fabs fraction 1

Percentage Accurate: 92.3% → 99.9%

Time: 10.0s

Alternatives: 6

Speedup: 1.0×

• 20.1% 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: 92.3% 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, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 200:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 200.0)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 200.0) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (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 (y <= 200.0d0) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (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 (y <= 200.0) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 200.0:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x * (z / y))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 200.0)
		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(z / y))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 200.0)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 200.0], 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[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 200

   1. Initial program 92.0%

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

      1. associate-*l/ 92.7%

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

      2. associate-*r/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in y around 0 95.9%

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

      1. +-commutative 95.9%

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

   6. Simplified 95.9%

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

   if 200 < y

   1. Initial program 92.6%

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

      1. associate-*l/ 92.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 200:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 2: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.05 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+221} \lor \neg \left(x \leq 2.1 \cdot 10^{+301}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (* z (/ x y)))))
   (if (<= x -2.3e+169)
     t_0
     (if (<= x -1.7e-55)
       t_1
       (if (<= x 4.05e-50)
         (fabs (/ 4.0 y))
         (if (or (<= x 6.1e+221) (not (<= x 2.1e+301))) 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((z * (x / y)));
	double tmp;
	if (x <= -2.3e+169) {
		tmp = t_0;
	} else if (x <= -1.7e-55) {
		tmp = t_1;
	} else if (x <= 4.05e-50) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 6.1e+221) || !(x <= 2.1e+301)) {
		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((z * (x / y)))
    if (x <= (-2.3d+169)) then
        tmp = t_0
    else if (x <= (-1.7d-55)) then
        tmp = t_1
    else if (x <= 4.05d-50) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 6.1d+221) .or. (.not. (x <= 2.1d+301))) 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((z * (x / y)));
	double tmp;
	if (x <= -2.3e+169) {
		tmp = t_0;
	} else if (x <= -1.7e-55) {
		tmp = t_1;
	} else if (x <= 4.05e-50) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 6.1e+221) || !(x <= 2.1e+301)) {
		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((z * (x / y)))
	tmp = 0
	if x <= -2.3e+169:
		tmp = t_0
	elif x <= -1.7e-55:
		tmp = t_1
	elif x <= 4.05e-50:
		tmp = math.fabs((4.0 / y))
	elif (x <= 6.1e+221) or not (x <= 2.1e+301):
		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(z * Float64(x / y)))
	tmp = 0.0
	if (x <= -2.3e+169)
		tmp = t_0;
	elseif (x <= -1.7e-55)
		tmp = t_1;
	elseif (x <= 4.05e-50)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 6.1e+221) || !(x <= 2.1e+301))
		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((z * (x / y)));
	tmp = 0.0;
	if (x <= -2.3e+169)
		tmp = t_0;
	elseif (x <= -1.7e-55)
		tmp = t_1;
	elseif (x <= 4.05e-50)
		tmp = abs((4.0 / y));
	elseif ((x <= 6.1e+221) || ~((x <= 2.1e+301)))
		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[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.3e+169], t$95$0, If[LessEqual[x, -1.7e-55], t$95$1, If[LessEqual[x, 4.05e-50], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 6.1e+221], N[Not[LessEqual[x, 2.1e+301]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999999e169 or 6.0999999999999998e221 < x < 2.0999999999999999e301

   1. Initial program 89.0%

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

      1. associate-*l/ 85.4%

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

      2. associate-*r/ 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in z around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. metadata-eval 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in x around inf 75.7%

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

   if -2.2999999999999999e169 < x < -1.69999999999999986e-55 or 4.0499999999999999e-50 < x < 6.0999999999999998e221 or 2.0999999999999999e301 < x

   1. Initial program 92.8%

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

      1. associate-*l/ 89.6%

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

      2. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in z around inf 58.0%

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

      1. associate-*r/ 58.0%

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

      2. associate-*r* 58.0%

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

      3. neg-mul-1 58.0%

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

      4. *-commutative 58.0%

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

      5. associate-*r/ 68.9%

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

      6. distribute-frac-neg 68.9%

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

   6. Simplified 68.9%

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

      1. add-sqr-sqrt 34.4%

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

      2. sqrt-unprod 57.6%

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

      3. sqr-neg 57.6%

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

      4. sqrt-unprod 34.3%

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

      5. add-sqr-sqrt 68.9%

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

      6. associate-*r/ 58.0%

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

      7. *-commutative 58.0%

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

      8. associate-/l* 63.9%

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

   8. Applied egg-rr 63.9%

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

      1. associate-/r/ 68.9%

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

   10. Applied egg-rr 68.9%

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

   if -1.69999999999999986e-55 < x < 4.0499999999999999e-50

   1. Initial program 93.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around 0 81.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+169}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-55}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4.05 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+221} \lor \neg \left(x \leq 2.1 \cdot 10^{+301}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x -5e+16)
   (fabs (- (/ x y) (* x (/ z y))))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e+16) {
		tmp = fabs(((x / y) - (x * (z / y))));
	} 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 <= (-5d+16)) then
        tmp = abs(((x / y) - (x * (z / y))))
    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 <= -5e+16) {
		tmp = Math.abs(((x / y) - (x * (z / y))));
	} 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 <= -5e+16:
		tmp = math.fabs(((x / y) - (x * (z / y))))
	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 <= -5e+16)
		tmp = abs(Float64(Float64(x / y) - Float64(x * Float64(z / y))));
	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 <= -5e+16)
		tmp = abs(((x / y) - (x * (z / y))));
	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[LessEqual[x, -5e+16], N[Abs[N[(N[(x / y), $MachinePrecision] - N[(x * N[(z / y), $MachinePrecision]), $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 < -5e16

   1. Initial program 93.9%

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

      1. associate-*l/ 85.6%

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

      2. associate-*r/ 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 96.9%

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

   if -5e16 < x

   1. Initial program 91.6%

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

      1. associate-*l/ 95.3%

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

      2. associate-*r/ 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. +-commutative 97.4%

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

   6. Simplified 97.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+37}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 95000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -7.8e+37)
   (fabs (* z (/ x y)))
   (if (<= z 95000000.0) (fabs (/ (- -4.0 x) y)) (fabs (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.8e+37) {
		tmp = fabs((z * (x / y)));
	} else if (z <= 95000000.0) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((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 (z <= (-7.8d+37)) then
        tmp = abs((z * (x / y)))
    else if (z <= 95000000.0d0) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((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 (z <= -7.8e+37) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= 95000000.0) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -7.8e+37:
		tmp = math.fabs((z * (x / y)))
	elif z <= 95000000.0:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -7.8e+37)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= 95000000.0)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.8e+37)
		tmp = abs((z * (x / y)));
	elseif (z <= 95000000.0)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -7.8e+37], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 95000000.0], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.7999999999999997e37

   1. Initial program 93.2%

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

      1. associate-*l/ 88.6%

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

      2. associate-*r/ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in z around inf 75.9%

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

      1. associate-*r/ 75.9%

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

      2. associate-*r* 75.9%

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

      3. neg-mul-1 75.9%

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

      4. *-commutative 75.9%

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

      5. associate-*r/ 80.9%

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

      6. distribute-frac-neg 80.9%

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

   6. Simplified 80.9%

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

      1. add-sqr-sqrt 38.0%

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

      2. sqrt-unprod 69.3%

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

      3. sqr-neg 69.3%

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

      4. sqrt-unprod 43.3%

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

      5. add-sqr-sqrt 80.9%

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

      6. associate-*r/ 75.9%

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

      7. *-commutative 75.9%

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

      8. associate-/l* 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. associate-/r/ 80.9%

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

   10. Applied egg-rr 80.9%

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

   if -7.7999999999999997e37 < z < 9.5e7

   1. Initial program 95.0%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around 0 97.3%

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

      1. +-commutative 97.3%

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

      2. associate-*r/ 97.3%

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

      3. +-commutative 97.3%

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

      4. distribute-lft-in 97.3%

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

      5. metadata-eval 97.3%

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

      6. neg-mul-1 97.3%

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

      7. sub-neg 97.3%

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

   6. Simplified 97.3%

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

   if 9.5e7 < z

   1. Initial program 84.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in z around inf 75.4%

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

      1. associate-*r/ 75.4%

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

      2. associate-*r* 75.4%

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

      3. neg-mul-1 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*r/ 81.9%

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

      6. distribute-frac-neg 81.9%

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

   6. Simplified 81.9%

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

      1. add-sqr-sqrt 41.0%

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

      2. sqrt-unprod 71.9%

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

      3. sqr-neg 71.9%

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

      4. sqrt-unprod 41.2%

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

      5. add-sqr-sqrt 81.9%

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

      6. associate-*r/ 75.4%

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

      7. *-commutative 75.4%

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

      8. associate-/l* 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. div-inv 84.4%

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

      2. clear-num 84.4%

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

      3. *-commutative 84.4%

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

   10. Applied egg-rr 84.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+37}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 95000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 5: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55) (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 <= -1.55) || !(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 <= (-1.55d0)) .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 <= -1.55) || !(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 <= -1.55) 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 <= -1.55) || !(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 <= -1.55) || ~((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, -1.55], 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 < -1.55000000000000004 or 4 < x

   1. Initial program 90.6%

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

      1. associate-*l/ 86.9%

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

      2. associate-*r/ 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in z around 0 58.5%

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

      1. associate-*r/ 58.5%

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

      2. metadata-eval 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in x around inf 58.1%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in x around 0 73.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 6: 40.3% 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 92.2%

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

   1. associate-*l/ 92.8%

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

   2. associate-*r/ 94.3%

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

3. Simplified 94.3%

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

4. Taylor expanded in x around 0 35.8%

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

5. Final simplification 35.8%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))