fabs fraction 1

Percentage Accurate: 91.7% → 99.7%

Time: 7.6s

Alternatives: 10

Speedup: 1.0×

• 21.1% 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: 91.7% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+78}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7e+78)
   (fabs (* (- 1.0 z) (/ x y)))
   (if (<= x 2e+20)
     (fabs (/ (- (+ 4.0 x) (* x z)) y))
     (fabs (/ x (/ y (- 1.0 z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e+78) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else if (x <= 2e+20) {
		tmp = fabs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = fabs((x / (y / (1.0 - z))));
	}
	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 <= (-7d+78)) then
        tmp = abs(((1.0d0 - z) * (x / y)))
    else if (x <= 2d+20) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y))
    else
        tmp = abs((x / (y / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e+78) {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	} else if (x <= 2e+20) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = Math.abs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7e+78:
		tmp = math.fabs(((1.0 - z) * (x / y)))
	elif x <= 2e+20:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y))
	else:
		tmp = math.fabs((x / (y / (1.0 - z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7e+78)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	elseif (x <= 2e+20)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7e+78)
		tmp = abs(((1.0 - z) * (x / y)));
	elseif (x <= 2e+20)
		tmp = abs((((4.0 + x) - (x * z)) / y));
	else
		tmp = abs((x / (y / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7e+78], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2e+20], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0000000000000003e78

   1. Initial program 77.7%

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

   2. Taylor expanded in y around 0 85.4%

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

   3. Taylor expanded in x around inf 85.4%

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

      1. *-rgt-identity 85.4%

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

      2. associate-*r/ 85.3%

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

      3. *-commutative 85.3%

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

      4. associate-*l* 99.8%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   if -7.0000000000000003e78 < x < 2e20

   1. Initial program 94.6%

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

   2. Taylor expanded in y around 0 99.9%

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

   if 2e20 < x

   1. Initial program 84.7%

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

      1. associate-*l/ 77.8%

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

      2. sub-div 86.9%

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

      3. clear-num 86.6%

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

      4. associate--l+ 86.6%

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

   3. Applied egg-rr 86.6%

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

   4. Taylor expanded in x around inf 86.9%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+78}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-11)
   (fabs (/ (- (+ 4.0 x) (* x z)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-11) {
		tmp = fabs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-11)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-11], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000018e-11

   1. Initial program 85.6%

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

   2. Taylor expanded in y around 0 94.3%

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

   if 5.00000000000000018e-11 < y

   1. Initial program 98.5%

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

   3. Simplified 99.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 3: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{4}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ t_2 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+268}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y))) (t_1 (fabs (/ x y))) (t_2 (fabs (* x (/ z y)))))
   (if (<= x -1.8e+60)
     t_1
     (if (<= x -2.3e-15)
       t_2
       (if (<= x 3.8e-97)
         t_0
         (if (<= x 3.75e-62)
           t_2
           (if (<= x 4.0) t_0 (if (<= x 5.8e+268) t_1 t_2))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((4.0 / y));
	double t_1 = fabs((x / y));
	double t_2 = fabs((x * (z / y)));
	double tmp;
	if (x <= -1.8e+60) {
		tmp = t_1;
	} else if (x <= -2.3e-15) {
		tmp = t_2;
	} else if (x <= 3.8e-97) {
		tmp = t_0;
	} else if (x <= 3.75e-62) {
		tmp = t_2;
	} else if (x <= 4.0) {
		tmp = t_0;
	} else if (x <= 5.8e+268) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = abs((4.0d0 / y))
    t_1 = abs((x / y))
    t_2 = abs((x * (z / y)))
    if (x <= (-1.8d+60)) then
        tmp = t_1
    else if (x <= (-2.3d-15)) then
        tmp = t_2
    else if (x <= 3.8d-97) then
        tmp = t_0
    else if (x <= 3.75d-62) then
        tmp = t_2
    else if (x <= 4.0d0) then
        tmp = t_0
    else if (x <= 5.8d+268) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((4.0 / y));
	double t_1 = Math.abs((x / y));
	double t_2 = Math.abs((x * (z / y)));
	double tmp;
	if (x <= -1.8e+60) {
		tmp = t_1;
	} else if (x <= -2.3e-15) {
		tmp = t_2;
	} else if (x <= 3.8e-97) {
		tmp = t_0;
	} else if (x <= 3.75e-62) {
		tmp = t_2;
	} else if (x <= 4.0) {
		tmp = t_0;
	} else if (x <= 5.8e+268) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((4.0 / y))
	t_1 = math.fabs((x / y))
	t_2 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -1.8e+60:
		tmp = t_1
	elif x <= -2.3e-15:
		tmp = t_2
	elif x <= 3.8e-97:
		tmp = t_0
	elif x <= 3.75e-62:
		tmp = t_2
	elif x <= 4.0:
		tmp = t_0
	elif x <= 5.8e+268:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(4.0 / y))
	t_1 = abs(Float64(x / y))
	t_2 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -1.8e+60)
		tmp = t_1;
	elseif (x <= -2.3e-15)
		tmp = t_2;
	elseif (x <= 3.8e-97)
		tmp = t_0;
	elseif (x <= 3.75e-62)
		tmp = t_2;
	elseif (x <= 4.0)
		tmp = t_0;
	elseif (x <= 5.8e+268)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((4.0 / y));
	t_1 = abs((x / y));
	t_2 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -1.8e+60)
		tmp = t_1;
	elseif (x <= -2.3e-15)
		tmp = t_2;
	elseif (x <= 3.8e-97)
		tmp = t_0;
	elseif (x <= 3.75e-62)
		tmp = t_2;
	elseif (x <= 4.0)
		tmp = t_0;
	elseif (x <= 5.8e+268)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.8e+60], t$95$1, If[LessEqual[x, -2.3e-15], t$95$2, If[LessEqual[x, 3.8e-97], t$95$0, If[LessEqual[x, 3.75e-62], t$95$2, If[LessEqual[x, 4.0], t$95$0, If[LessEqual[x, 5.8e+268], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999984e60 or 4 < x < 5.8000000000000002e268

   1. Initial program 83.1%

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

   2. Taylor expanded in z around 0 72.9%

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

      1. associate-*r/ 72.9%

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

      2. metadata-eval 72.9%

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

   4. Simplified 72.9%

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

   5. Taylor expanded in x around inf 72.5%

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

   if -1.79999999999999984e60 < x < -2.2999999999999999e-15 or 3.8000000000000001e-97 < x < 3.75000000000000015e-62 or 5.8000000000000002e268 < x

   1. Initial program 87.5%

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

   2. Taylor expanded in z around inf 63.8%

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

      1. mul-1-neg 63.8%

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

      2. associate-*l/ 78.6%

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

      3. distribute-rgt-neg-out 78.6%

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

   4. Simplified 78.6%

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

      1. clear-num 78.4%

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

      2. associate-*l/ 78.5%

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

      3. *-un-lft-identity 78.5%

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

      4. add-sqr-sqrt 37.2%

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

      5. sqrt-unprod 57.4%

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

      6. sqr-neg 57.4%

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

      7. sqrt-unprod 41.3%

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

      8. add-sqr-sqrt 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-/r/ 78.6%

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

   8. Applied egg-rr 78.6%

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

   if -2.2999999999999999e-15 < x < 3.8000000000000001e-97 or 3.75000000000000015e-62 < x < 4

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0 79.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-15}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-62}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+268}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 4: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{4}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ t_2 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.66 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+268}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y))) (t_1 (fabs (/ x y))) (t_2 (fabs (* x (/ z y)))))
   (if (<= x -6.8e+59)
     t_1
     (if (<= x -1.66e-16)
       (fabs (/ x (/ y z)))
       (if (<= x 3.8e-97)
         t_0
         (if (<= x 3.2e-62)
           t_2
           (if (<= x 4.0) t_0 (if (<= x 5.6e+268) t_1 t_2))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((4.0 / y));
	double t_1 = fabs((x / y));
	double t_2 = fabs((x * (z / y)));
	double tmp;
	if (x <= -6.8e+59) {
		tmp = t_1;
	} else if (x <= -1.66e-16) {
		tmp = fabs((x / (y / z)));
	} else if (x <= 3.8e-97) {
		tmp = t_0;
	} else if (x <= 3.2e-62) {
		tmp = t_2;
	} else if (x <= 4.0) {
		tmp = t_0;
	} else if (x <= 5.6e+268) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = abs((4.0d0 / y))
    t_1 = abs((x / y))
    t_2 = abs((x * (z / y)))
    if (x <= (-6.8d+59)) then
        tmp = t_1
    else if (x <= (-1.66d-16)) then
        tmp = abs((x / (y / z)))
    else if (x <= 3.8d-97) then
        tmp = t_0
    else if (x <= 3.2d-62) then
        tmp = t_2
    else if (x <= 4.0d0) then
        tmp = t_0
    else if (x <= 5.6d+268) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((4.0 / y));
	double t_1 = Math.abs((x / y));
	double t_2 = Math.abs((x * (z / y)));
	double tmp;
	if (x <= -6.8e+59) {
		tmp = t_1;
	} else if (x <= -1.66e-16) {
		tmp = Math.abs((x / (y / z)));
	} else if (x <= 3.8e-97) {
		tmp = t_0;
	} else if (x <= 3.2e-62) {
		tmp = t_2;
	} else if (x <= 4.0) {
		tmp = t_0;
	} else if (x <= 5.6e+268) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((4.0 / y))
	t_1 = math.fabs((x / y))
	t_2 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -6.8e+59:
		tmp = t_1
	elif x <= -1.66e-16:
		tmp = math.fabs((x / (y / z)))
	elif x <= 3.8e-97:
		tmp = t_0
	elif x <= 3.2e-62:
		tmp = t_2
	elif x <= 4.0:
		tmp = t_0
	elif x <= 5.6e+268:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(4.0 / y))
	t_1 = abs(Float64(x / y))
	t_2 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -6.8e+59)
		tmp = t_1;
	elseif (x <= -1.66e-16)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (x <= 3.8e-97)
		tmp = t_0;
	elseif (x <= 3.2e-62)
		tmp = t_2;
	elseif (x <= 4.0)
		tmp = t_0;
	elseif (x <= 5.6e+268)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((4.0 / y));
	t_1 = abs((x / y));
	t_2 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -6.8e+59)
		tmp = t_1;
	elseif (x <= -1.66e-16)
		tmp = abs((x / (y / z)));
	elseif (x <= 3.8e-97)
		tmp = t_0;
	elseif (x <= 3.2e-62)
		tmp = t_2;
	elseif (x <= 4.0)
		tmp = t_0;
	elseif (x <= 5.6e+268)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.8e+59], t$95$1, If[LessEqual[x, -1.66e-16], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.8e-97], t$95$0, If[LessEqual[x, 3.2e-62], t$95$2, If[LessEqual[x, 4.0], t$95$0, If[LessEqual[x, 5.6e+268], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.80000000000000012e59 or 4 < x < 5.5999999999999999e268

   1. Initial program 83.1%

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

   2. Taylor expanded in z around 0 72.9%

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

      1. associate-*r/ 72.9%

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

      2. metadata-eval 72.9%

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

   4. Simplified 72.9%

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

   5. Taylor expanded in x around inf 72.5%

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

   if -6.80000000000000012e59 < x < -1.6600000000000001e-16

   1. Initial program 91.4%

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

   2. Taylor expanded in z around inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. associate-*l/ 76.4%

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

      3. distribute-rgt-neg-out 76.4%

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

   4. Simplified 76.4%

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

      1. associate-*l/ 76.5%

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

      2. associate-/l* 76.5%

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

      3. add-sqr-sqrt 24.9%

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

      4. sqrt-unprod 44.5%

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

      5. sqr-neg 44.5%

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

      6. sqrt-unprod 51.1%

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

      7. add-sqr-sqrt 76.5%

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

   6. Applied egg-rr 76.5%

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

   if -1.6600000000000001e-16 < x < 3.8000000000000001e-97 or 3.20000000000000021e-62 < x < 4

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0 79.2%

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

   if 3.8000000000000001e-97 < x < 3.20000000000000021e-62 or 5.5999999999999999e268 < x

   1. Initial program 85.2%

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

   2. Taylor expanded in z around inf 56.1%

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

      1. mul-1-neg 56.1%

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

      2. associate-*l/ 79.9%

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

      3. distribute-rgt-neg-out 79.9%

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

   4. Simplified 79.9%

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

      1. clear-num 79.8%

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

      2. associate-*l/ 79.8%

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

      3. *-un-lft-identity 79.8%

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

      4. add-sqr-sqrt 44.6%

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

      5. sqrt-unprod 65.1%

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

      6. sqr-neg 65.1%

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

      7. sqrt-unprod 35.4%

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

      8. add-sqr-sqrt 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/r/ 80.0%

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

   8. Applied egg-rr 80.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+59}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.66 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+268}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 5: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{4}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-61}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+268}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y))) (t_1 (fabs (/ x y))))
   (if (<= x -6.5e+59)
     t_1
     (if (<= x -1.4e-16)
       (fabs (/ x (/ y z)))
       (if (<= x 3.4e-97)
         t_0
         (if (<= x 1.22e-61)
           (fabs (* x (/ z y)))
           (if (<= x 4.0)
             t_0
             (if (<= x 3.5e+268) t_1 (fabs (/ z (/ y x)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((4.0 / y));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -6.5e+59) {
		tmp = t_1;
	} else if (x <= -1.4e-16) {
		tmp = fabs((x / (y / z)));
	} else if (x <= 3.4e-97) {
		tmp = t_0;
	} else if (x <= 1.22e-61) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = t_0;
	} else if (x <= 3.5e+268) {
		tmp = t_1;
	} else {
		tmp = fabs((z / (y / x)));
	}
	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((4.0d0 / y))
    t_1 = abs((x / y))
    if (x <= (-6.5d+59)) then
        tmp = t_1
    else if (x <= (-1.4d-16)) then
        tmp = abs((x / (y / z)))
    else if (x <= 3.4d-97) then
        tmp = t_0
    else if (x <= 1.22d-61) then
        tmp = abs((x * (z / y)))
    else if (x <= 4.0d0) then
        tmp = t_0
    else if (x <= 3.5d+268) then
        tmp = t_1
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((4.0 / y));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -6.5e+59) {
		tmp = t_1;
	} else if (x <= -1.4e-16) {
		tmp = Math.abs((x / (y / z)));
	} else if (x <= 3.4e-97) {
		tmp = t_0;
	} else if (x <= 1.22e-61) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = t_0;
	} else if (x <= 3.5e+268) {
		tmp = t_1;
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((4.0 / y))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -6.5e+59:
		tmp = t_1
	elif x <= -1.4e-16:
		tmp = math.fabs((x / (y / z)))
	elif x <= 3.4e-97:
		tmp = t_0
	elif x <= 1.22e-61:
		tmp = math.fabs((x * (z / y)))
	elif x <= 4.0:
		tmp = t_0
	elif x <= 3.5e+268:
		tmp = t_1
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(4.0 / y))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -6.5e+59)
		tmp = t_1;
	elseif (x <= -1.4e-16)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (x <= 3.4e-97)
		tmp = t_0;
	elseif (x <= 1.22e-61)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 4.0)
		tmp = t_0;
	elseif (x <= 3.5e+268)
		tmp = t_1;
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((4.0 / y));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -6.5e+59)
		tmp = t_1;
	elseif (x <= -1.4e-16)
		tmp = abs((x / (y / z)));
	elseif (x <= 3.4e-97)
		tmp = t_0;
	elseif (x <= 1.22e-61)
		tmp = abs((x * (z / y)));
	elseif (x <= 4.0)
		tmp = t_0;
	elseif (x <= 3.5e+268)
		tmp = t_1;
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.5e+59], t$95$1, If[LessEqual[x, -1.4e-16], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.4e-97], t$95$0, If[LessEqual[x, 1.22e-61], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], t$95$0, If[LessEqual[x, 3.5e+268], t$95$1, N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.50000000000000021e59 or 4 < x < 3.49999999999999972e268

   1. Initial program 83.1%

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

   2. Taylor expanded in z around 0 72.9%

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

      1. associate-*r/ 72.9%

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

      2. metadata-eval 72.9%

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

   4. Simplified 72.9%

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

   5. Taylor expanded in x around inf 72.5%

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

   if -6.50000000000000021e59 < x < -1.4000000000000001e-16

   1. Initial program 91.4%

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

   2. Taylor expanded in z around inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. associate-*l/ 76.4%

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

      3. distribute-rgt-neg-out 76.4%

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

   4. Simplified 76.4%

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

      1. associate-*l/ 76.5%

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

      2. associate-/l* 76.5%

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

      3. add-sqr-sqrt 24.9%

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

      4. sqrt-unprod 44.5%

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

      5. sqr-neg 44.5%

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

      6. sqrt-unprod 51.1%

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

      7. add-sqr-sqrt 76.5%

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

   6. Applied egg-rr 76.5%

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

   if -1.4000000000000001e-16 < x < 3.3999999999999999e-97 or 1.22e-61 < x < 4

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0 79.2%

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

   if 3.3999999999999999e-97 < x < 1.22e-61

   1. Initial program 89.5%

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

   2. Taylor expanded in z around inf 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-*l/ 66.3%

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

      3. distribute-rgt-neg-out 66.3%

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

   4. Simplified 66.3%

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

      1. clear-num 66.3%

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

      2. associate-*l/ 66.3%

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

      3. *-un-lft-identity 66.3%

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

      4. add-sqr-sqrt 43.1%

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

      5. sqrt-unprod 66.3%

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

      6. sqr-neg 66.3%

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

      7. sqrt-unprod 23.4%

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

      8. add-sqr-sqrt 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. associate-/r/ 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 3.49999999999999972e268 < x

   1. Initial program 81.7%

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

   2. Taylor expanded in z around inf 39.2%

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

      1. mul-1-neg 39.2%

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

      2. associate-*l/ 91.1%

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

      3. distribute-rgt-neg-out 91.1%

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

   4. Simplified 91.1%

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

      1. clear-num 90.8%

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

      2. associate-*l/ 90.9%

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

      3. *-un-lft-identity 90.9%

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

      4. add-sqr-sqrt 45.8%

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

      5. sqrt-unprod 64.1%

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

      6. sqr-neg 64.1%

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

      7. sqrt-unprod 45.2%

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

      8. add-sqr-sqrt 90.9%

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

   6. Applied egg-rr 90.9%

      \[\leadsto \left|\color{blue}{\frac{z}{\frac{y}{x}}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+59}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-97}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-61}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+268}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.5e+36)
   (fabs (/ z (/ y x)))
   (if (<= z 5.5e+41) (fabs (+ (/ x y) (/ 4.0 y))) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.5e+36) {
		tmp = fabs((z / (y / x)));
	} else if (z <= 5.5e+41) {
		tmp = fabs(((x / y) + (4.0 / y)));
	} else {
		tmp = fabs((x / (y / z)));
	}
	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 <= (-3.5d+36)) then
        tmp = abs((z / (y / x)))
    else if (z <= 5.5d+41) then
        tmp = abs(((x / y) + (4.0d0 / y)))
    else
        tmp = abs((x / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.5e+36) {
		tmp = Math.abs((z / (y / x)));
	} else if (z <= 5.5e+41) {
		tmp = Math.abs(((x / y) + (4.0 / y)));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.5e+36:
		tmp = math.fabs((z / (y / x)))
	elif z <= 5.5e+41:
		tmp = math.fabs(((x / y) + (4.0 / y)))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.5e+36)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (z <= 5.5e+41)
		tmp = abs(Float64(Float64(x / y) + Float64(4.0 / y)));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.5e+36)
		tmp = abs((z / (y / x)));
	elseif (z <= 5.5e+41)
		tmp = abs(((x / y) + (4.0 / y)));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.5e+36], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 5.5e+41], N[Abs[N[(N[(x / y), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4999999999999998e36

   1. Initial program 97.7%

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

   2. Taylor expanded in z around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. associate-*l/ 77.5%

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

      3. distribute-rgt-neg-out 77.5%

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

   4. Simplified 77.5%

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

      1. clear-num 77.5%

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

      2. associate-*l/ 77.6%

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

      3. *-un-lft-identity 77.6%

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

      4. add-sqr-sqrt 77.3%

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

      5. sqrt-unprod 51.2%

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

      6. sqr-neg 51.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 77.6%

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

   6. Applied egg-rr 77.6%

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

   if -3.4999999999999998e36 < z < 5.5000000000000003e41

   1. Initial program 90.8%

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

   2. Taylor expanded in z around 0 96.4%

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

      1. associate-*r/ 96.4%

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

      2. metadata-eval 96.4%

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

   4. Simplified 96.4%

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

   if 5.5000000000000003e41 < z

   1. Initial program 77.2%

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

   2. Taylor expanded in z around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-*l/ 71.7%

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

      3. distribute-rgt-neg-out 71.7%

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

   4. Simplified 71.7%

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

      1. associate-*l/ 68.1%

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

      2. associate-/l* 79.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 50.8%

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

      5. sqr-neg 50.8%

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

      6. sqrt-unprod 79.4%

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

      7. add-sqr-sqrt 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 7: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e+40)
   (fabs (- (/ x y) (/ z (/ y x))))
   (if (<= z 1.1e+41) (fabs (+ (/ x y) (/ 4.0 y))) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+40) {
		tmp = fabs(((x / y) - (z / (y / x))));
	} else if (z <= 1.1e+41) {
		tmp = fabs(((x / y) + (4.0 / y)));
	} else {
		tmp = fabs((x / (y / z)));
	}
	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 <= (-1.9d+40)) then
        tmp = abs(((x / y) - (z / (y / x))))
    else if (z <= 1.1d+41) then
        tmp = abs(((x / y) + (4.0d0 / y)))
    else
        tmp = abs((x / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+40) {
		tmp = Math.abs(((x / y) - (z / (y / x))));
	} else if (z <= 1.1e+41) {
		tmp = Math.abs(((x / y) + (4.0 / y)));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.9e+40:
		tmp = math.fabs(((x / y) - (z / (y / x))))
	elif z <= 1.1e+41:
		tmp = math.fabs(((x / y) + (4.0 / y)))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e+40)
		tmp = abs(Float64(Float64(x / y) - Float64(z / Float64(y / x))));
	elseif (z <= 1.1e+41)
		tmp = abs(Float64(Float64(x / y) + Float64(4.0 / y)));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.9e+40)
		tmp = abs(((x / y) - (z / (y / x))));
	elseif (z <= 1.1e+41)
		tmp = abs(((x / y) + (4.0 / y)));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e+40], N[Abs[N[(N[(x / y), $MachinePrecision] - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.1e+41], N[Abs[N[(N[(x / y), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.90000000000000002e40

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. clear-num 97.6%

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

      3. un-div-inv 97.8%

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

   3. Applied egg-rr 97.8%

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

   4. Taylor expanded in x around inf 77.6%

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

   if -1.90000000000000002e40 < z < 1.09999999999999995e41

   1. Initial program 90.8%

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

   2. Taylor expanded in z around 0 96.4%

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

      1. associate-*r/ 96.4%

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

      2. metadata-eval 96.4%

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

   4. Simplified 96.4%

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

   if 1.09999999999999995e41 < z

   1. Initial program 77.2%

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

   2. Taylor expanded in z around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-*l/ 71.7%

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

      3. distribute-rgt-neg-out 71.7%

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

   4. Simplified 71.7%

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

      1. associate-*l/ 68.1%

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

      2. associate-/l* 79.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 50.8%

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

      5. sqr-neg 50.8%

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

      6. sqrt-unprod 79.4%

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

      7. add-sqr-sqrt 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 8: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.18e+36)
   (fabs (/ z (/ y x)))
   (if (<= z 3e+40) (fabs (/ (- -4.0 x) y)) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.18e+36) {
		tmp = fabs((z / (y / x)));
	} else if (z <= 3e+40) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((x / (y / z)));
	}
	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 <= (-1.18d+36)) then
        tmp = abs((z / (y / x)))
    else if (z <= 3d+40) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.18e+36) {
		tmp = Math.abs((z / (y / x)));
	} else if (z <= 3e+40) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.18e+36:
		tmp = math.fabs((z / (y / x)))
	elif z <= 3e+40:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.18e+36)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (z <= 3e+40)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.18e+36)
		tmp = abs((z / (y / x)));
	elseif (z <= 3e+40)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.18e+36], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3e+40], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.17999999999999997e36

   1. Initial program 97.7%

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

   2. Taylor expanded in z around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. associate-*l/ 77.5%

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

      3. distribute-rgt-neg-out 77.5%

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

   4. Simplified 77.5%

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

      1. clear-num 77.5%

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

      2. associate-*l/ 77.6%

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

      3. *-un-lft-identity 77.6%

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

      4. add-sqr-sqrt 77.3%

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

      5. sqrt-unprod 51.2%

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

      6. sqr-neg 51.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 77.6%

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

   6. Applied egg-rr 77.6%

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

   if -1.17999999999999997e36 < z < 3.0000000000000002e40

   1. Initial program 90.8%

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

   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 96.4%

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

      1. associate-*r/ 96.4%

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

      2. distribute-lft-in 96.4%

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

      3. metadata-eval 96.4%

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

      4. neg-mul-1 96.4%

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

      5. sub-neg 96.4%

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

   6. Simplified 96.4%

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

   if 3.0000000000000002e40 < z

   1. Initial program 77.2%

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

   2. Taylor expanded in z around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-*l/ 71.7%

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

      3. distribute-rgt-neg-out 71.7%

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

   4. Simplified 71.7%

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

      1. associate-*l/ 68.1%

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

      2. associate-/l* 79.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 50.8%

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

      5. sqr-neg 50.8%

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

      6. sqrt-unprod 79.4%

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

      7. add-sqr-sqrt 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 9: 68.3% 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 83.1%

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

   2. Taylor expanded in z around 0 68.7%

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

      1. associate-*r/ 68.7%

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

      2. metadata-eval 68.7%

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

   4. Simplified 68.7%

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

   5. Taylor expanded in x around inf 68.0%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 73.0%

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

4. Final simplification 70.4%

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

Alternative 10: 38.7% 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 89.1%

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

2. Taylor expanded in x around 0 38.3%

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

3. Final simplification 38.3%

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

Reproduce

herbie shell --seed 2023321 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))