fabs fraction 1

Percentage Accurate: 92.4% → 97.6%

Time: 9.1s

Alternatives: 9

Speedup: 1.0×

• 20.4% 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: 92.4% 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: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e-49)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-49) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Fortran

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 <= 6d-49) 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

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-49) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e-49:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x * (z / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e-49)
		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

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e-49)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e-49], 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 < 6e-49

   1. Initial program 87.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 86.3%

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

   3. Simplified 86.3%

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

      1. associate-*r/ 92.6%

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

      2. sub-div 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 6e-49 < y

   1. Initial program 98.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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/ 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 2: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (* x (/ z y)))))
   (if (<= x -2.5e+46)
     t_0
     (if (<= x -2.35e-11)
       t_1
       (if (<= x 2e-12) (fabs (/ 4.0 y)) (if (<= x 2.3e+143) t_1 t_0))))))

C

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 <= -2.5e+46) {
		tmp = t_0;
	} else if (x <= -2.35e-11) {
		tmp = t_1;
	} else if (x <= 2e-12) {
		tmp = fabs((4.0 / y));
	} else if (x <= 2.3e+143) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 <= (-2.5d+46)) then
        tmp = t_0
    else if (x <= (-2.35d-11)) then
        tmp = t_1
    else if (x <= 2d-12) then
        tmp = abs((4.0d0 / y))
    else if (x <= 2.3d+143) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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 <= -2.5e+46) {
		tmp = t_0;
	} else if (x <= -2.35e-11) {
		tmp = t_1;
	} else if (x <= 2e-12) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 2.3e+143) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -2.5e+46:
		tmp = t_0
	elif x <= -2.35e-11:
		tmp = t_1
	elif x <= 2e-12:
		tmp = math.fabs((4.0 / y))
	elif x <= 2.3e+143:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -2.5e+46)
		tmp = t_0;
	elseif (x <= -2.35e-11)
		tmp = t_1;
	elseif (x <= 2e-12)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 2.3e+143)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -2.5e+46)
		tmp = t_0;
	elseif (x <= -2.35e-11)
		tmp = t_1;
	elseif (x <= 2e-12)
		tmp = abs((4.0 / y));
	elseif (x <= 2.3e+143)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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, -2.5e+46], t$95$0, If[LessEqual[x, -2.35e-11], t$95$1, If[LessEqual[x, 2e-12], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.3e+143], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000001e46 or 2.3e143 < x

   1. Initial program 78.5%

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

      1. associate-*l/ 80.1%

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

      2. associate-*r/ 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*r/ 80.1%

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

      2. sub-div 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in x around inf 95.6%

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

      1. associate-/l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in z around 0 79.6%

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

   if -2.5000000000000001e46 < x < -2.34999999999999996e-11 or 1.99999999999999996e-12 < x < 2.3e143

   1. Initial program 97.7%

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

      1. associate-*l/ 95.6%

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

      2. associate-*r/ 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in z around inf 59.6%

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

      1. associate-*r/ 59.6%

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

      2. mul-1-neg 59.6%

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

      3. distribute-lft-neg-out 59.6%

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

      4. *-commutative 59.6%

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

   6. Simplified 59.6%

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

      1. div-inv 59.7%

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

      2. associate-*l* 61.7%

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

      3. add-sqr-sqrt 26.4%

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

      4. sqrt-unprod 55.7%

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

      5. sqr-neg 55.7%

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

      6. sqrt-unprod 35.2%

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

      7. add-sqr-sqrt 61.7%

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

      8. div-inv 61.7%

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

   8. Applied egg-rr 61.7%

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

   if -2.34999999999999996e-11 < x < 1.99999999999999996e-12

   1. Initial program 95.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 75.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-11}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -8 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+143}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -8e+43)
     t_0
     (if (<= x -1.45e-12)
       (fabs (* z (/ x y)))
       (if (<= x 3.5e-13)
         (fabs (/ 4.0 y))
         (if (<= x 4.6e+143) (fabs (* x (/ z y))) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -8e+43) {
		tmp = t_0;
	} else if (x <= -1.45e-12) {
		tmp = fabs((z * (x / y)));
	} else if (x <= 3.5e-13) {
		tmp = fabs((4.0 / y));
	} else if (x <= 4.6e+143) {
		tmp = fabs((x * (z / y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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((x / y))
    if (x <= (-8d+43)) then
        tmp = t_0
    else if (x <= (-1.45d-12)) then
        tmp = abs((z * (x / y)))
    else if (x <= 3.5d-13) then
        tmp = abs((4.0d0 / y))
    else if (x <= 4.6d+143) then
        tmp = abs((x * (z / y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -8e+43) {
		tmp = t_0;
	} else if (x <= -1.45e-12) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= 3.5e-13) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 4.6e+143) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -8e+43:
		tmp = t_0
	elif x <= -1.45e-12:
		tmp = math.fabs((z * (x / y)))
	elif x <= 3.5e-13:
		tmp = math.fabs((4.0 / y))
	elif x <= 4.6e+143:
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -8e+43)
		tmp = t_0;
	elseif (x <= -1.45e-12)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= 3.5e-13)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 4.6e+143)
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -8e+43)
		tmp = t_0;
	elseif (x <= -1.45e-12)
		tmp = abs((z * (x / y)));
	elseif (x <= 3.5e-13)
		tmp = abs((4.0 / y));
	elseif (x <= 4.6e+143)
		tmp = abs((x * (z / y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8e+43], t$95$0, If[LessEqual[x, -1.45e-12], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.5e-13], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.6e+143], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.00000000000000011e43 or 4.5999999999999999e143 < x

   1. Initial program 78.5%

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

      1. associate-*l/ 80.1%

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

      2. associate-*r/ 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*r/ 80.1%

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

      2. sub-div 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in x around inf 95.6%

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

      1. associate-/l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in z around 0 79.6%

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

   if -8.00000000000000011e43 < x < -1.4500000000000001e-12

   1. Initial program 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around inf 57.1%

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

      1. associate-*r/ 57.1%

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

      2. mul-1-neg 57.1%

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

      3. distribute-lft-neg-out 57.1%

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

      4. *-commutative 57.1%

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

   6. Simplified 57.1%

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

      1. distribute-rgt-neg-out 57.1%

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

      2. distribute-frac-neg 57.1%

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

      3. associate-*l/ 57.1%

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

      4. distribute-rgt-neg-in 57.1%

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

      5. add-sqr-sqrt 14.6%

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

      6. sqrt-unprod 45.1%

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

      7. sqr-neg 45.1%

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

      8. sqrt-unprod 42.3%

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

      9. add-sqr-sqrt 57.1%

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

   8. Applied egg-rr 57.1%

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

   if -1.4500000000000001e-12 < x < 3.5000000000000002e-13

   1. Initial program 95.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 75.4%

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

   if 3.5000000000000002e-13 < x < 4.5999999999999999e143

   1. Initial program 96.7%

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

      1. associate-*l/ 93.7%

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

      2. associate-*r/ 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in z around inf 60.8%

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

      1. associate-*r/ 60.8%

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

      2. mul-1-neg 60.8%

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

      3. distribute-lft-neg-out 60.8%

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

      4. *-commutative 60.8%

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

   6. Simplified 60.8%

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

      1. div-inv 60.9%

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

      2. associate-*l* 63.8%

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

      3. add-sqr-sqrt 32.0%

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

      4. sqrt-unprod 60.7%

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

      5. sqr-neg 60.7%

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

      6. sqrt-unprod 31.9%

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

      7. add-sqr-sqrt 63.8%

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

      8. div-inv 63.9%

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

   8. Applied egg-rr 63.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+143}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -7.2e+47)
     t_0
     (if (<= x -2.2e-11)
       (fabs (* z (/ x y)))
       (if (<= x 1.7e-10)
         (fabs (/ 4.0 y))
         (if (<= x 2.3e+143) (fabs (/ x (/ y z))) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -7.2e+47) {
		tmp = t_0;
	} else if (x <= -2.2e-11) {
		tmp = fabs((z * (x / y)));
	} else if (x <= 1.7e-10) {
		tmp = fabs((4.0 / y));
	} else if (x <= 2.3e+143) {
		tmp = fabs((x / (y / z)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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((x / y))
    if (x <= (-7.2d+47)) then
        tmp = t_0
    else if (x <= (-2.2d-11)) then
        tmp = abs((z * (x / y)))
    else if (x <= 1.7d-10) then
        tmp = abs((4.0d0 / y))
    else if (x <= 2.3d+143) then
        tmp = abs((x / (y / z)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -7.2e+47) {
		tmp = t_0;
	} else if (x <= -2.2e-11) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= 1.7e-10) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 2.3e+143) {
		tmp = Math.abs((x / (y / z)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -7.2e+47:
		tmp = t_0
	elif x <= -2.2e-11:
		tmp = math.fabs((z * (x / y)))
	elif x <= 1.7e-10:
		tmp = math.fabs((4.0 / y))
	elif x <= 2.3e+143:
		tmp = math.fabs((x / (y / z)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -7.2e+47)
		tmp = t_0;
	elseif (x <= -2.2e-11)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= 1.7e-10)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 2.3e+143)
		tmp = abs(Float64(x / Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -7.2e+47)
		tmp = t_0;
	elseif (x <= -2.2e-11)
		tmp = abs((z * (x / y)));
	elseif (x <= 1.7e-10)
		tmp = abs((4.0 / y));
	elseif (x <= 2.3e+143)
		tmp = abs((x / (y / z)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7.2e+47], t$95$0, If[LessEqual[x, -2.2e-11], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.7e-10], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.3e+143], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.20000000000000015e47 or 2.3e143 < x

   1. Initial program 78.5%

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

      1. associate-*l/ 80.1%

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

      2. associate-*r/ 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*r/ 80.1%

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

      2. sub-div 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in x around inf 95.6%

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

      1. associate-/l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in z around 0 79.6%

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

   if -7.20000000000000015e47 < x < -2.2000000000000002e-11

   1. Initial program 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around inf 57.1%

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

      1. associate-*r/ 57.1%

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

      2. mul-1-neg 57.1%

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

      3. distribute-lft-neg-out 57.1%

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

      4. *-commutative 57.1%

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

   6. Simplified 57.1%

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

      1. distribute-rgt-neg-out 57.1%

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

      2. distribute-frac-neg 57.1%

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

      3. associate-*l/ 57.1%

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

      4. distribute-rgt-neg-in 57.1%

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

      5. add-sqr-sqrt 14.6%

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

      6. sqrt-unprod 45.1%

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

      7. sqr-neg 45.1%

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

      8. sqrt-unprod 42.3%

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

      9. add-sqr-sqrt 57.1%

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

   8. Applied egg-rr 57.1%

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

   if -2.2000000000000002e-11 < x < 1.70000000000000007e-10

   1. Initial program 95.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 75.4%

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

   if 1.70000000000000007e-10 < x < 2.3e143

   1. Initial program 96.7%

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

      1. associate-*l/ 93.7%

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

      2. associate-*r/ 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in z around inf 60.8%

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

      1. associate-*r/ 60.8%

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

      2. mul-1-neg 60.8%

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

      3. distribute-lft-neg-out 60.8%

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

      4. *-commutative 60.8%

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

   6. Simplified 60.8%

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

      1. div-inv 60.9%

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

      2. associate-*l* 63.8%

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

      3. add-sqr-sqrt 32.0%

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

      4. sqrt-unprod 60.7%

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

      5. sqr-neg 60.7%

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

      6. sqrt-unprod 31.9%

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

      7. add-sqr-sqrt 63.8%

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

      8. div-inv 63.9%

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

   8. Applied egg-rr 63.9%

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

      1. clear-num 63.9%

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

      2. un-div-inv 64.0%

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

   10. Applied egg-rr 64.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+47}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4000000000:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4000000000.0)
   (fabs (/ (* x z) y))
   (if (<= z 2.5e+49) (fabs (/ (+ x 4.0) y)) (fabs (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4000000000.0) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 2.5e+49) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((z * (x / y)));
	}
	return tmp;
}

Fortran

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 <= (-4000000000.0d0)) then
        tmp = abs(((x * z) / y))
    else if (z <= 2.5d+49) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4000000000.0) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 2.5e+49) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4000000000.0:
		tmp = math.fabs(((x * z) / y))
	elif z <= 2.5e+49:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4000000000.0)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 2.5e+49)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4000000000.0)
		tmp = abs(((x * z) / y));
	elseif (z <= 2.5e+49)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4000000000.0], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e+49], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4e9

   1. Initial program 96.9%

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

      1. associate-*l/ 98.2%

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

      2. associate-*r/ 88.9%

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

   3. Simplified 88.9%

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

      1. sub-neg 88.9%

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

      2. +-commutative 88.9%

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

      3. distribute-lft-neg-in 88.9%

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

      4. div-inv 88.9%

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

      5. associate-*r* 98.3%

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

      6. fma-def 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. fma-udef 98.3%

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

      2. distribute-lft-neg-out 98.3%

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

      3. distribute-rgt-neg-out 98.3%

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

      4. div-inv 98.2%

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

      5. associate-/l* 88.9%

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

      6. frac-add 71.0%

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

      7. add-sqr-sqrt 71.0%

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

      8. sqrt-unprod 67.9%

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

      9. sqr-neg 67.9%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 71.0%

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

   7. Applied egg-rr 71.0%

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

   8. Taylor expanded in z around inf 75.1%

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

   if -4e9 < z < 2.5000000000000002e49

   1. Initial program 91.6%

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

      1. associate-*l/ 95.1%

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

      2. associate-*r/ 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in z around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. *-rgt-identity 97.0%

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

      3. associate-*r/ 96.9%

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

      4. distribute-rgt-in 96.9%

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

      5. associate-*l/ 97.0%

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

      6. *-lft-identity 97.0%

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

   6. Simplified 97.0%

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

   if 2.5000000000000002e49 < z

   1. Initial program 80.2%

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

      1. associate-*l/ 79.1%

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

      2. associate-*r/ 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in z around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. mul-1-neg 77.4%

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

      3. distribute-lft-neg-out 77.4%

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

      4. *-commutative 77.4%

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

   6. Simplified 77.4%

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

      1. distribute-rgt-neg-out 77.4%

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

      2. distribute-frac-neg 77.4%

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

      3. associate-*l/ 80.2%

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

      4. distribute-rgt-neg-in 80.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 55.7%

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

      7. sqr-neg 55.7%

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

      8. sqrt-unprod 80.0%

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

      9. add-sqr-sqrt 80.2%

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

   8. Applied egg-rr 80.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4000000000:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2100000000:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x \cdot z}}\right|\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2100000000.0)
   (fabs (/ 1.0 (/ y (* x z))))
   (if (<= z 3.2e+46) (fabs (/ (+ x 4.0) y)) (fabs (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2100000000.0) {
		tmp = fabs((1.0 / (y / (x * z))));
	} else if (z <= 3.2e+46) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((z * (x / y)));
	}
	return tmp;
}

Fortran

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 <= (-2100000000.0d0)) then
        tmp = abs((1.0d0 / (y / (x * z))))
    else if (z <= 3.2d+46) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2100000000.0) {
		tmp = Math.abs((1.0 / (y / (x * z))));
	} else if (z <= 3.2e+46) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2100000000.0:
		tmp = math.fabs((1.0 / (y / (x * z))))
	elif z <= 3.2e+46:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2100000000.0)
		tmp = abs(Float64(1.0 / Float64(y / Float64(x * z))));
	elseif (z <= 3.2e+46)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2100000000.0)
		tmp = abs((1.0 / (y / (x * z))));
	elseif (z <= 3.2e+46)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2100000000.0], N[Abs[N[(1.0 / N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.2e+46], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e9

   1. Initial program 96.9%

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

      1. associate-*l/ 98.2%

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

      2. associate-*r/ 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in z around inf 75.1%

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

      1. associate-*r/ 75.1%

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

      2. mul-1-neg 75.1%

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

      3. distribute-lft-neg-out 75.1%

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

      4. *-commutative 75.1%

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

   6. Simplified 75.1%

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

      1. clear-num 75.2%

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

      2. inv-pow 75.2%

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

      3. add-sqr-sqrt 75.0%

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

      4. sqrt-unprod 55.9%

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

      5. sqr-neg 55.9%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. unpow-1 75.2%

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

      2. *-commutative 75.2%

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

   10. Simplified 75.2%

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

   if -2.1e9 < z < 3.1999999999999998e46

   1. Initial program 91.6%

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

      1. associate-*l/ 95.1%

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

      2. associate-*r/ 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in z around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. *-rgt-identity 97.0%

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

      3. associate-*r/ 96.9%

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

      4. distribute-rgt-in 96.9%

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

      5. associate-*l/ 97.0%

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

      6. *-lft-identity 97.0%

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

   6. Simplified 97.0%

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

   if 3.1999999999999998e46 < z

   1. Initial program 80.2%

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

      1. associate-*l/ 79.1%

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

      2. associate-*r/ 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in z around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. mul-1-neg 77.4%

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

      3. distribute-lft-neg-out 77.4%

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

      4. *-commutative 77.4%

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

   6. Simplified 77.4%

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

      1. distribute-rgt-neg-out 77.4%

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

      2. distribute-frac-neg 77.4%

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

      3. associate-*l/ 80.2%

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

      4. distribute-rgt-neg-in 80.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 55.7%

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

      7. sqr-neg 55.7%

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

      8. sqrt-unprod 80.0%

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

      9. add-sqr-sqrt 80.2%

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

   8. Applied egg-rr 80.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2100000000:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x \cdot z}}\right|\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) - (x * z)) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 90.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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. associate-*r/ 89.7%

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

3. Simplified 89.7%

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

   1. associate-*r/ 92.6%

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

   2. sub-div 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 8: 68.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

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

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

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

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

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

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

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 84.6%

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

      1. associate-*l/ 84.9%

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

      2. associate-*r/ 88.6%

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

   3. Simplified 88.6%

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

      1. associate-*r/ 84.9%

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

      2. sub-div 96.2%

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

   5. Applied egg-rr 96.2%

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

   6. Taylor expanded in x around inf 94.8%

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

      1. associate-/l* 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in z around 0 65.3%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 96.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around 0 72.3%

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

4. Final simplification 68.9%

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

Alternative 9: 40.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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. associate-*r/ 89.7%

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

3. Simplified 89.7%

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

4. Taylor expanded in x around 0 39.9%

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

5. Final simplification 39.9%

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

Reproduce

herbie shell --seed 2023275 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))