 4\right)}}{-y}} \]

      17. frac-2neg 50.0%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{x + 4}{y}}} \]

      18. sqrt-unprod 39.2%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y}} \cdot \sqrt{\frac{x + 4}{y}}} \]

      19. add-sqr-sqrt 40.3%

          \[\leadsto \color{blue}{\frac{x + 4}{y}} \]

   9. Applied egg-rr 40.3%

      \[\leadsto \color{blue}{\frac{x + 4}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 62.1% accurate, 11.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;z \cdot \frac{x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -0.88) (* z (/ 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 <= -0.88) {
		tmp = z * (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 <= (-0.88d0)) then
        tmp = z * (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 <= -0.88) {
		tmp = z * (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 <= -0.88:
		tmp = z * (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 <= -0.88)
		tmp = Float64(z * 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 <= -0.88)
		tmp = z * (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, -0.88], N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.880000000000000004

   1. Initial program 83.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 45.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 45.9%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 46.4%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. sub-neg 46.4%

         \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      5. +-commutative 46.4%

         \[\leadsto \color{blue}{\left(-\frac{x}{y} \cdot z\right) + \frac{x + 4}{y}} \]

      6. associate-*l/ 48.2%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \frac{x + 4}{y} \]

      7. associate-*r/ 52.5%

         \[\leadsto \left(-\color{blue}{x \cdot \frac{z}{y}}\right) + \frac{x + 4}{y} \]

      8. frac-2neg 52.5%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\frac{-\left(x + 4\right)}{-y}} \]

      9. +-commutative 52.5%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{-\color{blue}{\left(4 + x\right)}}{-y} \]

      10. distribute-neg-in 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{-y} \]

      11. metadata-eval 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4} + \left(-x\right)}{-y} \]

      12. sub-neg 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4 - x}}{-y} \]

      13. distribute-frac-neg2 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\left(-\frac{-4 - x}{y}\right)} \]

      14. distribute-neg-in 52.5%

          \[\leadsto \color{blue}{-\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      15. fma-undefine 60.2%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      16. neg-sub0 60.2%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 52.5%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 48.2%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 46.4%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 21.7%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 21.7%

         \[\leadsto 0 - \frac{\color{blue}{z \cdot x}}{y} \]

      2. associate-*r/ 28.9%

         \[\leadsto 0 - \color{blue}{z \cdot \frac{x}{y}} \]

   7. Simplified 28.9%

      \[\leadsto 0 - \color{blue}{z \cdot \frac{x}{y}} \]
   8. Step-by-step derivation

      1. sub0-neg 28.9%

         \[\leadsto \color{blue}{-z \cdot \frac{x}{y}} \]

      2. *-commutative 28.9%

         \[\leadsto -\color{blue}{\frac{x}{y} \cdot z} \]

      3. distribute-lft-neg-in 28.9%

         \[\leadsto \color{blue}{\left(-\frac{x}{y}\right) \cdot z} \]

      4. add-sqr-sqrt 11.1%

         \[\leadsto \left(-\frac{x}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right) \cdot z \]

      5. sqrt-unprod 32.1%

         \[\leadsto \left(-\frac{x}{\color{blue}{\sqrt{y \cdot y}}}\right) \cdot z \]

      6. sqr-neg 32.1%

         \[\leadsto \left(-\frac{x}{\sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}}\right) \cdot z \]

      7. sqrt-unprod 25.5%

         \[\leadsto \left(-\frac{x}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right) \cdot z \]

      8. add-sqr-sqrt 39.9%

         \[\leadsto \left(-\frac{x}{\color{blue}{-y}}\right) \cdot z \]

      9. distribute-frac-neg 39.9%

         \[\leadsto \color{blue}{\frac{-x}{-y}} \cdot z \]

      10. frac-2neg 39.9%

          \[\leadsto \color{blue}{\frac{x}{y}} \cdot z \]

   9. Applied egg-rr 39.9%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot z} \]

   if -0.880000000000000004 < x

   1. Initial program 95.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 95.3%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 97.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 95.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 96.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 96.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 96.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 96.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 96.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 96.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 74.1%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 74.1%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 74.1%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 74.1%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 74.1%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 74.1%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 74.1%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 73.6%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right|} \cdot \sqrt{\left|\frac{-4 - x}{y}\right|}} \]

      2. sqrt-unprod 50.3%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right| \cdot \left|\frac{-4 - x}{y}\right|}} \]

      3. neg-fabs 50.3%

         \[\leadsto \sqrt{\color{blue}{\left|-\frac{-4 - x}{y}\right|} \cdot \left|\frac{-4 - x}{y}\right|} \]

      4. neg-fabs 50.3%

         \[\leadsto \sqrt{\left|-\frac{-4 - x}{y}\right| \cdot \color{blue}{\left|-\frac{-4 - x}{y}\right|}} \]

      5. sqr-abs 50.3%

         \[\leadsto \sqrt{\color{blue}{\left(-\frac{-4 - x}{y}\right) \cdot \left(-\frac{-4 - x}{y}\right)}} \]

      6. distribute-frac-neg2 50.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 - x}{-y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      7. sub-neg 50.3%

         \[\leadsto \sqrt{\frac{\color{blue}{-4 + \left(-x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      8. metadata-eval 50.3%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      9. distribute-neg-in 50.3%

         \[\leadsto \sqrt{\frac{\color{blue}{-\left(4 + x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      10. +-commutative 50.3%

          \[\leadsto \sqrt{\frac{-\color{blue}{\left(x + 4\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      11. frac-2neg 50.3%

          \[\leadsto \sqrt{\color{blue}{\frac{x + 4}{y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      12. distribute-frac-neg2 50.3%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 50.3%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-4 + \left(-x\right)}}{-y}} \]

      14. metadata-eval 50.3%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y}} \]

      15. distribute-neg-in 50.3%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-\left(4 + x\right)}}{-y}} \]

      16. +-commutative 50.3%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{-\color{blue}{\left(x + 4\right)}}{-y}} \]

      17. frac-2neg 50.3%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{x + 4}{y}}} \]

      18. sqrt-unprod 39.4%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y}} \cdot \sqrt{\frac{x + 4}{y}}} \]

      19. add-sqr-sqrt 40.5%

          \[\leadsto \color{blue}{\frac{x + 4}{y}} \]

   9. Applied egg-rr 40.5%

      \[\leadsto \color{blue}{\frac{x + 4}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.3% 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 91.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 93.3%

         \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]

      2. sub-div 96.7%

         \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]

   4. Applied egg-rr 96.7%

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}} \cdot \sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}}}\right| \]

      2. fabs-sqr 52.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}} \cdot \sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}}} \]

      3. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

      4. frac-2neg 53.5%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + 4\right) - x \cdot z\right)}{-y}} \]

      5. sub-neg 53.5%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + 4\right) + \left(-x \cdot z\right)\right)}}{-y} \]

      6. distribute-rgt-neg-out 53.5%

         \[\leadsto \frac{-\left(\left(x + 4\right) + \color{blue}{x \cdot \left(-z\right)}\right)}{-y} \]

      7. distribute-neg-in 53.5%

         \[\leadsto \frac{\color{blue}{\left(-\left(x + 4\right)\right) + \left(-x \cdot \left(-z\right)\right)}}{-y} \]

      8. +-commutative 53.5%

         \[\leadsto \frac{\left(-\color{blue}{\left(4 + x\right)}\right) + \left(-x \cdot \left(-z\right)\right)}{-y} \]

      9. distribute-neg-in 53.5%

         \[\leadsto \frac{\color{blue}{\left(\left(-4\right) + \left(-x\right)\right)} + \left(-x \cdot \left(-z\right)\right)}{-y} \]

      10. metadata-eval 53.5%

          \[\leadsto \frac{\left(\color{blue}{-4} + \left(-x\right)\right) + \left(-x \cdot \left(-z\right)\right)}{-y} \]

      11. sub-neg 53.5%

          \[\leadsto \frac{\color{blue}{\left(-4 - x\right)} + \left(-x \cdot \left(-z\right)\right)}{-y} \]

      12. distribute-rgt-neg-out 53.5%

          \[\leadsto \frac{\left(-4 - x\right) + \left(-\color{blue}{\left(-x \cdot z\right)}\right)}{-y} \]

      13. remove-double-neg 53.5%

          \[\leadsto \frac{\left(-4 - x\right) + \color{blue}{x \cdot z}}{-y} \]

   6. Applied egg-rr 53.5%

      \[\leadsto \color{blue}{\frac{\left(-4 - x\right) + x \cdot z}{-y}} \]

   7. Taylor expanded in x around 0 28.5%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 94.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 54.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 54.7%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 55.2%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. sub-neg 55.2%

         \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      5. +-commutative 55.2%

         \[\leadsto \color{blue}{\left(-\frac{x}{y} \cdot z\right) + \frac{x + 4}{y}} \]

      6. associate-*l/ 51.4%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \frac{x + 4}{y} \]

      7. associate-*r/ 55.1%

         \[\leadsto \left(-\color{blue}{x \cdot \frac{z}{y}}\right) + \frac{x + 4}{y} \]

      8. frac-2neg 55.1%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\frac{-\left(x + 4\right)}{-y}} \]

      9. +-commutative 55.1%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{-\color{blue}{\left(4 + x\right)}}{-y} \]

      10. distribute-neg-in 55.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{-y} \]

      11. metadata-eval 55.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4} + \left(-x\right)}{-y} \]

      12. sub-neg 55.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4 - x}}{-y} \]

      13. distribute-frac-neg2 55.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\left(-\frac{-4 - x}{y}\right)} \]

      14. distribute-neg-in 55.1%

          \[\leadsto \color{blue}{-\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      15. fma-undefine 57.1%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      16. neg-sub0 57.1%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 55.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 51.4%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 55.2%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in x around inf 55.4%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 59.0%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 59.0%

         \[\leadsto 0 - x \cdot \frac{\color{blue}{z + \left(-1\right)}}{y} \]

      3. metadata-eval 59.0%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 59.0%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]

   8. Taylor expanded in z around 0 37.1%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 40.4% 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 92.2%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 93.1%

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]

   2. sub-div 96.6%

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]

4. Applied egg-rr 96.6%

   \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]
5. Step-by-step derivation

   1. add-sqr-sqrt 52.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}} \cdot \sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}}}\right| \]

   2. fabs-sqr 52.9%

      \[\leadsto \color{blue}{\sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}} \cdot \sqrt{\frac{\left(x + 4\right) - x \cdot z}{y}}} \]

   3. add-sqr-sqrt 53.9%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   4. frac-2neg 53.9%

      \[\leadsto \color{blue}{\frac{-\left(\left(x + 4\right) - x \cdot z\right)}{-y}} \]

   5. sub-neg 53.9%

      \[\leadsto \frac{-\color{blue}{\left(\left(x + 4\right) + \left(-x \cdot z\right)\right)}}{-y} \]

   6. distribute-rgt-neg-out 53.9%

      \[\leadsto \frac{-\left(\left(x + 4\right) + \color{blue}{x \cdot \left(-z\right)}\right)}{-y} \]

   7. distribute-neg-in 53.9%

      \[\leadsto \frac{\color{blue}{\left(-\left(x + 4\right)\right) + \left(-x \cdot \left(-z\right)\right)}}{-y} \]

   8. +-commutative 53.9%

      \[\leadsto \frac{\left(-\color{blue}{\left(4 + x\right)}\right) + \left(-x \cdot \left(-z\right)\right)}{-y} \]

   9. distribute-neg-in 53.9%

      \[\leadsto \frac{\color{blue}{\left(\left(-4\right) + \left(-x\right)\right)} + \left(-x \cdot \left(-z\right)\right)}{-y} \]

   10. metadata-eval 53.9%

       \[\leadsto \frac{\left(\color{blue}{-4} + \left(-x\right)\right) + \left(-x \cdot \left(-z\right)\right)}{-y} \]

   11. sub-neg 53.9%

       \[\leadsto \frac{\color{blue}{\left(-4 - x\right)} + \left(-x \cdot \left(-z\right)\right)}{-y} \]

   12. distribute-rgt-neg-out 53.9%

       \[\leadsto \frac{\left(-4 - x\right) + \left(-\color{blue}{\left(-x \cdot z\right)}\right)}{-y} \]

   13. remove-double-neg 53.9%

       \[\leadsto \frac{\left(-4 - x\right) + \color{blue}{x \cdot z}}{-y} \]

6. Applied egg-rr 53.9%

   \[\leadsto \color{blue}{\frac{\left(-4 - x\right) + x \cdot z}{-y}} \]

7. Taylor expanded in x around 0 23.5%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))