fabs fraction 1

Percentage Accurate: 91.8% → 99.7%

Time: 7.4s

Alternatives: 10

Speedup: 1.0×

• 20.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 9.2 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 9.2e-46)
   (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 <= 9.2e-46) {
		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 <= 9.2e-46)
		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, 9.2e-46], 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 < 9.1999999999999997e-46

   1. Initial program 85.8%

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

      1. associate-*l/ 88.8%

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

      2. sub-div 95.1%

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

   4. Applied egg-rr 95.1%

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

   if 9.1999999999999997e-46 < y

   1. Initial program 96.5%

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

      1. fabs-sub 96.5%

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

      2. associate-*l/ 94.1%

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

      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 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ t_1 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -8 \cdot 10^{+105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4900000000:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -8e+105)
     t_0
     (if (<= x -3.8e+80)
       t_1
       (if (<= x -2e-78)
         t_0
         (if (<= x 4900000000.0)
           (fabs (/ 4.0 y_m))
           (if (<= x 4.4e+70) t_0 t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -8e+105) {
		tmp = t_0;
	} else if (x <= -3.8e+80) {
		tmp = t_1;
	} else if (x <= -2e-78) {
		tmp = t_0;
	} else if (x <= 4900000000.0) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 4.4e+70) {
		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((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-8d+105)) then
        tmp = t_0
    else if (x <= (-3.8d+80)) then
        tmp = t_1
    else if (x <= (-2d-78)) then
        tmp = t_0
    else if (x <= 4900000000.0d0) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 4.4d+70) 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((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -8e+105) {
		tmp = t_0;
	} else if (x <= -3.8e+80) {
		tmp = t_1;
	} else if (x <= -2e-78) {
		tmp = t_0;
	} else if (x <= 4900000000.0) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 4.4e+70) {
		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((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -8e+105:
		tmp = t_0
	elif x <= -3.8e+80:
		tmp = t_1
	elif x <= -2e-78:
		tmp = t_0
	elif x <= 4900000000.0:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 4.4e+70:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -8e+105)
		tmp = t_0;
	elseif (x <= -3.8e+80)
		tmp = t_1;
	elseif (x <= -2e-78)
		tmp = t_0;
	elseif (x <= 4900000000.0)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 4.4e+70)
		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((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -8e+105)
		tmp = t_0;
	elseif (x <= -3.8e+80)
		tmp = t_1;
	elseif (x <= -2e-78)
		tmp = t_0;
	elseif (x <= 4900000000.0)
		tmp = abs((4.0 / y_m));
	elseif (x <= 4.4e+70)
		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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8e+105], t$95$0, If[LessEqual[x, -3.8e+80], t$95$1, If[LessEqual[x, -2e-78], t$95$0, If[LessEqual[x, 4900000000.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.4e+70], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.9999999999999995e105 or -3.79999999999999997e80 < x < -2e-78 or 4.9e9 < x < 4.40000000000000001e70

   1. Initial program 88.8%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in z around inf 54.0%

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

      1. associate-*r/ 54.0%

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

      2. neg-mul-1 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

   7. Simplified 54.0%

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

      1. *-commutative 54.0%

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

      2. associate-/l* 72.5%

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

      3. add-sqr-sqrt 39.4%

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

      4. sqrt-unprod 50.4%

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

      5. sqr-neg 50.4%

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

      6. sqrt-unprod 32.8%

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

      7. add-sqr-sqrt 72.5%

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

   9. Applied egg-rr 72.5%

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

   if -7.9999999999999995e105 < x < -3.79999999999999997e80 or 4.40000000000000001e70 < x

   1. Initial program 85.1%

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

   3. Taylor expanded in z around 0 73.8%

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

   4. Taylor expanded in x around inf 73.8%

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

   if -2e-78 < x < 4.9e9

   1. Initial program 91.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 70.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+105}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4900000000:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+70}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y\_m}\right|\\ t_1 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-172}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;z \leq 2.7:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ z y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= z -1.0)
     t_0
     (if (<= z 2.15e-303)
       t_1
       (if (<= z 3e-172) (fabs (/ 4.0 y_m)) (if (<= z 2.7) t_1 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((x / y_m));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.15e-303) {
		tmp = t_1;
	} else if (z <= 3e-172) {
		tmp = fabs((4.0 / y_m));
	} else if (z <= 2.7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x * (z / y_m)))
    t_1 = abs((x / y_m))
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 2.15d-303) then
        tmp = t_1
    else if (z <= 3d-172) then
        tmp = abs((4.0d0 / y_m))
    else if (z <= 2.7d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x * (z / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.15e-303) {
		tmp = t_1;
	} else if (z <= 3e-172) {
		tmp = Math.abs((4.0 / y_m));
	} else if (z <= 2.7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x * (z / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 2.15e-303:
		tmp = t_1
	elif z <= 3e-172:
		tmp = math.fabs((4.0 / y_m))
	elif z <= 2.7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x * Float64(z / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.15e-303)
		tmp = t_1;
	elseif (z <= 3e-172)
		tmp = abs(Float64(4.0 / y_m));
	elseif (z <= 2.7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x * (z / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.15e-303)
		tmp = t_1;
	elseif (z <= 3e-172)
		tmp = abs((4.0 / y_m));
	elseif (z <= 2.7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 2.15e-303], t$95$1, If[LessEqual[z, 3e-172], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 2.7000000000000002 < z

   1. Initial program 87.3%

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

   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 70.1%

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

      1. associate-*r/ 70.1%

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

      2. neg-mul-1 70.1%

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

      3. distribute-rgt-neg-in 70.1%

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

   7. Simplified 70.1%

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

      1. distribute-rgt-neg-out 70.1%

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

      2. distribute-frac-neg 70.1%

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

      3. distribute-frac-neg2 70.1%

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

      4. associate-*r/ 77.5%

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

      5. *-commutative 77.5%

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

      6. add-sqr-sqrt 42.3%

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

      7. sqrt-unprod 54.2%

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

      8. sqr-neg 54.2%

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

      9. sqrt-unprod 35.0%

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

      10. add-sqr-sqrt 77.5%

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

   9. Applied egg-rr 77.5%

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

   if -1 < z < 2.14999999999999991e-303 or 2.99999999999999984e-172 < z < 2.7000000000000002

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0 99.0%

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

   4. Taylor expanded in x around inf 62.4%

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

   if 2.14999999999999991e-303 < z < 2.99999999999999984e-172

   1. Initial program 71.4%

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

   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 64.6%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-303}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-172}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 2.7:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-175}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;z \leq 2.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= z -1.0)
     (fabs (/ x (/ y_m z)))
     (if (<= z 8.5e-303)
       t_0
       (if (<= z 4.8e-175)
         (fabs (/ 4.0 y_m))
         (if (<= z 2.6) t_0 (fabs (* x (/ z y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (z <= -1.0) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 8.5e-303) {
		tmp = t_0;
	} else if (z <= 4.8e-175) {
		tmp = fabs((4.0 / y_m));
	} else if (z <= 2.6) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (z <= (-1.0d0)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 8.5d-303) then
        tmp = t_0
    else if (z <= 4.8d-175) then
        tmp = abs((4.0d0 / y_m))
    else if (z <= 2.6d0) then
        tmp = t_0
    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 t_0 = Math.abs((x / y_m));
	double tmp;
	if (z <= -1.0) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 8.5e-303) {
		tmp = t_0;
	} else if (z <= 4.8e-175) {
		tmp = Math.abs((4.0 / y_m));
	} else if (z <= 2.6) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if z <= -1.0:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 8.5e-303:
		tmp = t_0
	elif z <= 4.8e-175:
		tmp = math.fabs((4.0 / y_m))
	elif z <= 2.6:
		tmp = t_0
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (z <= -1.0)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 8.5e-303)
		tmp = t_0;
	elseif (z <= 4.8e-175)
		tmp = abs(Float64(4.0 / y_m));
	elseif (z <= 2.6)
		tmp = t_0;
	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)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 8.5e-303)
		tmp = t_0;
	elseif (z <= 4.8e-175)
		tmp = abs((4.0 / y_m));
	elseif (z <= 2.6)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.0], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 8.5e-303], t$95$0, If[LessEqual[z, 4.8e-175], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.6], t$95$0, N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1

   1. Initial program 92.1%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in z around inf 72.4%

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

      1. mul-1-neg 72.4%

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

      2. distribute-frac-neg2 72.4%

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

      3. associate-/l* 77.8%

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

   7. Simplified 77.8%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.9%

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

      3. add-sqr-sqrt 42.0%

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

      4. sqrt-unprod 54.7%

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

      5. sqr-neg 54.7%

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

      6. sqrt-unprod 35.6%

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

      7. add-sqr-sqrt 77.9%

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

   9. Applied egg-rr 77.9%

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

   if -1 < z < 8.5e-303 or 4.8e-175 < z < 2.60000000000000009

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0 99.0%

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

   4. Taylor expanded in x around inf 62.4%

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

   if 8.5e-303 < z < 4.8e-175

   1. Initial program 71.4%

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

   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 64.6%

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

   if 2.60000000000000009 < z

   1. Initial program 82.2%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in z around inf 67.7%

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

      1. associate-*r/ 67.7%

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

      2. neg-mul-1 67.7%

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

      3. distribute-rgt-neg-in 67.7%

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

   7. Simplified 67.7%

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

      1. distribute-rgt-neg-out 67.7%

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

      2. distribute-frac-neg 67.7%

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

      3. distribute-frac-neg2 67.7%

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

      4. associate-*r/ 77.2%

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

      5. *-commutative 77.2%

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

      6. add-sqr-sqrt 42.6%

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

      7. sqrt-unprod 53.8%

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

      8. sqr-neg 53.8%

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

      9. sqrt-unprod 34.4%

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

      10. add-sqr-sqrt 77.2%

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

   9. Applied egg-rr 77.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-303}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-175}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 2.6:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{if}\;z \leq -9 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 9000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+92}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- -4.0 x) y_m))))
   (if (<= z -9e+35)
     (fabs (/ x (/ y_m z)))
     (if (<= z 9000000000000.0)
       t_0
       (if (<= z 1.45e+92)
         (fabs (* z (/ x y_m)))
         (if (<= z 2.55e+157) t_0 (fabs (* x (/ z y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((-4.0 - x) / y_m));
	double tmp;
	if (z <= -9e+35) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 9000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.45e+92) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 2.55e+157) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = abs((((-4.0d0) - x) / y_m))
    if (z <= (-9d+35)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 9000000000000.0d0) then
        tmp = t_0
    else if (z <= 1.45d+92) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 2.55d+157) then
        tmp = t_0
    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 t_0 = Math.abs(((-4.0 - x) / y_m));
	double tmp;
	if (z <= -9e+35) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 9000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.45e+92) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 2.55e+157) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((-4.0 - x) / y_m))
	tmp = 0
	if z <= -9e+35:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 9000000000000.0:
		tmp = t_0
	elif z <= 1.45e+92:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 2.55e+157:
		tmp = t_0
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(-4.0 - x) / y_m))
	tmp = 0.0
	if (z <= -9e+35)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 9000000000000.0)
		tmp = t_0;
	elseif (z <= 1.45e+92)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 2.55e+157)
		tmp = t_0;
	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)
	t_0 = abs(((-4.0 - x) / y_m));
	tmp = 0.0;
	if (z <= -9e+35)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 9000000000000.0)
		tmp = t_0;
	elseif (z <= 1.45e+92)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 2.55e+157)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -9e+35], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 9000000000000.0], t$95$0, If[LessEqual[z, 1.45e+92], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.55e+157], t$95$0, N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.9999999999999993e35

   1. Initial program 90.9%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in z around inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. distribute-frac-neg2 78.1%

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

      3. associate-/l* 84.5%

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

   7. Simplified 84.5%

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

      1. clear-num 84.3%

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

      2. un-div-inv 84.5%

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

      3. add-sqr-sqrt 46.4%

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

      4. sqrt-unprod 57.8%

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

      5. sqr-neg 57.8%

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

      6. sqrt-unprod 37.8%

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

      7. add-sqr-sqrt 84.5%

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

   9. Applied egg-rr 84.5%

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

   if -8.9999999999999993e35 < z < 9e12 or 1.45e92 < z < 2.55e157

   1. Initial program 91.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in z around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. rem-square-sqrt 51.5%

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

      3. fabs-sqr 51.5%

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

      4. rem-square-sqrt 93.9%

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

      5. fabs-neg 93.9%

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

      6. distribute-neg-frac 93.9%

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

      7. distribute-neg-in 93.9%

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

      8. metadata-eval 93.9%

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

      9. +-commutative 93.9%

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

      10. sub-neg 93.9%

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

      11. rem-square-sqrt 41.8%

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

      12. fabs-sqr 41.8%

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

      13. rem-square-sqrt 93.9%

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

   7. Simplified 93.9%

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

   if 9e12 < z < 1.45e92

   1. Initial program 92.5%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in z around inf 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. *-commutative 66.2%

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

      2. associate-/l* 78.7%

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

      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 78.7%

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

      5. sqr-neg 78.7%

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

      6. sqrt-unprod 78.1%

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

      7. add-sqr-sqrt 78.7%

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

   9. Applied egg-rr 78.7%

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

   if 2.55e157 < z

   1. Initial program 74.0%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in z around inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

      3. distribute-rgt-neg-in 86.3%

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

   7. Simplified 86.3%

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

      1. distribute-rgt-neg-out 86.3%

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

      2. distribute-frac-neg 86.3%

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

      3. distribute-frac-neg2 86.3%

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

      4. associate-*r/ 97.2%

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

      5. *-commutative 97.2%

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

      6. add-sqr-sqrt 50.1%

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

      7. sqrt-unprod 68.6%

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

      8. sqr-neg 68.6%

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

      9. sqrt-unprod 46.8%

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

      10. add-sqr-sqrt 97.2%

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

   9. Applied egg-rr 97.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 9000000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+92}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+157}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 2000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+92}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- -4.0 x) y_m))))
   (if (<= z -2.1e+37)
     (fabs (/ x (/ y_m z)))
     (if (<= z 2000000000000.0)
       t_0
       (if (<= z 1.5e+92)
         (fabs (* (- 1.0 z) (/ x y_m)))
         (if (<= z 2.55e+157) t_0 (fabs (* x (/ z y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((-4.0 - x) / y_m));
	double tmp;
	if (z <= -2.1e+37) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 2000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.5e+92) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else if (z <= 2.55e+157) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = abs((((-4.0d0) - x) / y_m))
    if (z <= (-2.1d+37)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 2000000000000.0d0) then
        tmp = t_0
    else if (z <= 1.5d+92) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else if (z <= 2.55d+157) then
        tmp = t_0
    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 t_0 = Math.abs(((-4.0 - x) / y_m));
	double tmp;
	if (z <= -2.1e+37) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 2000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.5e+92) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else if (z <= 2.55e+157) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((-4.0 - x) / y_m))
	tmp = 0
	if z <= -2.1e+37:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 2000000000000.0:
		tmp = t_0
	elif z <= 1.5e+92:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	elif z <= 2.55e+157:
		tmp = t_0
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(-4.0 - x) / y_m))
	tmp = 0.0
	if (z <= -2.1e+37)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 2000000000000.0)
		tmp = t_0;
	elseif (z <= 1.5e+92)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	elseif (z <= 2.55e+157)
		tmp = t_0;
	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)
	t_0 = abs(((-4.0 - x) / y_m));
	tmp = 0.0;
	if (z <= -2.1e+37)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 2000000000000.0)
		tmp = t_0;
	elseif (z <= 1.5e+92)
		tmp = abs(((1.0 - z) * (x / y_m)));
	elseif (z <= 2.55e+157)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2.1e+37], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2000000000000.0], t$95$0, If[LessEqual[z, 1.5e+92], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.55e+157], t$95$0, N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.1000000000000001e37

   1. Initial program 90.9%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in z around inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. distribute-frac-neg2 78.1%

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

      3. associate-/l* 84.5%

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

   7. Simplified 84.5%

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

      1. clear-num 84.3%

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

      2. un-div-inv 84.5%

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

      3. add-sqr-sqrt 46.4%

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

      4. sqrt-unprod 57.8%

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

      5. sqr-neg 57.8%

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

      6. sqrt-unprod 37.8%

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

      7. add-sqr-sqrt 84.5%

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

   9. Applied egg-rr 84.5%

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

   if -2.1000000000000001e37 < z < 2e12 or 1.50000000000000007e92 < z < 2.55e157

   1. Initial program 91.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in z around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. rem-square-sqrt 51.5%

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

      3. fabs-sqr 51.5%

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

      4. rem-square-sqrt 93.9%

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

      5. fabs-neg 93.9%

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

      6. distribute-neg-frac 93.9%

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

      7. distribute-neg-in 93.9%

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

      8. metadata-eval 93.9%

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

      9. +-commutative 93.9%

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

      10. sub-neg 93.9%

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

      11. rem-square-sqrt 41.8%

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

      12. fabs-sqr 41.8%

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

      13. rem-square-sqrt 93.9%

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

   7. Simplified 93.9%

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

   if 2e12 < z < 1.50000000000000007e92

   1. Initial program 92.5%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in x around inf 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* 79.7%

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

      4. distribute-lft-neg-in 79.7%

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

      5. neg-sub0 79.7%

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

      6. associate-+l- 79.7%

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

      7. neg-sub0 79.7%

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

      8. +-commutative 79.7%

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

      9. unsub-neg 79.7%

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

   7. Simplified 79.7%

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

   if 2.55e157 < z

   1. Initial program 74.0%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in z around inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

      3. distribute-rgt-neg-in 86.3%

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

   7. Simplified 86.3%

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

      1. distribute-rgt-neg-out 86.3%

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

      2. distribute-frac-neg 86.3%

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

      3. distribute-frac-neg2 86.3%

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

      4. associate-*r/ 97.2%

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

      5. *-commutative 97.2%

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

      6. add-sqr-sqrt 50.1%

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

      7. sqrt-unprod 68.6%

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

      8. sqr-neg 68.6%

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

      9. sqrt-unprod 46.8%

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

      10. add-sqr-sqrt 97.2%

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

   9. Applied egg-rr 97.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 2000000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+92}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+157}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.65 \cdot 10^{-118}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.65e-118)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.65e-118) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.65d-118) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.65e-118) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 1.65e-118:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.65e-118)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.65e-118)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.65e-118], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.65e-118

   1. Initial program 84.5%

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

      1. associate-*l/ 87.7%

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

      2. sub-div 94.6%

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

   4. Applied egg-rr 94.6%

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

   if 1.65e-118 < y

   1. Initial program 97.1%

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

      1. associate-*l/ 95.0%

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

      2. associate-*r/ 98.9%

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

      3. clear-num 98.8%

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

      4. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-118}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.2)
   (fabs (* (- 1.0 z) (/ x 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 <= -8.2) {
		tmp = fabs(((1.0 - z) * (x / 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 <= (-8.2d0)) then
        tmp = abs(((1.0d0 - z) * (x / 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 <= -8.2) {
		tmp = Math.abs(((1.0 - z) * (x / 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 <= -8.2:
		tmp = math.fabs(((1.0 - z) * (x / 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 <= -8.2)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / 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 <= -8.2)
		tmp = abs(((1.0 - z) * (x / 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[LessEqual[x, -8.2], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / 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 < -8.1999999999999993

   1. Initial program 85.1%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf 88.4%

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

      1. mul-1-neg 88.4%

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

      2. *-commutative 88.4%

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

      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 -8.1999999999999993 < x

   1. Initial program 91.2%

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

      1. associate-*l/ 94.8%

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

      2. sub-div 97.7%

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

   4. Applied egg-rr 97.7%

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

4. Final simplification 98.4%

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

Alternative 9: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -8.2) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -8.2) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.2d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -8.2) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -8.2) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -8.2) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -8.2) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -8.2], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.1999999999999993 or 4 < x

   1. Initial program 86.7%

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

   3. Taylor expanded in z around 0 60.9%

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

   4. Taylor expanded in x around inf 60.5%

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

   if -8.1999999999999993 < x < 4

   1. Initial program 92.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 68.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.2% 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 89.2%

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

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 x around 0 33.3%

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

6. Final simplification 33.3%

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

Reproduce

herbie shell --seed 2024080 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))