fabs fraction 1

Percentage Accurate: 91.9% → 99.3%

Time: 9.4s

Alternatives: 23

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-99)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-99) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-99)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 4e-99], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e-99

   1. Initial program 88.7%

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

      1. associate-*l/ 89.2%

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

      2. sub-div 95.6%

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

   4. Applied egg-rr 95.6%

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

   if 4.0000000000000001e-99 < y

   1. Initial program 97.4%

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

      1. fabs-sub 97.4%

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

      2. associate-*l/ 91.2%

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

      3. associate-*r/ 99.9%

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

      4. fma-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{1}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* z (/ x y_m)))))
   (if (<= t_0 2e+301)
     (fabs t_0)
     (fabs (* (fma x z (- -4.0 x)) (/ 1.0 y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - (z * (x / y_m));
	double tmp;
	if (t_0 <= 2e+301) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((fma(x, z, (-4.0 - x)) * (1.0 / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_0 <= 2e+301)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) * Float64(1.0 / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+301], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 2.00000000000000011e301

   1. Initial program 98.0%

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

   if 2.00000000000000011e301 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 46.9%

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

   3. Simplified 97.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, z, -4 - x\right) \cdot \frac{1}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-99)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-99) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (z * (x / y_m))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 4d-99) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (z * (x / y_m))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-99) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (z * (x / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 4e-99:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (z * (x / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-99)
		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(z * Float64(x / y_m))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 4e-99)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (z * (x / y_m))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 4e-99], 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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e-99

   1. Initial program 88.7%

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

      1. associate-*l/ 89.2%

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

      2. sub-div 95.6%

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

   4. Applied egg-rr 95.6%

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

   if 4.0000000000000001e-99 < y

   1. Initial program 97.4%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-99)
   (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 <= 4e-99) {
		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 <= 4d-99) 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 <= 4e-99) {
		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 <= 4e-99:
		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 <= 4e-99)
		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 <= 4e-99)
		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, 4e-99], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e-99

   1. Initial program 88.7%

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

      1. associate-*l/ 89.2%

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

      2. sub-div 95.6%

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

   4. Applied egg-rr 95.6%

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

   if 4.0000000000000001e-99 < y

   1. Initial program 97.4%

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

      1. associate-*l/ 91.2%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.8%

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

      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. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}} - \frac{x + 4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-24)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ z (/ y_m x)) (/ (+ x 4.0) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-24) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((z / (y_m / x)) - ((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 (y_m <= 4d-24) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((z / (y_m / x)) - ((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 (y_m <= 4e-24) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((z / (y_m / x)) - ((x + 4.0) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 4e-24:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((z / (y_m / x)) - ((x + 4.0) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-24)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(z / Float64(y_m / x)) - 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 (y_m <= 4e-24)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((z / (y_m / x)) - ((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[y$95$m, 4e-24], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999969e-24

   1. Initial program 89.8%

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

      1. associate-*l/ 90.2%

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

      2. sub-div 96.0%

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

   4. Applied egg-rr 96.0%

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

   if 3.99999999999999969e-24 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0 88.7%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r/ 95.7%

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

   5. Simplified 95.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - \frac{x + 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} + 4 \cdot \frac{1}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -6.9e+49)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (fabs (+ (/ x y_m) (* 4.0 (/ 1.0 y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -6.9e+49) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = fabs(((x / y_m) + (4.0 * (1.0 / y_m))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-6.9d+49)) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = abs(((x / y_m) + (4.0d0 * (1.0d0 / y_m))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -6.9e+49) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = Math.abs(((x / y_m) + (4.0 * (1.0 / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -6.9e+49:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = math.fabs(((x / y_m) + (4.0 * (1.0 / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -6.9e+49)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(x / y_m) + Float64(4.0 * Float64(1.0 / y_m))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -6.9e+49)
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = abs(((x / y_m) + (4.0 * (1.0 / y_m))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -6.9e+49], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] + N[(4.0 * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9000000000000003e49

   1. Initial program 90.3%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-/l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. neg-sub0 99.9%

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

      6. associate-+l- 99.9%

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

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -6.9000000000000003e49 < x

   1. Initial program 91.9%

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

   3. Taylor expanded in z around 0 72.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + 4 \cdot \frac{1}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-89}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m} - \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 1.35e-89)
   (fabs (/ (- -4.0 x) y_m))
   (fabs (- (* z (/ x y_m)) (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e-89) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((z * (x / y_m)) - (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.35d-89) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((z * (x / y_m)) - (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e-89) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((z * (x / y_m)) - (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 1.35e-89:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((z * (x / y_m)) - (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 1.35e-89)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(z * Float64(x / y_m)) - Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 1.35e-89)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((z * (x / y_m)) - (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 1.35e-89], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] - N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999994e-89

   1. Initial program 94.5%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around 0 77.5%

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

      1. +-commutative 77.5%

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

      2. rem-square-sqrt 41.6%

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

      3. fabs-sqr 41.6%

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

      4. rem-square-sqrt 77.5%

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

      5. fabs-neg 77.5%

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

      6. distribute-neg-frac 77.5%

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

      7. distribute-neg-in 77.5%

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

      8. metadata-eval 77.5%

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

      9. +-commutative 77.5%

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

      10. sub-neg 77.5%

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

      11. rem-square-sqrt 35.4%

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

      12. fabs-sqr 35.4%

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

      13. rem-square-sqrt 77.5%

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

   7. Simplified 77.5%

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

   if 1.34999999999999994e-89 < x

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf 75.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-89}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y} - \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42 \cdot 10^{-89}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} - \frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 1.42e-89)
   (fabs (/ (- -4.0 x) y_m))
   (fabs (- (/ x y_m) (/ z (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1.42e-89) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((x / y_m) - (z / (y_m / x))));
	}
	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 <= 1.42d-89) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((x / y_m) - (z / (y_m / x))))
    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 <= 1.42e-89) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((x / y_m) - (z / (y_m / x))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 1.42e-89:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((x / y_m) - (z / (y_m / x))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 1.42e-89)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(x / y_m) - Float64(z / Float64(y_m / x))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 1.42e-89)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((x / y_m) - (z / (y_m / x))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 1.42e-89], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] - N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.42e-89

   1. Initial program 94.5%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around 0 77.5%

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

      1. +-commutative 77.5%

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

      2. rem-square-sqrt 41.6%

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

      3. fabs-sqr 41.6%

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

      4. rem-square-sqrt 77.5%

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

      5. fabs-neg 77.5%

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

      6. distribute-neg-frac 77.5%

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

      7. distribute-neg-in 77.5%

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

      8. metadata-eval 77.5%

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

      9. +-commutative 77.5%

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

      10. sub-neg 77.5%

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

      11. rem-square-sqrt 35.4%

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

      12. fabs-sqr 35.4%

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

      13. rem-square-sqrt 77.5%

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

   7. Simplified 77.5%

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

   if 1.42e-89 < x

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0 85.6%

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

      1. associate-*r/ 90.8%

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

      2. *-commutative 90.8%

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

      3. associate-/r/ 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in x around inf 74.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42 \cdot 10^{-89}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{y\_m}{x \cdot \left(1 - z\right)}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 5.6e-90)
   (fabs (/ (- -4.0 x) y_m))
   (fabs (/ 1.0 (/ y_m (* x (- 1.0 z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 5.6e-90) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((1.0 / (y_m / (x * (1.0 - 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 <= 5.6d-90) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((1.0d0 / (y_m / (x * (1.0d0 - 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 <= 5.6e-90) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((1.0 / (y_m / (x * (1.0 - z)))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 5.6e-90:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((1.0 / (y_m / (x * (1.0 - z)))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 5.6e-90)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(1.0 / Float64(y_m / Float64(x * Float64(1.0 - z)))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 5.6e-90)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((1.0 / (y_m / (x * (1.0 - z)))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 5.6e-90], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(1.0 / N[(y$95$m / N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-90

   1. Initial program 95.0%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in z around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. rem-square-sqrt 41.9%

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

      3. fabs-sqr 41.9%

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

      4. rem-square-sqrt 77.9%

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

      5. fabs-neg 77.9%

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

      6. distribute-neg-frac 77.9%

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

      7. distribute-neg-in 77.9%

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

      8. metadata-eval 77.9%

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

      9. +-commutative 77.9%

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

      10. sub-neg 77.9%

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

      11. rem-square-sqrt 35.5%

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

      12. fabs-sqr 35.5%

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

      13. rem-square-sqrt 77.9%

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

   7. Simplified 77.9%

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

   if 5.5999999999999998e-90 < x

   1. Initial program 85.3%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around inf 83.8%

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

      1. mul-1-neg 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-/l* 87.8%

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

      4. distribute-lft-neg-in 87.8%

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

      5. neg-sub0 87.8%

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

      6. associate-+l- 87.8%

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

      7. neg-sub0 87.8%

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

      8. +-commutative 87.8%

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

      9. unsub-neg 87.8%

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

   7. Simplified 87.8%

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

      1. associate-*r/ 83.8%

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

      2. clear-num 83.7%

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

      3. *-commutative 83.7%

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

   9. Applied egg-rr 83.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x \cdot \left(1 - z\right)}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z 4.5e+17) (fabs (/ (- -4.0 x) y_m)) (fabs (* (/ x y_m) (- 1.0 z)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= 4.5e+17) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((x / y_m) * (1.0 - 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 (z <= 4.5d+17) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((x / y_m) * (1.0d0 - 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 (z <= 4.5e+17) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= 4.5e+17:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= 4.5e+17)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= 4.5e+17)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((x / y_m) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, 4.5e+17], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.5e17

   1. Initial program 93.9%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. rem-square-sqrt 41.4%

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

      3. fabs-sqr 41.4%

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

      4. rem-square-sqrt 76.6%

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

      5. fabs-neg 76.6%

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

      6. distribute-neg-frac 76.6%

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

      7. distribute-neg-in 76.6%

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

      8. metadata-eval 76.6%

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

      9. +-commutative 76.6%

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

      10. sub-neg 76.6%

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

      11. rem-square-sqrt 34.8%

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

      12. fabs-sqr 34.8%

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

      13. rem-square-sqrt 76.6%

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

   7. Simplified 76.6%

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

   if 4.5e17 < z

   1. Initial program 81.6%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in x around inf 66.2%

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

      1. mul-1-neg 66.2%

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

      2. *-commutative 66.2%

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

      3. associate-/l* 75.7%

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

      4. distribute-lft-neg-in 75.7%

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

      5. neg-sub0 75.7%

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

      6. associate-+l- 75.7%

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

      7. neg-sub0 75.7%

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

      8. +-commutative 75.7%

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

      9. unsub-neg 75.7%

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

   7. Simplified 75.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot z\right) \cdot \frac{1}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z 2.05e+19) (fabs (/ (- -4.0 x) y_m)) (fabs (* (* x z) (/ 1.0 y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= 2.05e+19) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((x * z) * (1.0 / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.05d+19) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((x * z) * (1.0d0 / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= 2.05e+19) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((x * z) * (1.0 / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= 2.05e+19:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((x * z) * (1.0 / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= 2.05e+19)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(x * z) * Float64(1.0 / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= 2.05e+19)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((x * z) * (1.0 / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, 2.05e+19], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * z), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e19

   1. Initial program 93.9%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. rem-square-sqrt 41.4%

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

      3. fabs-sqr 41.4%

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

      4. rem-square-sqrt 76.6%

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

      5. fabs-neg 76.6%

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

      6. distribute-neg-frac 76.6%

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

      7. distribute-neg-in 76.6%

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

      8. metadata-eval 76.6%

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

      9. +-commutative 76.6%

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

      10. sub-neg 76.6%

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

      11. rem-square-sqrt 34.8%

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

      12. fabs-sqr 34.8%

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

      13. rem-square-sqrt 76.6%

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

   7. Simplified 76.6%

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

   if 2.05e19 < z

   1. Initial program 81.6%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in z around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

      3. distribute-rgt-neg-in 66.2%

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

   7. Simplified 66.2%

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

      1. clear-num 66.2%

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

      2. associate-/r/ 66.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 58.4%

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

      5. sqr-neg 58.4%

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

      6. sqrt-unprod 66.2%

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

      7. add-sqr-sqrt 66.3%

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

   9. Applied egg-rr 66.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4}{y\_m} - \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -6.9e+49)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (fabs (- (/ -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 <= -6.9e+49) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = fabs(((-4.0 / y_m) - (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 <= (-6.9d+49)) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = abs((((-4.0d0) / y_m) - (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 <= -6.9e+49) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = Math.abs(((-4.0 / y_m) - (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -6.9e+49:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = math.fabs(((-4.0 / y_m) - (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -6.9e+49)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(-4.0 / y_m) - 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 <= -6.9e+49)
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = abs(((-4.0 / y_m) - (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -6.9e+49], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 / y$95$m), $MachinePrecision] - N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9000000000000003e49

   1. Initial program 90.3%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-/l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. neg-sub0 99.9%

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

      6. associate-+l- 99.9%

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

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -6.9000000000000003e49 < x

   1. Initial program 91.9%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in z around 0 72.3%

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

      1. +-commutative 72.3%

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

      2. rem-square-sqrt 39.8%

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

      3. fabs-sqr 39.8%

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

      4. rem-square-sqrt 72.3%

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

      5. fabs-neg 72.3%

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

      6. distribute-neg-frac 72.3%

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

      7. distribute-neg-in 72.3%

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

      8. metadata-eval 72.3%

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

      9. +-commutative 72.3%

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

      10. sub-neg 72.3%

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

      11. rem-square-sqrt 32.1%

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

      12. fabs-sqr 32.1%

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

      13. rem-square-sqrt 72.3%

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

   7. Simplified 72.3%

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

      1. div-sub 72.3%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4}{y} - \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-65}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 4.6e-65) (fabs (/ 4.0 y_m)) (fabs (* z (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.6e-65) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.6d-65) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.6e-65) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 4.6e-65:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 4.6e-65)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 4.6e-65)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 4.6e-65], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999999e-65

   1. Initial program 94.6%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in x around 0 54.9%

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

   if 4.5999999999999999e-65 < x

   1. Initial program 85.5%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in z around inf 56.2%

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

      1. associate-*r/ 56.2%

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

      2. neg-mul-1 56.2%

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

      3. distribute-rgt-neg-in 56.2%

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

   7. Simplified 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-/l* 70.3%

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

      3. add-sqr-sqrt 38.2%

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

      4. sqrt-unprod 53.4%

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

      5. sqr-neg 53.4%

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

      6. sqrt-unprod 32.0%

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

      7. add-sqr-sqrt 70.3%

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

   9. Applied egg-rr 70.3%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-65}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 5.6e-90) (fabs (/ 4.0 y_m)) (fabs (* x (/ z y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 5.6e-90) {
		tmp = fabs((4.0 / 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 (x <= 5.6d-90) then
        tmp = abs((4.0d0 / 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 (x <= 5.6e-90) {
		tmp = Math.abs((4.0 / 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 x <= 5.6e-90:
		tmp = math.fabs((4.0 / 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 (x <= 5.6e-90)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(x * Float64(z / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 5.6e-90)
		tmp = abs((4.0 / 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[x, 5.6e-90], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-90

   1. Initial program 95.0%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in x around 0 55.2%

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

   if 5.5999999999999998e-90 < x

   1. Initial program 85.3%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in z around inf 56.5%

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

      1. associate-*r/ 56.5%

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

      2. neg-mul-1 56.5%

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

      3. distribute-rgt-neg-in 56.5%

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

   7. Simplified 56.5%

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

      1. distribute-rgt-neg-out 56.5%

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

      2. distribute-frac-neg 56.5%

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

      3. distribute-frac-neg2 56.5%

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

      4. associate-*r/ 61.7%

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

      5. *-commutative 61.7%

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

      6. add-sqr-sqrt 28.2%

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

      7. sqrt-unprod 47.1%

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

      8. sqr-neg 47.1%

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

      9. sqrt-unprod 33.4%

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

      10. add-sqr-sqrt 61.7%

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

   9. Applied egg-rr 61.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 5.6e-90) (fabs (/ 4.0 y_m)) (fabs (/ x (/ y_m z)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 5.6e-90) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((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 (x <= 5.6d-90) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((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 (x <= 5.6e-90) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 5.6e-90:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 5.6e-90)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(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 (x <= 5.6e-90)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((x / (y_m / z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 5.6e-90], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-90

   1. Initial program 95.0%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in x around 0 55.2%

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

   if 5.5999999999999998e-90 < x

   1. Initial program 85.3%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in z around inf 56.5%

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

      1. mul-1-neg 56.5%

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

      2. distribute-frac-neg2 56.5%

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

      3. associate-/l* 61.7%

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

   7. Simplified 61.7%

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

      1. clear-num 61.6%

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

      2. un-div-inv 61.7%

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

      3. add-sqr-sqrt 28.2%

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

      4. sqrt-unprod 47.1%

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

      5. sqr-neg 47.1%

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

      6. sqrt-unprod 33.4%

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

      7. add-sqr-sqrt 61.7%

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

   9. Applied egg-rr 61.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 1.4e-63) (fabs (/ 4.0 y_m)) (fabs (/ z (/ y_m x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4e-63) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	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 <= 1.4d-63) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((z / (y_m / x)))
    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 <= 1.4e-63) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 1.4e-63:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 1.4e-63)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 1.4e-63)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 1.4e-63], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4000000000000001e-63

   1. Initial program 94.6%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in x around 0 54.9%

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

   if 1.4000000000000001e-63 < x

   1. Initial program 85.5%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in z around inf 56.2%

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

      1. associate-*r/ 56.2%

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

      2. neg-mul-1 56.2%

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

      3. distribute-rgt-neg-in 56.2%

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

   7. Simplified 56.2%

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

      1. distribute-rgt-neg-out 56.2%

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

      2. distribute-frac-neg 56.2%

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

      3. distribute-frac-neg2 56.2%

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

      4. associate-*r/ 61.7%

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

      5. *-commutative 61.7%

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

      6. add-sqr-sqrt 27.5%

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

      7. sqrt-unprod 48.4%

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

      8. sqr-neg 48.4%

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

      9. sqrt-unprod 34.1%

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

      10. add-sqr-sqrt 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. associate-/r/ 67.1%

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

   11. Applied egg-rr 67.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -4.3e+70) (fabs (/ x (/ y_m z))) (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -4.3e+70) {
		tmp = fabs((x / (y_m / z)));
	} 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 <= (-4.3d+70)) then
        tmp = abs((x / (y_m / z)))
    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 <= -4.3e+70) {
		tmp = Math.abs((x / (y_m / z)));
	} 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 <= -4.3e+70:
		tmp = math.fabs((x / (y_m / z)))
	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 <= -4.3e+70)
		tmp = abs(Float64(x / Float64(y_m / z)));
	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 <= -4.3e+70)
		tmp = abs((x / (y_m / z)));
	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[LessEqual[z, -4.3e+70], N[Abs[N[(x / N[(y$95$m / z), $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 < -4.3000000000000001e70

   1. Initial program 93.6%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in z around inf 71.5%

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

      1. mul-1-neg 71.5%

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

      2. distribute-frac-neg2 71.5%

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

      3. associate-/l* 84.6%

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

   7. Simplified 84.6%

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

      1. clear-num 84.4%

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

      2. un-div-inv 84.7%

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

      3. add-sqr-sqrt 42.1%

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

      4. sqrt-unprod 51.4%

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

      5. sqr-neg 51.4%

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

      6. sqrt-unprod 42.4%

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

      7. add-sqr-sqrt 84.7%

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

   9. Applied egg-rr 84.7%

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

   if -4.3000000000000001e70 < z

   1. Initial program 91.0%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in z around 0 80.7%

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

      1. +-commutative 80.7%

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

      2. rem-square-sqrt 44.0%

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

      3. fabs-sqr 44.0%

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

      4. rem-square-sqrt 80.7%

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

      5. fabs-neg 80.7%

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

      6. distribute-neg-frac 80.7%

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

      7. distribute-neg-in 80.7%

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

      8. metadata-eval 80.7%

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

      9. +-commutative 80.7%

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

      10. sub-neg 80.7%

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

      11. rem-square-sqrt 36.2%

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

      12. fabs-sqr 36.2%

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

      13. rem-square-sqrt 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z 1.7e+19) (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 <= 1.7e+19) {
		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 <= 1.7d+19) 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 <= 1.7e+19) {
		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 <= 1.7e+19:
		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 <= 1.7e+19)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(z / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= 1.7e+19)
		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, 1.7e+19], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7e19

   1. Initial program 93.9%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. rem-square-sqrt 41.4%

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

      3. fabs-sqr 41.4%

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

      4. rem-square-sqrt 76.6%

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

      5. fabs-neg 76.6%

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

      6. distribute-neg-frac 76.6%

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

      7. distribute-neg-in 76.6%

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

      8. metadata-eval 76.6%

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

      9. +-commutative 76.6%

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

      10. sub-neg 76.6%

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

      11. rem-square-sqrt 34.8%

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

      12. fabs-sqr 34.8%

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

      13. rem-square-sqrt 76.6%

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

   7. Simplified 76.6%

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

   if 1.7e19 < z

   1. Initial program 81.6%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in z around inf 66.2%

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

      1. mul-1-neg 66.2%

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

      2. distribute-frac-neg2 66.2%

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

      3. associate-/l* 77.5%

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

   7. Simplified 77.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z 6.2e+18) (fabs (/ (- -4.0 x) y_m)) (fabs (* z (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= 6.2e+18) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 6.2d+18) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= 6.2e+18) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= 6.2e+18:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= 6.2e+18)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= 6.2e+18)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, 6.2e+18], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.2e18

   1. Initial program 93.9%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. rem-square-sqrt 41.4%

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

      3. fabs-sqr 41.4%

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

      4. rem-square-sqrt 76.6%

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

      5. fabs-neg 76.6%

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

      6. distribute-neg-frac 76.6%

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

      7. distribute-neg-in 76.6%

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

      8. metadata-eval 76.6%

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

      9. +-commutative 76.6%

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

      10. sub-neg 76.6%

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

      11. rem-square-sqrt 34.8%

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

      12. fabs-sqr 34.8%

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

      13. rem-square-sqrt 76.6%

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

   7. Simplified 76.6%

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

   if 6.2e18 < z

   1. Initial program 81.6%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in z around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

      3. distribute-lft-neg-in 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*r/ 75.7%

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

   7. Simplified 75.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z 1e+19) (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 <= 1e+19) {
		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 <= 1d+19) 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 <= 1e+19) {
		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 <= 1e+19:
		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 <= 1e+19)
		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 <= 1e+19)
		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, 1e+19], 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 2 regimes

2. if z < 1e19

   1. Initial program 93.9%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. rem-square-sqrt 41.4%

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

      3. fabs-sqr 41.4%

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

      4. rem-square-sqrt 76.6%

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

      5. fabs-neg 76.6%

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

      6. distribute-neg-frac 76.6%

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

      7. distribute-neg-in 76.6%

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

      8. metadata-eval 76.6%

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

      9. +-commutative 76.6%

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

      10. sub-neg 76.6%

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

      11. rem-square-sqrt 34.8%

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

      12. fabs-sqr 34.8%

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

      13. rem-square-sqrt 76.6%

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

   7. Simplified 76.6%

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

   if 1e19 < z

   1. Initial program 81.6%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in z around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

      3. distribute-rgt-neg-in 66.2%

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

   7. Simplified 66.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-*l/ 89.9%

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

   2. sub-div 94.1%

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

4. Applied egg-rr 94.1%

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

5. Final simplification 94.1%

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

Alternative 22: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(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 = abs((4.0 / y_m));
	else
		tmp = abs((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[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 95.0%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around 0 53.7%

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

   if 4 < x

   1. Initial program 82.2%

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

   3. Taylor expanded in z around 0 58.4%

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

   4. Taylor expanded in x around inf 57.3%

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

4. Final simplification 54.7%

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

Alternative 23: 39.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Simplified 94.1%

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

5. Taylor expanded in x around 0 40.8%

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

6. Final simplification 40.8%

   \[\leadsto \left|\frac{4}{y}\right| \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))