fabs fraction 1

Percentage Accurate: 91.6% → 99.6%

Time: 6.0s

Alternatives: 8

Speedup: 1.0×

• 20.6% 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: 91.6% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-78}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e-78)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-78) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.6e-78)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 1.6e-78], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e-78

   1. Initial program 90.4%

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

      1. associate-*l/ 91.3%

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

      2. sub-div 97.9%

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

   3. Applied egg-rr 97.9%

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

   if 1.6e-78 < y

   1. Initial program 88.0%

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

   3. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-78}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+68} \lor \neg \left(z \leq -1.1 \cdot 10^{+36}\right) \land \left(z \leq -1.7 \cdot 10^{+16} \lor \neg \left(z \leq 1.3 \cdot 10^{+68}\right)\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.26e+68)
         (and (not (<= z -1.1e+36)) (or (<= z -1.7e+16) (not (<= z 1.3e+68)))))
   (fabs (* x (/ z y)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.26e+68) || (!(z <= -1.1e+36) && ((z <= -1.7e+16) || !(z <= 1.3e+68)))) {
		tmp = fabs((x * (z / y)));
	} else {
		tmp = fabs(((-4.0 - x) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.26d+68)) .or. (.not. (z <= (-1.1d+36))) .and. (z <= (-1.7d+16)) .or. (.not. (z <= 1.3d+68))) then
        tmp = abs((x * (z / y)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.26e+68) || (!(z <= -1.1e+36) && ((z <= -1.7e+16) || !(z <= 1.3e+68)))) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (z <= -1.26e+68) or (not (z <= -1.1e+36) and ((z <= -1.7e+16) or not (z <= 1.3e+68))):
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.26e+68) || (!(z <= -1.1e+36) && ((z <= -1.7e+16) || !(z <= 1.3e+68))))
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.26e+68) || (~((z <= -1.1e+36)) && ((z <= -1.7e+16) || ~((z <= 1.3e+68)))))
		tmp = abs((x * (z / y)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[z, -1.26e+68], And[N[Not[LessEqual[z, -1.1e+36]], $MachinePrecision], Or[LessEqual[z, -1.7e+16], N[Not[LessEqual[z, 1.3e+68]], $MachinePrecision]]]], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.26e68 or -1.1e36 < z < -1.7e16 or 1.2999999999999999e68 < z

   1. Initial program 82.7%

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

   2. Taylor expanded in z around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. associate-*r/ 85.7%

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

      3. distribute-rgt-neg-out 85.7%

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

      4. distribute-neg-frac 85.7%

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

   4. Simplified 85.7%

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

      1. associate-*r/ 80.5%

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

      2. associate-*l/ 80.2%

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

      3. expm1-log1p-u 46.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot \left(-z\right)\right)\right)}\right| \]

      4. expm1-udef 37.0%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot \left(-z\right)\right)} - 1}\right| \]

      5. associate-*l/ 32.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot \left(-z\right)}{y}}\right)} - 1\right| \]

      6. div-inv 32.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(-z\right)\right) \cdot \frac{1}{y}}\right)} - 1\right| \]

      7. associate-*r* 35.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(\left(-z\right) \cdot \frac{1}{y}\right)}\right)} - 1\right| \]

      8. div-inv 35.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{-z}{y}}\right)} - 1\right| \]

      9. add-sqr-sqrt 20.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}{y}\right)} - 1\right| \]

      10. sqrt-unprod 26.1%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}{y}\right)} - 1\right| \]

      11. sqr-neg 26.1%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\sqrt{\color{blue}{z \cdot z}}}{y}\right)} - 1\right| \]

      12. sqrt-unprod 16.2%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{y}\right)} - 1\right| \]

      13. add-sqr-sqrt 32.4%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{z}}{y}\right)} - 1\right| \]

   6. Applied egg-rr 32.4%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 49.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)\right)}\right| \]

      2. expm1-log1p 85.7%

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

   8. Simplified 85.7%

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

   if -1.26e68 < z < -1.1e36 or -1.7e16 < z < 1.2999999999999999e68

   1. Initial program 94.2%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in z around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. distribute-lft-in 95.9%

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

      3. metadata-eval 95.9%

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

      4. neg-mul-1 95.9%

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

      5. sub-neg 95.9%

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

   6. Simplified 95.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+68} \lor \neg \left(z \leq -1.1 \cdot 10^{+36}\right) \land \left(z \leq -1.7 \cdot 10^{+16} \lor \neg \left(z \leq 1.3 \cdot 10^{+68}\right)\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 3: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (* x (/ z y)))))
   (if (<= x -3.2e+32)
     t_0
     (if (<= x -1.25e-15)
       t_1
       (if (<= x 3e-27) (fabs (/ 4.0 y)) (if (<= x 5.8e+170) t_1 t_0))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((x * (z / y)));
	double tmp;
	if (x <= -3.2e+32) {
		tmp = t_0;
	} else if (x <= -1.25e-15) {
		tmp = t_1;
	} else if (x <= 3e-27) {
		tmp = fabs((4.0 / y));
	} else if (x <= 5.8e+170) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((x * (z / y)))
    if (x <= (-3.2d+32)) then
        tmp = t_0
    else if (x <= (-1.25d-15)) then
        tmp = t_1
    else if (x <= 3d-27) then
        tmp = abs((4.0d0 / y))
    else if (x <= 5.8d+170) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((x * (z / y)));
	double tmp;
	if (x <= -3.2e+32) {
		tmp = t_0;
	} else if (x <= -1.25e-15) {
		tmp = t_1;
	} else if (x <= 3e-27) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 5.8e+170) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -3.2e+32:
		tmp = t_0
	elif x <= -1.25e-15:
		tmp = t_1
	elif x <= 3e-27:
		tmp = math.fabs((4.0 / y))
	elif x <= 5.8e+170:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -3.2e+32)
		tmp = t_0;
	elseif (x <= -1.25e-15)
		tmp = t_1;
	elseif (x <= 3e-27)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 5.8e+170)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -3.2e+32)
		tmp = t_0;
	elseif (x <= -1.25e-15)
		tmp = t_1;
	elseif (x <= 3e-27)
		tmp = abs((4.0 / y));
	elseif (x <= 5.8e+170)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.2e+32], t$95$0, If[LessEqual[x, -1.25e-15], t$95$1, If[LessEqual[x, 3e-27], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.8e+170], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1999999999999999e32 or 5.8000000000000001e170 < x

   1. Initial program 79.5%

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

      1. associate-*l/ 82.1%

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

      2. sub-div 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in x around inf 96.5%

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

   5. Taylor expanded in z around 0 82.3%

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

   if -3.1999999999999999e32 < x < -1.25e-15 or 3.0000000000000001e-27 < x < 5.8000000000000001e170

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 71.0%

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

      1. mul-1-neg 71.0%

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

      2. associate-*r/ 79.0%

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

      3. distribute-rgt-neg-out 79.0%

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

      4. distribute-neg-frac 79.0%

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

   4. Simplified 79.0%

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

      1. associate-*r/ 71.0%

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

      2. associate-*l/ 79.2%

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

      3. expm1-log1p-u 43.2%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot \left(-z\right)\right)\right)}\right| \]

      4. expm1-udef 34.9%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot \left(-z\right)\right)} - 1}\right| \]

      5. associate-*l/ 29.5%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot \left(-z\right)}{y}}\right)} - 1\right| \]

      6. div-inv 29.5%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(-z\right)\right) \cdot \frac{1}{y}}\right)} - 1\right| \]

      7. associate-*r* 34.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(\left(-z\right) \cdot \frac{1}{y}\right)}\right)} - 1\right| \]

      8. div-inv 34.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{-z}{y}}\right)} - 1\right| \]

      9. add-sqr-sqrt 16.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}{y}\right)} - 1\right| \]

      10. sqrt-unprod 23.6%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}{y}\right)} - 1\right| \]

      11. sqr-neg 23.6%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\sqrt{\color{blue}{z \cdot z}}}{y}\right)} - 1\right| \]

      12. sqrt-unprod 14.6%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{y}\right)} - 1\right| \]

      13. add-sqr-sqrt 33.2%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{z}}{y}\right)} - 1\right| \]

   6. Applied egg-rr 33.2%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 41.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)\right)}\right| \]

      2. expm1-log1p 79.0%

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

   8. Simplified 79.0%

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

   if -1.25e-15 < x < 3.0000000000000001e-27

   1. Initial program 92.7%

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

   2. Taylor expanded in x around 0 78.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-15}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+170}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+59}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+59)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+59) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d+59) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x / (y / z))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+59) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 1e+59:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x / (y / z))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+59)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x / Float64(y / z))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+59)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x / (y / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 1e+59], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999972e58

   1. Initial program 91.4%

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

      1. associate-*l/ 92.2%

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

      2. sub-div 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 9.99999999999999972e58 < y

   1. Initial program 82.6%

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

   3. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+59}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 5: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+171}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))))
   (if (<= x -4.2e-16)
     t_0
     (if (<= x 1.15e-27)
       (fabs (/ 4.0 y))
       (if (<= x 3.2e+171) t_0 (fabs (/ x y)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double tmp;
	if (x <= -4.2e-16) {
		tmp = t_0;
	} else if (x <= 1.15e-27) {
		tmp = fabs((4.0 / y));
	} else if (x <= 3.2e+171) {
		tmp = t_0;
	} else {
		tmp = fabs((x / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    if (x <= (-4.2d-16)) then
        tmp = t_0
    else if (x <= 1.15d-27) then
        tmp = abs((4.0d0 / y))
    else if (x <= 3.2d+171) then
        tmp = t_0
    else
        tmp = abs((x / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double tmp;
	if (x <= -4.2e-16) {
		tmp = t_0;
	} else if (x <= 1.15e-27) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 3.2e+171) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	tmp = 0
	if x <= -4.2e-16:
		tmp = t_0
	elif x <= 1.15e-27:
		tmp = math.fabs((4.0 / y))
	elif x <= 3.2e+171:
		tmp = t_0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	tmp = 0.0
	if (x <= -4.2e-16)
		tmp = t_0;
	elseif (x <= 1.15e-27)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 3.2e+171)
		tmp = t_0;
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	tmp = 0.0;
	if (x <= -4.2e-16)
		tmp = t_0;
	elseif (x <= 1.15e-27)
		tmp = abs((4.0 / y));
	elseif (x <= 3.2e+171)
		tmp = t_0;
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.2e-16], t$95$0, If[LessEqual[x, 1.15e-27], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.2e+171], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2000000000000002e-16 or 1.15e-27 < x < 3.20000000000000011e171

   1. Initial program 91.0%

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

   2. Taylor expanded in z around inf 57.9%

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

      1. mul-1-neg 57.9%

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

      2. associate-*r/ 64.4%

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

      3. distribute-rgt-neg-out 64.4%

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

      4. distribute-neg-frac 64.4%

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

   4. Simplified 64.4%

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

      1. associate-*r/ 57.9%

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

      2. associate-/l* 64.3%

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

      3. add-sqr-sqrt 32.7%

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

      4. sqrt-unprod 52.0%

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

      5. sqr-neg 52.0%

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

      6. sqrt-unprod 31.5%

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

      7. add-sqr-sqrt 64.3%

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

      8. associate-/l* 57.9%

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

   6. Applied egg-rr 57.9%

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

      1. associate-/l* 64.3%

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

      2. associate-/r/ 73.7%

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

   8. Applied egg-rr 73.7%

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

   if -4.2000000000000002e-16 < x < 1.15e-27

   1. Initial program 92.7%

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

   2. Taylor expanded in x around 0 78.5%

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

   if 3.20000000000000011e171 < x

   1. Initial program 72.4%

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

      1. associate-*l/ 79.3%

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

      2. sub-div 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-16}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+171}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 95.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (fabs (/ (- (+ x 4.0) (* x z)) y)))

C

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

Fortran

NOTE: y should be positive before calling this function
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) - (x * z)) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 89.7%

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

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

3. Applied egg-rr 97.3%

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

4. Final simplification 97.3%

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

Alternative 7: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -1.5) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 4 < x

   1. Initial program 85.8%

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

      1. associate-*l/ 84.4%

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

      2. sub-div 94.4%

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

   3. Applied egg-rr 94.4%

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

   4. Taylor expanded in x around inf 93.7%

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

   5. Taylor expanded in z around 0 66.6%

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

   if -1.5 < x < 4

   1. Initial program 93.2%

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

   2. Taylor expanded in x around 0 73.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 8: 39.7% accurate, 1.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (fabs (/ 4.0 y)))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in x around 0 41.5%

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

3. Final simplification 41.5%

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

Reproduce

herbie shell --seed 2023309 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))