fabs fraction 1

Percentage Accurate: 92.0% → 99.9%

Time: 9.0s

Alternatives: 9

Speedup: 1.0×

• 20.9% 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: 92.0% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2700000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2700000.0)
   (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 <= 2700000.0) {
		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 <= 2700000.0)
		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, 2700000.0], 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 < 2.7e6

   1. Initial program 89.3%

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

      1. associate-*l/ 93.1%

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

      2. sub-div 99.0%

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

   4. Applied egg-rr 99.0%

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

   if 2.7e6 < y

   1. Initial program 96.2%

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

      1. fabs-sub 96.2%

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

      2. associate-*l/ 90.7%

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

      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. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2700000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{-112}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -2.45e+82)
     t_0
     (if (<= x -1.06e-42)
       t_1
       (if (<= x 1e-112) (fabs (/ 4.0 y_m)) (if (<= x 2.95e+54) t_1 t_0))))))

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 <= -2.45e+82) {
		tmp = t_0;
	} else if (x <= -1.06e-42) {
		tmp = t_1;
	} else if (x <= 1e-112) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 2.95e+54) {
		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 / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-2.45d+82)) then
        tmp = t_0
    else if (x <= (-1.06d-42)) then
        tmp = t_1
    else if (x <= 1d-112) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 2.95d+54) 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 / y_m));
	double t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -2.45e+82) {
		tmp = t_0;
	} else if (x <= -1.06e-42) {
		tmp = t_1;
	} else if (x <= 1e-112) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 2.95e+54) {
		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 / y_m))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -2.45e+82:
		tmp = t_0
	elif x <= -1.06e-42:
		tmp = t_1
	elif x <= 1e-112:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 2.95e+54:
		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 / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -2.45e+82)
		tmp = t_0;
	elseif (x <= -1.06e-42)
		tmp = t_1;
	elseif (x <= 1e-112)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 2.95e+54)
		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 / y_m));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -2.45e+82)
		tmp = t_0;
	elseif (x <= -1.06e-42)
		tmp = t_1;
	elseif (x <= 1e-112)
		tmp = abs((4.0 / y_m));
	elseif (x <= 2.95e+54)
		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 / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.45e+82], t$95$0, If[LessEqual[x, -1.06e-42], t$95$1, If[LessEqual[x, 1e-112], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.95e+54], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.45e82 or 2.9499999999999999e54 < x

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. clear-num 79.9%

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

      3. un-div-inv 79.9%

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

   4. Applied egg-rr 79.9%

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

      1. frac-sub 75.4%

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

      2. associate-/r* 85.1%

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

      3. *-commutative 85.1%

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

   6. Applied egg-rr 85.1%

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

   7. Taylor expanded in z around 0 77.8%

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

   8. Taylor expanded in x around inf 77.9%

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

   if -2.45e82 < x < -1.0600000000000001e-42 or 9.9999999999999995e-113 < x < 2.9499999999999999e54

   1. Initial program 99.9%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 61.8%

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

      1. associate-*r/ 61.8%

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

      2. neg-mul-1 61.8%

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

      3. distribute-lft-neg-in 61.8%

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

      4. *-commutative 61.8%

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

      5. associate-*r/ 61.9%

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

   7. Simplified 61.9%

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

      1. clear-num 61.8%

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

      2. un-div-inv 61.7%

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

      3. add-sqr-sqrt 22.3%

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

      4. sqrt-unprod 61.7%

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

      5. sqr-neg 61.7%

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

      6. sqrt-unprod 39.3%

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

      7. add-sqr-sqrt 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. associate-/r/ 57.3%

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

       2. associate-*l/ 61.8%

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

       3. associate-/l* 61.9%

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

   11. Simplified 61.9%

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

   if -1.0600000000000001e-42 < x < 9.9999999999999995e-113

   1. Initial program 96.1%

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

   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 78.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+82}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-42}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 10^{-112}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+54}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.3e+93)
   (fabs (/ x (/ y_m z)))
   (if (<= z 9.5e+21)
     (fabs (/ (- x -4.0) y_m))
     (fabs (* (- 1.0 z) (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.3e+93) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 9.5e+21) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.3d+93)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 9.5d+21) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.3e+93) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 9.5e+21) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.3e+93:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 9.5e+21:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.3e+93)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 9.5e+21)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.3e+93)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 9.5e+21)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = abs(((1.0 - z) * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.3e+93], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 9.5e+21], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e93

   1. Initial program 93.0%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around inf 82.8%

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

      1. mul-1-neg 82.8%

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

      2. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

      1. clear-num 82.9%

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

      2. un-div-inv 83.1%

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

   9. Applied egg-rr 83.1%

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

   if -1.3e93 < z < 9.500000000000001e21

   1. Initial program 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate-*l/ 98.0%

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

      3. associate-*r/ 98.1%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.5%

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

      1. associate-*r/ 95.5%

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

      2. distribute-lft-in 95.5%

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

      3. metadata-eval 95.5%

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

      4. neg-mul-1 95.5%

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

      5. sub-neg 95.5%

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

   7. Simplified 95.5%

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

   if 9.500000000000001e21 < z

   1. Initial program 85.0%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around -inf 78.9%

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

      1. *-commutative 78.9%

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

      2. associate-/l* 83.7%

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

      3. neg-mul-1 83.7%

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

      4. unsub-neg 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+94}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -7.5e+94)
   (fabs (/ (* x z) y_m))
   (if (<= z 1.8e+15) (fabs (/ (- x -4.0) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -7.5e+94) {
		tmp = fabs(((x * z) / y_m));
	} else if (z <= 1.8e+15) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7.5d+94)) then
        tmp = abs(((x * z) / y_m))
    else if (z <= 1.8d+15) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -7.5e+94) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (z <= 1.8e+15) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -7.5e+94:
		tmp = math.fabs(((x * z) / y_m))
	elif z <= 1.8e+15:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -7.5e+94)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (z <= 1.8e+15)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -7.5e+94)
		tmp = abs(((x * z) / y_m));
	elseif (z <= 1.8e+15)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -7.5e+94], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.8e+15], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.49999999999999978e94

   1. Initial program 93.0%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. neg-mul-1 82.8%

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

      3. distribute-lft-neg-in 82.8%

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

      4. *-commutative 82.8%

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

      5. associate-*r/ 82.5%

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

   7. Simplified 82.5%

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

      1. add-sqr-sqrt 33.9%

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

      2. sqrt-unprod 71.8%

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

      3. sqr-neg 71.8%

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

      4. sqrt-unprod 48.5%

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

      5. add-sqr-sqrt 82.5%

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

      6. associate-/l* 82.8%

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

      7. *-commutative 82.8%

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

   9. Applied egg-rr 82.8%

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

   if -7.49999999999999978e94 < z < 1.8e15

   1. Initial program 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate-*l/ 98.0%

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

      3. associate-*r/ 98.1%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.5%

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

      1. associate-*r/ 95.5%

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

      2. distribute-lft-in 95.5%

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

      3. metadata-eval 95.5%

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

      4. neg-mul-1 95.5%

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

      5. sub-neg 95.5%

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

   7. Simplified 95.5%

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

   if 1.8e15 < z

   1. Initial program 85.0%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in z around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. neg-mul-1 78.9%

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

      3. distribute-lft-neg-in 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*r/ 83.7%

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

   7. Simplified 83.7%

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

      1. clear-num 83.4%

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

      2. un-div-inv 83.5%

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

      3. add-sqr-sqrt 33.2%

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

      4. sqrt-unprod 64.2%

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

      5. sqr-neg 64.2%

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

      6. sqrt-unprod 50.0%

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

      7. add-sqr-sqrt 83.5%

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

   9. Applied egg-rr 83.5%

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

       1. associate-/r/ 76.2%

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

       2. associate-*l/ 78.9%

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

       3. associate-/l* 83.7%

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

   11. Simplified 83.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+94}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -3.45 \cdot 10^{+93}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -3.45e+93)
   (fabs (* x (/ z y_m)))
   (if (<= z 1.6e+25) (fabs (/ (- x -4.0) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -3.45e+93) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= 1.6e+25) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.45d+93)) then
        tmp = abs((x * (z / y_m)))
    else if (z <= 1.6d+25) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -3.45e+93) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= 1.6e+25) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -3.45e+93:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= 1.6e+25:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -3.45e+93)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= 1.6e+25)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -3.45e+93)
		tmp = abs((x * (z / y_m)));
	elseif (z <= 1.6e+25)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -3.45e+93], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.6e+25], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4499999999999998e93

   1. Initial program 93.0%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around inf 82.8%

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

      1. mul-1-neg 82.8%

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

      2. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

   if -3.4499999999999998e93 < z < 1.6e25

   1. Initial program 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate-*l/ 98.0%

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

      3. associate-*r/ 98.1%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.5%

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

      1. associate-*r/ 95.5%

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

      2. distribute-lft-in 95.5%

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

      3. metadata-eval 95.5%

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

      4. neg-mul-1 95.5%

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

      5. sub-neg 95.5%

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

   7. Simplified 95.5%

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

   if 1.6e25 < z

   1. Initial program 85.0%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in z around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. neg-mul-1 78.9%

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

      3. distribute-lft-neg-in 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*r/ 83.7%

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

   7. Simplified 83.7%

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

      1. clear-num 83.4%

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

      2. un-div-inv 83.5%

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

      3. add-sqr-sqrt 33.2%

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

      4. sqrt-unprod 64.2%

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

      5. sqr-neg 64.2%

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

      6. sqrt-unprod 50.0%

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

      7. add-sqr-sqrt 83.5%

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

   9. Applied egg-rr 83.5%

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

       1. associate-/r/ 76.2%

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

       2. associate-*l/ 78.9%

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

       3. associate-/l* 83.7%

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

   11. Simplified 83.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.45 \cdot 10^{+93}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+92}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.95e+92)
   (fabs (/ x (/ y_m z)))
   (if (<= z 1.85e+25) (fabs (/ (- x -4.0) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.95e+92) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 1.85e+25) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.95d+92)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 1.85d+25) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.95e+92) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 1.85e+25) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.95e+92:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 1.85e+25:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.95e+92)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 1.85e+25)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.95e+92)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 1.85e+25)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.95e+92], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.85e+25], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95000000000000006e92

   1. Initial program 93.0%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around inf 82.8%

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

      1. mul-1-neg 82.8%

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

      2. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

      1. clear-num 82.9%

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

      2. un-div-inv 83.1%

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

   9. Applied egg-rr 83.1%

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

   if -1.95000000000000006e92 < z < 1.8499999999999999e25

   1. Initial program 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate-*l/ 98.0%

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

      3. associate-*r/ 98.1%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.5%

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

      1. associate-*r/ 95.5%

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

      2. distribute-lft-in 95.5%

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

      3. metadata-eval 95.5%

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

      4. neg-mul-1 95.5%

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

      5. sub-neg 95.5%

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

   7. Simplified 95.5%

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

   if 1.8499999999999999e25 < z

   1. Initial program 85.0%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in z around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. neg-mul-1 78.9%

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

      3. distribute-lft-neg-in 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*r/ 83.7%

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

   7. Simplified 83.7%

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

      1. clear-num 83.4%

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

      2. un-div-inv 83.5%

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

      3. add-sqr-sqrt 33.2%

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

      4. sqrt-unprod 64.2%

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

      5. sqr-neg 64.2%

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

      6. sqrt-unprod 50.0%

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

      7. add-sqr-sqrt 83.5%

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

   9. Applied egg-rr 83.5%

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

       1. associate-/r/ 76.2%

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

       2. associate-*l/ 78.9%

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

       3. associate-/l* 83.7%

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

   11. Simplified 83.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+92}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -2.3) (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 <= -2.3) || !(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 <= (-2.3d0)) .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 <= -2.3) || !(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 <= -2.3) 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 <= -2.3) || !(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 <= -2.3) || ~((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, -2.3], 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 < -2.2999999999999998 or 4 < x

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. clear-num 83.7%

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

      3. un-div-inv 83.8%

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

   4. Applied egg-rr 83.8%

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

      1. frac-sub 75.4%

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

      2. associate-/r* 85.5%

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

      3. *-commutative 85.5%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in z around 0 71.4%

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

   8. Taylor expanded in x around inf 70.1%

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

   if -2.2999999999999998 < x < 4

   1. Initial program 97.1%

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

   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 66.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

   1. associate-*l/ 92.6%

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

   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%

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

5. Final simplification 97.3%

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

Alternative 9: 40.0% 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 90.7%

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

3. Simplified 97.2%

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

5. Taylor expanded in x around 0 36.8%

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

6. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2024039 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))