fabs fraction 1

Percentage Accurate: 91.9% → 99.8%

Time: 6.3s

Alternatives: 9

Speedup: 1.0×

• 21.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-39}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-39)
   (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 <= 8e-39) {
		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 <= 8e-39)
		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, 8e-39], 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 < 7.99999999999999943e-39

   1. Initial program 89.5%

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

      1. associate-*l/ 90.3%

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

      2. sub-div 97.4%

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

   3. Applied egg-rr 97.4%

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

   if 7.99999999999999943e-39 < y

   1. Initial program 95.0%

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

      1. fabs-sub 95.0%

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

      2. associate-*l/ 90.7%

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

      3. *-commutative 90.7%

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

      4. associate-*l/ 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-39}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-41}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 -2.15e+79)
     t_0
     (if (<= x -2.6e-73)
       t_1
       (if (<= x 1.06e-41) (fabs (/ 4.0 y)) (if (<= x 3.1e+93) 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 <= -2.15e+79) {
		tmp = t_0;
	} else if (x <= -2.6e-73) {
		tmp = t_1;
	} else if (x <= 1.06e-41) {
		tmp = fabs((4.0 / y));
	} else if (x <= 3.1e+93) {
		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 <= (-2.15d+79)) then
        tmp = t_0
    else if (x <= (-2.6d-73)) then
        tmp = t_1
    else if (x <= 1.06d-41) then
        tmp = abs((4.0d0 / y))
    else if (x <= 3.1d+93) 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 <= -2.15e+79) {
		tmp = t_0;
	} else if (x <= -2.6e-73) {
		tmp = t_1;
	} else if (x <= 1.06e-41) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 3.1e+93) {
		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 <= -2.15e+79:
		tmp = t_0
	elif x <= -2.6e-73:
		tmp = t_1
	elif x <= 1.06e-41:
		tmp = math.fabs((4.0 / y))
	elif x <= 3.1e+93:
		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 <= -2.15e+79)
		tmp = t_0;
	elseif (x <= -2.6e-73)
		tmp = t_1;
	elseif (x <= 1.06e-41)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 3.1e+93)
		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 <= -2.15e+79)
		tmp = t_0;
	elseif (x <= -2.6e-73)
		tmp = t_1;
	elseif (x <= 1.06e-41)
		tmp = abs((4.0 / y));
	elseif (x <= 3.1e+93)
		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, -2.15e+79], t$95$0, If[LessEqual[x, -2.6e-73], t$95$1, If[LessEqual[x, 1.06e-41], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.1e+93], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1500000000000002e79 or 3.10000000000000019e93 < x

   1. Initial program 81.5%

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

   2. Taylor expanded in z around 0 68.5%

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

      1. associate-*r/ 68.5%

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

      2. metadata-eval 68.5%

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

   4. Simplified 68.5%

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

   5. Taylor expanded in x around inf 68.5%

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

   if -2.1500000000000002e79 < x < -2.6000000000000001e-73 or 1.06e-41 < x < 3.10000000000000019e93

   1. Initial program 94.2%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in z around inf 65.9%

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

      1. associate-*l/ 71.0%

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

      2. *-commutative 71.0%

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

   6. Simplified 71.0%

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

   if -2.6000000000000001e-73 < x < 1.06e-41

   1. Initial program 96.9%

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

   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 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-73}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-41}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+26} \lor \neg \left(x \leq 10^{+47}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1e+26) (not (<= x 1e+47)))
   (fabs (* (/ x y) (+ z -1.0)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e+26) || !(x <= 1e+47)) {
		tmp = fabs(((x / y) * (z + -1.0)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / 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 <= (-1d+26)) .or. (.not. (x <= 1d+47))) then
        tmp = abs(((x / y) * (z + (-1.0d0))))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / 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 <= -1e+26) || !(x <= 1e+47)) {
		tmp = Math.abs(((x / y) * (z + -1.0)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -1e+26) or not (x <= 1e+47):
		tmp = math.fabs(((x / y) * (z + -1.0)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1e+26) || !(x <= 1e+47))
		tmp = abs(Float64(Float64(x / y) * Float64(z + -1.0)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1e+26) || ~((x <= 1e+47)))
		tmp = abs(((x / y) * (z + -1.0)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -1e+26], N[Not[LessEqual[x, 1e+47]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000005e26 or 1e47 < x

   1. Initial program 83.5%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in x around inf 89.2%

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

   5. Taylor expanded in z around 0 76.6%

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

      1. associate-*r/ 83.5%

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

      2. distribute-rgt-in 99.9%

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

   7. Simplified 99.9%

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

   if -1.00000000000000005e26 < x < 1e47

   1. Initial program 96.3%

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

      1. associate-*l/ 99.9%

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

      2. sub-div 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+26} \lor \neg \left(x \leq 10^{+47}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 4: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-71} \lor \neg \left(x \leq 2.55 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.2e-71) (not (<= x 2.55e-40)))
   (fabs (* (/ x y) (+ z -1.0)))
   (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.2e-71) || !(x <= 2.55e-40)) {
		tmp = fabs(((x / y) * (z + -1.0)));
	} 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 <= (-9.2d-71)) .or. (.not. (x <= 2.55d-40))) then
        tmp = abs(((x / y) * (z + (-1.0d0))))
    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 <= -9.2e-71) || !(x <= 2.55e-40)) {
		tmp = Math.abs(((x / y) * (z + -1.0)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -9.2e-71) or not (x <= 2.55e-40):
		tmp = math.fabs(((x / y) * (z + -1.0)))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.2e-71) || !(x <= 2.55e-40))
		tmp = abs(Float64(Float64(x / y) * Float64(z + -1.0)));
	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 <= -9.2e-71) || ~((x <= 2.55e-40)))
		tmp = abs(((x / y) * (z + -1.0)));
	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, -9.2e-71], N[Not[LessEqual[x, 2.55e-40]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.1999999999999994e-71 or 2.55000000000000019e-40 < x

   1. Initial program 86.9%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around inf 86.0%

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

   5. Taylor expanded in z around 0 76.7%

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

      1. associate-*r/ 81.1%

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

      2. distribute-rgt-in 93.3%

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

   7. Simplified 93.3%

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

   if -9.1999999999999994e-71 < x < 2.55000000000000019e-40

   1. Initial program 96.1%

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

   2. Taylor expanded in x around 0 77.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-71} \lor \neg \left(x \leq 2.55 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 5: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{-71} \lor \neg \left(x \leq 3.45 \cdot 10^{-43}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.35e-71) (not (<= x 3.45e-43)))
   (fabs (* z (/ x y)))
   (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.35e-71) || !(x <= 3.45e-43)) {
		tmp = fabs((z * (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 <= (-3.35d-71)) .or. (.not. (x <= 3.45d-43))) then
        tmp = abs((z * (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 <= -3.35e-71) || !(x <= 3.45e-43)) {
		tmp = Math.abs((z * (x / y)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -3.35e-71) or not (x <= 3.45e-43):
		tmp = math.fabs((z * (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 <= -3.35e-71) || !(x <= 3.45e-43))
		tmp = abs(Float64(z * 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 <= -3.35e-71) || ~((x <= 3.45e-43)))
		tmp = abs((z * (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, -3.35e-71], N[Not[LessEqual[x, 3.45e-43]], $MachinePrecision]], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3499999999999999e-71 or 3.44999999999999982e-43 < x

   1. Initial program 86.9%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in z around inf 53.5%

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

      1. add-sqr-sqrt 32.7%

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

      2. sqrt-unprod 49.7%

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

      3. sqr-neg 49.7%

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

      4. sqrt-unprod 20.8%

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

      5. add-sqr-sqrt 53.5%

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

      6. associate-*r/ 67.5%

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

      7. *-commutative 67.5%

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

      8. add-sqr-sqrt 26.8%

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

      9. sqrt-unprod 55.8%

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

      10. sqr-neg 55.8%

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

      11. sqrt-unprod 40.6%

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

      12. add-sqr-sqrt 67.5%

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

   6. Applied egg-rr 67.5%

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

   if -3.3499999999999999e-71 < x < 3.44999999999999982e-43

   1. Initial program 96.1%

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

   2. Taylor expanded in x around 0 77.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{-71} \lor \neg \left(x \leq 3.45 \cdot 10^{-43}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 6: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-71} \lor \neg \left(x \leq 2.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.6e-71) (not (<= x 2.9e-40)))
   (fabs (/ z (/ y x)))
   (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.6e-71) || !(x <= 2.9e-40)) {
		tmp = fabs((z / (y / x)));
	} 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 <= (-5.6d-71)) .or. (.not. (x <= 2.9d-40))) then
        tmp = abs((z / (y / x)))
    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 <= -5.6e-71) || !(x <= 2.9e-40)) {
		tmp = Math.abs((z / (y / x)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -5.6e-71) or not (x <= 2.9e-40):
		tmp = math.fabs((z / (y / x)))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.6e-71) || !(x <= 2.9e-40))
		tmp = abs(Float64(z / Float64(y / x)));
	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 <= -5.6e-71) || ~((x <= 2.9e-40)))
		tmp = abs((z / (y / x)));
	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, -5.6e-71], N[Not[LessEqual[x, 2.9e-40]], $MachinePrecision]], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.60000000000000001e-71 or 2.8999999999999999e-40 < x

   1. Initial program 86.9%

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

   2. Taylor expanded in z around inf 53.5%

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

      1. associate-*r/ 53.5%

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

      2. mul-1-neg 53.5%

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

      3. distribute-rgt-neg-out 53.5%

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

      4. associate-*r/ 67.5%

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

      5. distribute-frac-neg 67.5%

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

      6. mul-1-neg 67.5%

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

      7. metadata-eval 67.5%

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

      8. times-frac 67.5%

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

      9. *-lft-identity 67.5%

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

      10. neg-mul-1 67.5%

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

   4. Simplified 67.5%

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

      1. associate-*r/ 53.5%

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

      2. add-sqr-sqrt 20.8%

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

      3. sqrt-unprod 49.7%

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

      4. sqr-neg 49.7%

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

      5. sqrt-unprod 32.7%

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

      6. add-sqr-sqrt 53.5%

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

      7. associate-/l* 67.5%

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

   6. Applied egg-rr 67.5%

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

   if -5.60000000000000001e-71 < x < 2.8999999999999999e-40

   1. Initial program 96.1%

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

   2. Taylor expanded in x around 0 77.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-71} \lor \neg \left(x \leq 2.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 7: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+91}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -4.4e+91)
   (fabs (/ z (/ y x)))
   (if (<= z 2.5e+102) (fabs (/ (- -4.0 x) y)) (fabs (* z (/ x y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.4e+91) {
		tmp = fabs((z / (y / x)));
	} else if (z <= 2.5e+102) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z * (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 <= (-4.4d+91)) then
        tmp = abs((z / (y / x)))
    else if (z <= 2.5d+102) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z * (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 <= -4.4e+91) {
		tmp = Math.abs((z / (y / x)));
	} else if (z <= 2.5e+102) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -4.4e+91:
		tmp = math.fabs((z / (y / x)))
	elif z <= 2.5e+102:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -4.4e+91)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (z <= 2.5e+102)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.4e+91)
		tmp = abs((z / (y / x)));
	elseif (z <= 2.5e+102)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -4.4e+91], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e+102], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.39999999999999999e91

   1. Initial program 94.2%

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

   2. Taylor expanded in z around inf 65.2%

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

      1. associate-*r/ 65.2%

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

      2. mul-1-neg 65.2%

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

      3. distribute-rgt-neg-out 65.2%

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

      4. associate-*r/ 73.3%

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

      5. distribute-frac-neg 73.3%

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

      6. mul-1-neg 73.3%

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

      7. metadata-eval 73.3%

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

      8. times-frac 73.3%

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

      9. *-lft-identity 73.3%

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

      10. neg-mul-1 73.3%

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

   4. Simplified 73.3%

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

      1. associate-*r/ 65.2%

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

      2. add-sqr-sqrt 29.2%

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

      3. sqrt-unprod 48.9%

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

      4. sqr-neg 48.9%

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

      5. sqrt-unprod 35.9%

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

      6. add-sqr-sqrt 65.2%

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

      7. associate-/l* 73.5%

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

   6. Applied egg-rr 73.5%

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

   if -4.39999999999999999e91 < z < 2.5e102

   1. Initial program 94.0%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 90.2%

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

      1. associate-*r/ 90.2%

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

      2. distribute-lft-in 90.2%

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

      3. metadata-eval 90.2%

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

      4. neg-mul-1 90.2%

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

      5. sub-neg 90.2%

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

   6. Simplified 90.2%

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

   if 2.5e102 < z

   1. Initial program 81.2%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in z around inf 81.0%

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

      1. add-sqr-sqrt 37.8%

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

      2. sqrt-unprod 69.4%

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

      3. sqr-neg 69.4%

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

      4. sqrt-unprod 43.0%

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

      5. add-sqr-sqrt 81.0%

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

      6. associate-*r/ 82.2%

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

      7. *-commutative 82.2%

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

      8. add-sqr-sqrt 45.9%

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

      9. sqrt-unprod 72.5%

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

      10. sqr-neg 72.5%

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

      11. sqrt-unprod 36.1%

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

      12. add-sqr-sqrt 82.2%

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

   6. Applied egg-rr 82.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+91}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 8: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.2) (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 <= -10.2) || !(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 <= (-10.2d0)) .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 <= -10.2) || !(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 <= -10.2) 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 <= -10.2) || !(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 <= -10.2) || ~((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, -10.2], 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 < -10.199999999999999 or 4 < x

   1. Initial program 85.0%

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

   2. Taylor expanded in z around 0 60.6%

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

      1. associate-*r/ 60.6%

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

      2. metadata-eval 60.6%

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

   4. Simplified 60.6%

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

   5. Taylor expanded in x around inf 60.6%

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

   if -10.199999999999999 < x < 4

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0 69.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 9: 39.5% 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 91.1%

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

2. Taylor expanded in x around 0 40.4%

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

3. Final simplification 40.4%

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

Reproduce

herbie shell --seed 2023244 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))