fabs fraction 1

Percentage Accurate: 91.7% → 99.8%

Time: 8.1s

Alternatives: 11

Speedup: 0.9×

• 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: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-35)
   (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 <= 2e-35) {
		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 <= 2e-35)
		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, 2e-35], 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 < 2.00000000000000002e-35

   1. Initial program 88.4%

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

      1. associate-*l/ 89.7%

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

      2. sub-div 97.3%

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

   4. Applied egg-rr 97.3%

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

   if 2.00000000000000002e-35 < y

   1. Initial program 96.0%

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

      1. fabs-sub 96.0%

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

      2. associate-*l/ 93.3%

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

      3. associate-*r/ 99.8%

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

      4. fma-neg 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. unsub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ \mathbf{if}\;\left|t\_0 - z \cdot \frac{x}{y\_m}\right| \leq 2 \cdot 10^{-93}:\\ \;\;\;\;\left|t\_0 - \frac{x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(\frac{4}{x} + \left(1 - z\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)))
   (if (<= (fabs (- t_0 (* z (/ x y_m)))) 2e-93)
     (fabs (- t_0 (/ (* x z) y_m)))
     (fabs (* (/ x y_m) (+ (/ 4.0 x) (- 1.0 z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if (fabs((t_0 - (z * (x / y_m)))) <= 2e-93) {
		tmp = fabs((t_0 - ((x * z) / y_m)));
	} else {
		tmp = fabs(((x / y_m) * ((4.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    if (abs((t_0 - (z * (x / y_m)))) <= 2d-93) then
        tmp = abs((t_0 - ((x * z) / y_m)))
    else
        tmp = abs(((x / y_m) * ((4.0d0 / 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 t_0 = (x + 4.0) / y_m;
	double tmp;
	if (Math.abs((t_0 - (z * (x / y_m)))) <= 2e-93) {
		tmp = Math.abs((t_0 - ((x * z) / y_m)));
	} else {
		tmp = Math.abs(((x / y_m) * ((4.0 / x) + (1.0 - z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	tmp = 0
	if math.fabs((t_0 - (z * (x / y_m)))) <= 2e-93:
		tmp = math.fabs((t_0 - ((x * z) / y_m)))
	else:
		tmp = math.fabs(((x / y_m) * ((4.0 / x) + (1.0 - z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	tmp = 0.0
	if (abs(Float64(t_0 - Float64(z * Float64(x / y_m)))) <= 2e-93)
		tmp = abs(Float64(t_0 - Float64(Float64(x * z) / y_m)));
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(Float64(4.0 / x) + Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	tmp = 0.0;
	if (abs((t_0 - (z * (x / y_m)))) <= 2e-93)
		tmp = abs((t_0 - ((x * z) / y_m)));
	else
		tmp = abs(((x / y_m) * ((4.0 / x) + (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-93], N[Abs[N[(t$95$0 - N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(N[(4.0 / x), $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1.9999999999999998e-93

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1.9999999999999998e-93 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 90.4%

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

      1. associate-*l/ 88.7%

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

      2. sub-div 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in x around inf 95.3%

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

      1. associate--l+ 95.3%

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

      2. associate-*r/ 95.3%

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

      3. metadata-eval 95.3%

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

   7. Simplified 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l- 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{-93}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(\frac{4}{x} + \left(1 - z\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right| \leq 2 \cdot 10^{-93}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(\frac{4}{x} + \left(1 - z\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m)))) 2e-93)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (* (/ x y_m) (+ (/ 4.0 x) (- 1.0 z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (fabs((((x + 4.0) / y_m) - (z * (x / y_m)))) <= 2e-93) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((x / y_m) * ((4.0 / 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 (abs((((x + 4.0d0) / y_m) - (z * (x / y_m)))) <= 2d-93) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((x / y_m) * ((4.0d0 / 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 (Math.abs((((x + 4.0) / y_m) - (z * (x / y_m)))) <= 2e-93) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((x / y_m) * ((4.0 / x) + (1.0 - z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if math.fabs((((x + 4.0) / y_m) - (z * (x / y_m)))) <= 2e-93:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((x / y_m) * ((4.0 / x) + (1.0 - z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m)))) <= 2e-93)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(Float64(4.0 / 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 (abs((((x + 4.0) / y_m) - (z * (x / y_m)))) <= 2e-93)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((x / y_m) * ((4.0 / x) + (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-93], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(N[(4.0 / x), $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1.9999999999999998e-93

   1. Initial program 91.1%

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

      1. associate-*l/ 100.0%

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

      2. sub-div 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 1.9999999999999998e-93 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 90.4%

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

      1. associate-*l/ 88.7%

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

      2. sub-div 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in x around inf 95.3%

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

      1. associate--l+ 95.3%

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

      2. associate-*r/ 95.3%

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

      3. metadata-eval 95.3%

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

   7. Simplified 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l- 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{-93}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(\frac{4}{x} + \left(1 - z\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0056:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+51} \lor \neg \left(x \leq 1.1 \cdot 10^{+85}\right) \land \left(x \leq 5.6 \cdot 10^{+136} \lor \neg \left(x \leq 5.5 \cdot 10^{+213}\right) \land x \leq 1.45 \cdot 10^{+233}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -6.5e+161)
     t_0
     (if (<= x -1.08e-51)
       t_1
       (if (<= x 0.0056)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 1.1e+51)
                 (and (not (<= x 1.1e+85))
                      (or (<= x 5.6e+136)
                          (and (not (<= x 5.5e+213)) (<= x 1.45e+233)))))
           t_0
           t_1))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -6.5e+161) {
		tmp = t_0;
	} else if (x <= -1.08e-51) {
		tmp = t_1;
	} else if (x <= 0.0056) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 1.1e+51) || (!(x <= 1.1e+85) && ((x <= 5.6e+136) || (!(x <= 5.5e+213) && (x <= 1.45e+233))))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-6.5d+161)) then
        tmp = t_0
    else if (x <= (-1.08d-51)) then
        tmp = t_1
    else if (x <= 0.0056d0) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 1.1d+51) .or. (.not. (x <= 1.1d+85)) .and. (x <= 5.6d+136) .or. (.not. (x <= 5.5d+213)) .and. (x <= 1.45d+233)) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -6.5e+161) {
		tmp = t_0;
	} else if (x <= -1.08e-51) {
		tmp = t_1;
	} else if (x <= 0.0056) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 1.1e+51) || (!(x <= 1.1e+85) && ((x <= 5.6e+136) || (!(x <= 5.5e+213) && (x <= 1.45e+233))))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -6.5e+161:
		tmp = t_0
	elif x <= -1.08e-51:
		tmp = t_1
	elif x <= 0.0056:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 1.1e+51) or (not (x <= 1.1e+85) and ((x <= 5.6e+136) or (not (x <= 5.5e+213) and (x <= 1.45e+233)))):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -6.5e+161)
		tmp = t_0;
	elseif (x <= -1.08e-51)
		tmp = t_1;
	elseif (x <= 0.0056)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 1.1e+51) || (!(x <= 1.1e+85) && ((x <= 5.6e+136) || (!(x <= 5.5e+213) && (x <= 1.45e+233)))))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -6.5e+161)
		tmp = t_0;
	elseif (x <= -1.08e-51)
		tmp = t_1;
	elseif (x <= 0.0056)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 1.1e+51) || (~((x <= 1.1e+85)) && ((x <= 5.6e+136) || (~((x <= 5.5e+213)) && (x <= 1.45e+233)))))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.5e+161], t$95$0, If[LessEqual[x, -1.08e-51], t$95$1, If[LessEqual[x, 0.0056], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 1.1e+51], And[N[Not[LessEqual[x, 1.1e+85]], $MachinePrecision], Or[LessEqual[x, 5.6e+136], And[N[Not[LessEqual[x, 5.5e+213]], $MachinePrecision], LessEqual[x, 1.45e+233]]]]], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e161 or 0.00559999999999999994 < x < 1.09999999999999996e51 or 1.1000000000000001e85 < x < 5.6000000000000004e136 or 5.50000000000000059e213 < x < 1.45000000000000006e233

   1. Initial program 90.1%

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

   3. Taylor expanded in z around 0 70.6%

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

   4. Taylor expanded in x around inf 69.9%

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

   if -6.5e161 < x < -1.08000000000000004e-51 or 1.09999999999999996e51 < x < 1.1000000000000001e85 or 5.6000000000000004e136 < x < 5.50000000000000059e213 or 1.45000000000000006e233 < x

   1. Initial program 82.9%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in z around inf 61.5%

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

      1. associate-*r/ 61.5%

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

      2. neg-mul-1 61.5%

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

      3. distribute-rgt-neg-in 61.5%

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

   7. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-/l* 71.2%

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

      3. add-sqr-sqrt 27.6%

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

      4. sqrt-unprod 60.7%

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

      5. sqr-neg 60.7%

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

      6. sqrt-unprod 43.5%

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

      7. add-sqr-sqrt 71.2%

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

   9. Applied egg-rr 71.2%

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

   if -1.08000000000000004e-51 < x < 0.00559999999999999994

   1. Initial program 96.6%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 79.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+161}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{-51}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 0.0056:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+51} \lor \neg \left(x \leq 1.1 \cdot 10^{+85}\right) \land \left(x \leq 5.6 \cdot 10^{+136} \lor \neg \left(x \leq 5.5 \cdot 10^{+213}\right) \land x \leq 1.45 \cdot 10^{+233}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0056:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -6.5e+158)
     t_0
     (if (<= x -9.8e-52)
       t_1
       (if (<= x 0.0056)
         (fabs (/ 4.0 y_m))
         (if (<= x 1.3e+53)
           t_0
           (if (<= x 8e+85)
             t_1
             (if (<= x 1.4e+132)
               t_0
               (if (<= x 6.8e+213)
                 t_1
                 (if (<= x 4.8e+232) t_0 (fabs (/ z (/ y_m x)))))))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -6.5e+158) {
		tmp = t_0;
	} else if (x <= -9.8e-52) {
		tmp = t_1;
	} else if (x <= 0.0056) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 1.3e+53) {
		tmp = t_0;
	} else if (x <= 8e+85) {
		tmp = t_1;
	} else if (x <= 1.4e+132) {
		tmp = t_0;
	} else if (x <= 6.8e+213) {
		tmp = t_1;
	} else if (x <= 4.8e+232) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-6.5d+158)) then
        tmp = t_0
    else if (x <= (-9.8d-52)) then
        tmp = t_1
    else if (x <= 0.0056d0) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 1.3d+53) then
        tmp = t_0
    else if (x <= 8d+85) then
        tmp = t_1
    else if (x <= 1.4d+132) then
        tmp = t_0
    else if (x <= 6.8d+213) then
        tmp = t_1
    else if (x <= 4.8d+232) then
        tmp = t_0
    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 t_0 = Math.abs((x / y_m));
	double t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -6.5e+158) {
		tmp = t_0;
	} else if (x <= -9.8e-52) {
		tmp = t_1;
	} else if (x <= 0.0056) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 1.3e+53) {
		tmp = t_0;
	} else if (x <= 8e+85) {
		tmp = t_1;
	} else if (x <= 1.4e+132) {
		tmp = t_0;
	} else if (x <= 6.8e+213) {
		tmp = t_1;
	} else if (x <= 4.8e+232) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -6.5e+158:
		tmp = t_0
	elif x <= -9.8e-52:
		tmp = t_1
	elif x <= 0.0056:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 1.3e+53:
		tmp = t_0
	elif x <= 8e+85:
		tmp = t_1
	elif x <= 1.4e+132:
		tmp = t_0
	elif x <= 6.8e+213:
		tmp = t_1
	elif x <= 4.8e+232:
		tmp = t_0
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -6.5e+158)
		tmp = t_0;
	elseif (x <= -9.8e-52)
		tmp = t_1;
	elseif (x <= 0.0056)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 1.3e+53)
		tmp = t_0;
	elseif (x <= 8e+85)
		tmp = t_1;
	elseif (x <= 1.4e+132)
		tmp = t_0;
	elseif (x <= 6.8e+213)
		tmp = t_1;
	elseif (x <= 4.8e+232)
		tmp = t_0;
	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)
	t_0 = abs((x / y_m));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -6.5e+158)
		tmp = t_0;
	elseif (x <= -9.8e-52)
		tmp = t_1;
	elseif (x <= 0.0056)
		tmp = abs((4.0 / y_m));
	elseif (x <= 1.3e+53)
		tmp = t_0;
	elseif (x <= 8e+85)
		tmp = t_1;
	elseif (x <= 1.4e+132)
		tmp = t_0;
	elseif (x <= 6.8e+213)
		tmp = t_1;
	elseif (x <= 4.8e+232)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.5e+158], t$95$0, If[LessEqual[x, -9.8e-52], t$95$1, If[LessEqual[x, 0.0056], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3e+53], t$95$0, If[LessEqual[x, 8e+85], t$95$1, If[LessEqual[x, 1.4e+132], t$95$0, If[LessEqual[x, 6.8e+213], t$95$1, If[LessEqual[x, 4.8e+232], t$95$0, N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.5000000000000001e158 or 0.00559999999999999994 < x < 1.29999999999999999e53 or 8.0000000000000001e85 < x < 1.4e132 or 6.79999999999999983e213 < x < 4.8000000000000003e232

   1. Initial program 90.1%

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

   3. Taylor expanded in z around 0 70.6%

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

   4. Taylor expanded in x around inf 69.9%

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

   if -6.5000000000000001e158 < x < -9.80000000000000037e-52 or 1.29999999999999999e53 < x < 8.0000000000000001e85 or 1.4e132 < x < 6.79999999999999983e213

   1. Initial program 87.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in z around inf 61.2%

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

      1. associate-*r/ 61.2%

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

      2. neg-mul-1 61.2%

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

      3. distribute-rgt-neg-in 61.2%

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

   7. Simplified 61.2%

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

      1. *-commutative 61.2%

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

      2. associate-/l* 70.2%

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

      3. add-sqr-sqrt 25.1%

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

      4. sqrt-unprod 57.0%

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

      5. sqr-neg 57.0%

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

      6. sqrt-unprod 44.9%

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

      7. add-sqr-sqrt 70.2%

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

   9. Applied egg-rr 70.2%

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

   if -9.80000000000000037e-52 < x < 0.00559999999999999994

   1. Initial program 96.6%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 79.9%

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

   if 4.8000000000000003e232 < x

   1. Initial program 64.9%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in z around inf 62.5%

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

      1. associate-*r/ 62.5%

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

      2. neg-mul-1 62.5%

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

      3. distribute-rgt-neg-in 62.5%

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

   7. Simplified 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-/l* 74.8%

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

      3. add-sqr-sqrt 36.5%

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

      4. sqrt-unprod 74.6%

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

      5. sqr-neg 74.6%

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

      6. sqrt-unprod 38.3%

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

      7. add-sqr-sqrt 74.8%

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

   9. Applied egg-rr 74.8%

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

       1. clear-num 74.8%

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

       2. un-div-inv 74.8%

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

   11. Applied egg-rr 74.8%

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

Alternative 6: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x \cdot z}{y\_m}\right|\\ t_1 := \left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+163}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+259}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* x z) y_m))) (t_1 (fabs (/ (- -4.0 x) y_m))))
   (if (<= z -2.05e+197)
     t_0
     (if (<= z 2.35e+112)
       t_1
       (if (<= z 7e+163)
         (fabs (* x (/ z y_m)))
         (if (<= z 4.5e+178)
           t_1
           (if (<= z 6.2e+259) (fabs (* z (/ x y_m))) t_0)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x * z) / y_m));
	double t_1 = fabs(((-4.0 - x) / y_m));
	double tmp;
	if (z <= -2.05e+197) {
		tmp = t_0;
	} else if (z <= 2.35e+112) {
		tmp = t_1;
	} else if (z <= 7e+163) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= 4.5e+178) {
		tmp = t_1;
	} else if (z <= 6.2e+259) {
		tmp = fabs((z * (x / y_m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(((x * z) / y_m))
    t_1 = abs((((-4.0d0) - x) / y_m))
    if (z <= (-2.05d+197)) then
        tmp = t_0
    else if (z <= 2.35d+112) then
        tmp = t_1
    else if (z <= 7d+163) then
        tmp = abs((x * (z / y_m)))
    else if (z <= 4.5d+178) then
        tmp = t_1
    else if (z <= 6.2d+259) then
        tmp = abs((z * (x / y_m)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x * z) / y_m));
	double t_1 = Math.abs(((-4.0 - x) / y_m));
	double tmp;
	if (z <= -2.05e+197) {
		tmp = t_0;
	} else if (z <= 2.35e+112) {
		tmp = t_1;
	} else if (z <= 7e+163) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= 4.5e+178) {
		tmp = t_1;
	} else if (z <= 6.2e+259) {
		tmp = Math.abs((z * (x / y_m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x * z) / y_m))
	t_1 = math.fabs(((-4.0 - x) / y_m))
	tmp = 0
	if z <= -2.05e+197:
		tmp = t_0
	elif z <= 2.35e+112:
		tmp = t_1
	elif z <= 7e+163:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= 4.5e+178:
		tmp = t_1
	elif z <= 6.2e+259:
		tmp = math.fabs((z * (x / y_m)))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x * z) / y_m))
	t_1 = abs(Float64(Float64(-4.0 - x) / y_m))
	tmp = 0.0
	if (z <= -2.05e+197)
		tmp = t_0;
	elseif (z <= 2.35e+112)
		tmp = t_1;
	elseif (z <= 7e+163)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= 4.5e+178)
		tmp = t_1;
	elseif (z <= 6.2e+259)
		tmp = abs(Float64(z * Float64(x / y_m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x * z) / y_m));
	t_1 = abs(((-4.0 - x) / y_m));
	tmp = 0.0;
	if (z <= -2.05e+197)
		tmp = t_0;
	elseif (z <= 2.35e+112)
		tmp = t_1;
	elseif (z <= 7e+163)
		tmp = abs((x * (z / y_m)));
	elseif (z <= 4.5e+178)
		tmp = t_1;
	elseif (z <= 6.2e+259)
		tmp = abs((z * (x / y_m)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2.05e+197], t$95$0, If[LessEqual[z, 2.35e+112], t$95$1, If[LessEqual[z, 7e+163], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.5e+178], t$95$1, If[LessEqual[z, 6.2e+259], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.05000000000000015e197 or 6.2000000000000007e259 < z

   1. Initial program 81.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in z around inf 90.7%

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

      1. mul-1-neg 90.7%

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

      2. distribute-frac-neg2 90.7%

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

      3. associate-/l* 72.7%

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

   7. Simplified 72.7%

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

      1. associate-*r/ 90.7%

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

      2. add-sqr-sqrt 49.1%

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

      3. sqrt-unprod 69.3%

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

      4. sqr-neg 69.3%

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

      5. sqrt-unprod 41.3%

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

      6. add-sqr-sqrt 90.7%

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

   9. Applied egg-rr 90.7%

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

   if -2.05000000000000015e197 < z < 2.34999999999999999e112 or 7.0000000000000005e163 < z < 4.4999999999999997e178

   1. Initial program 92.4%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in z around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. rem-square-sqrt 49.0%

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

      3. fabs-sqr 49.0%

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

      4. rem-square-sqrt 84.3%

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

      5. fabs-neg 84.3%

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

      6. distribute-neg-frac 84.3%

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

      7. distribute-neg-in 84.3%

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

      8. metadata-eval 84.3%

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

      9. +-commutative 84.3%

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

      10. sub-neg 84.3%

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

      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 84.3%

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

   7. Simplified 84.3%

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

   if 2.34999999999999999e112 < z < 7.0000000000000005e163

   1. Initial program 76.6%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in z around inf 65.3%

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

      1. associate-*r/ 65.3%

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

      2. neg-mul-1 65.3%

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

      3. distribute-rgt-neg-in 65.3%

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

   7. Simplified 65.3%

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

      1. distribute-rgt-neg-out 65.3%

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

      2. distribute-frac-neg 65.3%

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

      3. distribute-frac-neg2 65.3%

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

      4. associate-*r/ 94.6%

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

      5. *-commutative 94.6%

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

      6. add-sqr-sqrt 48.4%

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

      7. sqrt-unprod 57.0%

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

      8. sqr-neg 57.0%

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

      9. sqrt-unprod 45.8%

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

      10. add-sqr-sqrt 94.6%

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

   9. Applied egg-rr 94.6%

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

   if 4.4999999999999997e178 < z < 6.2000000000000007e259

   1. Initial program 99.9%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in z around inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. neg-mul-1 67.9%

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

      3. distribute-rgt-neg-in 67.9%

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

   7. Simplified 67.9%

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

      1. *-commutative 67.9%

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

      2. associate-/l* 91.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 28.0%

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

      5. sqr-neg 28.0%

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

      6. sqrt-unprod 91.5%

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

      7. add-sqr-sqrt 91.9%

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

   9. Applied egg-rr 91.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+197}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+163}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+259}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+87} \lor \neg \left(x \leq 0.0056\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1e+87) (not (<= x 0.0056)))
   (fabs (* x (/ (- 1.0 z) y_m)))
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1e+87) || !(x <= 0.0056)) {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1d+87)) .or. (.not. (x <= 0.0056d0))) then
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1e+87) || !(x <= 0.0056)) {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1e+87) or not (x <= 0.0056):
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1e+87) || !(x <= 0.0056))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1e+87) || ~((x <= 0.0056)))
		tmp = abs((x * ((1.0 - z) / y_m)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1e+87], N[Not[LessEqual[x, 0.0056]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999996e86 or 0.00559999999999999994 < x

   1. Initial program 82.6%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around inf 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/l* 99.5%

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

      3. associate-*r* 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. mul-1-neg 99.5%

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

      7. neg-sub0 99.5%

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

      8. associate-+l- 99.5%

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

      9. neg-sub0 99.5%

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

      10. +-commutative 99.5%

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

      11. unsub-neg 99.5%

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

   7. Simplified 99.5%

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

   if -9.9999999999999996e86 < x < 0.00559999999999999994

   1. Initial program 96.0%

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

      1. associate-*l/ 99.2%

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

      2. sub-div 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+87} \lor \neg \left(x \leq 0.0056\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-10} \lor \neg \left(x \leq 1.5 \cdot 10^{-38}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x + 4\right) \cdot \frac{1}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -2e-10) (not (<= x 1.5e-38)))
   (fabs (* x (/ (- 1.0 z) y_m)))
   (fabs (* (+ x 4.0) (/ 1.0 y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -2e-10) || !(x <= 1.5e-38)) {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = fabs(((x + 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 <= (-2d-10)) .or. (.not. (x <= 1.5d-38))) then
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    else
        tmp = abs(((x + 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 <= -2e-10) || !(x <= 1.5e-38)) {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = Math.abs(((x + 4.0) * (1.0 / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -2e-10) or not (x <= 1.5e-38):
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	else:
		tmp = math.fabs(((x + 4.0) * (1.0 / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -2e-10) || !(x <= 1.5e-38))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	else
		tmp = abs(Float64(Float64(x + 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 <= -2e-10) || ~((x <= 1.5e-38)))
		tmp = abs((x * ((1.0 - z) / y_m)));
	else
		tmp = abs(((x + 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[Or[LessEqual[x, -2e-10], N[Not[LessEqual[x, 1.5e-38]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x + 4.0), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000007e-10 or 1.49999999999999994e-38 < x

   1. Initial program 85.8%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-/l* 93.5%

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

      3. associate-*r* 93.5%

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

      4. *-commutative 93.5%

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

      5. associate-*r/ 93.5%

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

      6. mul-1-neg 93.5%

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

      7. neg-sub0 93.5%

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

      8. associate-+l- 93.5%

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

      9. neg-sub0 93.5%

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

      10. +-commutative 93.5%

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

      11. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   if -2.00000000000000007e-10 < x < 1.49999999999999994e-38

   1. Initial program 96.5%

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

   3. Taylor expanded in z around 0 80.4%

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

      1. div-inv 80.4%

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

      2. distribute-rgt-out 80.4%

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

      3. +-commutative 80.4%

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

   5. Applied egg-rr 80.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-10} \lor \neg \left(x \leq 1.5 \cdot 10^{-38}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x + 4\right) \cdot \frac{1}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-9} \lor \neg \left(x \leq 3.4 \cdot 10^{-39}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.7e-9) (not (<= x 3.4e-39)))
   (fabs (* x (/ (- 1.0 z) y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.7e-9) || !(x <= 3.4e-39)) {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.7d-9)) .or. (.not. (x <= 3.4d-39))) then
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.7e-9) || !(x <= 3.4e-39)) {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.7e-9) or not (x <= 3.4e-39):
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.7e-9) || !(x <= 3.4e-39))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.7e-9) || ~((x <= 3.4e-39)))
		tmp = abs((x * ((1.0 - z) / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.7e-9], N[Not[LessEqual[x, 3.4e-39]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-9 or 3.3999999999999999e-39 < x

   1. Initial program 85.8%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-/l* 93.5%

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

      3. associate-*r* 93.5%

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

      4. *-commutative 93.5%

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

      5. associate-*r/ 93.5%

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

      6. mul-1-neg 93.5%

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

      7. neg-sub0 93.5%

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

      8. associate-+l- 93.5%

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

      9. neg-sub0 93.5%

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

      10. +-commutative 93.5%

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

      11. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   if -1.6999999999999999e-9 < x < 3.3999999999999999e-39

   1. Initial program 96.5%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. rem-square-sqrt 44.6%

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

      3. fabs-sqr 44.6%

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

      4. rem-square-sqrt 80.4%

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

      5. fabs-neg 80.4%

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

      6. distribute-neg-frac 80.4%

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

      7. distribute-neg-in 80.4%

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

      8. metadata-eval 80.4%

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

      9. +-commutative 80.4%

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

      10. sub-neg 80.4%

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

      11. rem-square-sqrt 35.2%

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

      12. fabs-sqr 35.2%

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

      13. rem-square-sqrt 80.4%

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

   7. Simplified 80.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-9} \lor \neg \left(x \leq 3.4 \cdot 10^{-39}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -380000000000 \lor \neg \left(x \leq 1.75 \cdot 10^{-10}\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -380000000000.0) (not (<= x 1.75e-10)))
   (fabs (/ x y_m))
   (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -380000000000.0) || !(x <= 1.75e-10)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-380000000000.0d0)) .or. (.not. (x <= 1.75d-10))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -380000000000.0) || !(x <= 1.75e-10)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -380000000000.0) or not (x <= 1.75e-10):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -380000000000.0) || !(x <= 1.75e-10))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -380000000000.0) || ~((x <= 1.75e-10)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -380000000000.0], N[Not[LessEqual[x, 1.75e-10]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e11 or 1.7499999999999999e-10 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in z around 0 63.1%

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

   4. Taylor expanded in x around inf 61.1%

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

   if -3.8e11 < x < 1.7499999999999999e-10

   1. Initial program 96.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 74.9%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380000000000 \lor \neg \left(x \leq 1.75 \cdot 10^{-10}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.4% 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 90.5%

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

3. Simplified 96.0%

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

5. Taylor expanded in x around 0 41.3%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))