fabs fraction 1

Percentage Accurate: 92.0% → 99.5%

Time: 7.1s

Alternatives: 10

Speedup: 1.0×

• 20.9% 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: 92.0% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+98} \lor \neg \left(x \leq 5 \cdot 10^{+37}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4e+98) (not (<= x 5e+37)))
   (fabs (/ x (/ y (- 1.0 z))))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4e+98) || !(x <= 5e+37)) {
		tmp = fabs((x / (y / (1.0 - z))));
	} else {
		tmp = fabs((((x + 4.0) - (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 ((x <= (-4d+98)) .or. (.not. (x <= 5d+37))) then
        tmp = abs((x / (y / (1.0d0 - z))))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4e+98) || !(x <= 5e+37)) {
		tmp = Math.abs((x / (y / (1.0 - z))));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4e+98) or not (x <= 5e+37):
		tmp = math.fabs((x / (y / (1.0 - z))))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4e+98) || !(x <= 5e+37))
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4e+98) || ~((x <= 5e+37)))
		tmp = abs((x / (y / (1.0 - z))));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4e+98], N[Not[LessEqual[x, 5e+37]], $MachinePrecision]], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999999e98 or 4.99999999999999989e37 < x

   1. Initial program 86.4%

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

      1. associate-*l/ 78.6%

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

      2. sub-div 88.2%

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

   3. Applied egg-rr 88.2%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   if -3.99999999999999999e98 < x < 4.99999999999999989e37

   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. sub-div 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+98} \lor \neg \left(x \leq 5 \cdot 10^{+37}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-20)
   (fabs (/ (- (+ x 4.0) (* 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 <= 4e-20) {
		tmp = fabs((((x + 4.0) - (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 <= 4e-20)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e-20], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999978e-20

   1. Initial program 90.5%

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

      1. associate-*l/ 91.4%

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

      2. sub-div 96.9%

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

   3. Applied egg-rr 96.9%

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

   if 3.99999999999999978e-20 < y

   1. Initial program 96.0%

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

   3. Simplified 99.9%

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

4. Final simplification 97.7%

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

Alternative 3: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -10:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -10.0)
   (fabs (* (- 1.0 z) (/ x y)))
   (if (<= z 8.2e+75)
     (fabs (/ (- x -4.0) y))
     (fabs (* x (+ (/ z y) (/ -1.0 y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -10.0) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else if (z <= 8.2e+75) {
		tmp = fabs(((x - -4.0) / y));
	} else {
		tmp = fabs((x * ((z / y) + (-1.0 / 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 (z <= (-10.0d0)) then
        tmp = abs(((1.0d0 - z) * (x / y)))
    else if (z <= 8.2d+75) then
        tmp = abs(((x - (-4.0d0)) / y))
    else
        tmp = abs((x * ((z / y) + ((-1.0d0) / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -10.0) {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	} else if (z <= 8.2e+75) {
		tmp = Math.abs(((x - -4.0) / y));
	} else {
		tmp = Math.abs((x * ((z / y) + (-1.0 / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -10.0:
		tmp = math.fabs(((1.0 - z) * (x / y)))
	elif z <= 8.2e+75:
		tmp = math.fabs(((x - -4.0) / y))
	else:
		tmp = math.fabs((x * ((z / y) + (-1.0 / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -10.0)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	elseif (z <= 8.2e+75)
		tmp = abs(Float64(Float64(x - -4.0) / y));
	else
		tmp = abs(Float64(x * Float64(Float64(z / y) + Float64(-1.0 / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -10.0)
		tmp = abs(((1.0 - z) * (x / y)));
	elseif (z <= 8.2e+75)
		tmp = abs(((x - -4.0) / y));
	else
		tmp = abs((x * ((z / y) + (-1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -10.0], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 8.2e+75], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(z / y), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -10

   1. Initial program 94.6%

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

      1. associate-*l/ 86.2%

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

      2. sub-div 86.1%

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

   3. Applied egg-rr 86.1%

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

   4. Taylor expanded in x around inf 73.3%

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

      1. associate-/l* 80.2%

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

      2. associate-/r/ 81.7%

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

   6. Applied egg-rr 81.7%

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

   if -10 < z < 8.1999999999999997e75

   1. Initial program 95.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. associate-*r/ 98.3%

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

      2. distribute-lft-in 98.3%

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

      3. metadata-eval 98.3%

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

      4. neg-mul-1 98.3%

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

      5. sub-neg 98.3%

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

   6. Simplified 98.3%

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

   if 8.1999999999999997e75 < z

   1. Initial program 76.4%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in x around inf 85.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-20)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-20) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (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 (y <= 5d-20) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x / (y / z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-20) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e-20:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x / (y / z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-20)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x / Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-20)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x / (y / z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-20], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e-20

   1. Initial program 90.5%

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

      1. associate-*l/ 91.4%

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

      2. sub-div 96.9%

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

   3. Applied egg-rr 96.9%

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

   if 4.9999999999999999e-20 < y

   1. Initial program 96.0%

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

   3. Simplified 99.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 5: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -9200000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-43}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -9200000000000.0)
     t_0
     (if (<= x -6.4e-43)
       (fabs (* z (/ x y)))
       (if (<= x 4.0) (fabs (/ 4.0 y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -9200000000000.0) {
		tmp = t_0;
	} else if (x <= -6.4e-43) {
		tmp = fabs((z * (x / y)));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / 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 <= (-9200000000000.0d0)) then
        tmp = t_0
    else if (x <= (-6.4d-43)) then
        tmp = abs((z * (x / y)))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / 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 <= -9200000000000.0) {
		tmp = t_0;
	} else if (x <= -6.4e-43) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -9200000000000.0:
		tmp = t_0
	elif x <= -6.4e-43:
		tmp = math.fabs((z * (x / y)))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -9200000000000.0)
		tmp = t_0;
	elseif (x <= -6.4e-43)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / 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 <= -9200000000000.0)
		tmp = t_0;
	elseif (x <= -6.4e-43)
		tmp = abs((z * (x / y)));
	elseif (x <= 4.0)
		tmp = abs((4.0 / 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, -9200000000000.0], t$95$0, If[LessEqual[x, -6.4e-43], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.2e12 or 4 < x

   1. Initial program 89.3%

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

      1. associate-*l/ 83.1%

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

      2. sub-div 90.7%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in x around inf 89.8%

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

   5. Taylor expanded in z around 0 72.0%

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

   if -9.2e12 < x < -6.3999999999999997e-43

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. associate-*r/ 87.3%

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

      3. distribute-rgt-neg-out 87.3%

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

      4. distribute-neg-frac 87.3%

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

   4. Simplified 87.3%

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

      1. expm1-log1p-u 65.7%

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

      2. expm1-udef 56.9%

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

      3. add-sqr-sqrt 31.2%

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

      4. sqrt-unprod 16.0%

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

      5. sqr-neg 16.0%

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

      6. sqrt-unprod 0.3%

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

      7. add-sqr-sqrt 18.5%

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

   6. Applied egg-rr 18.5%

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

      1. expm1-def 27.3%

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

      2. expm1-log1p 87.3%

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

      3. associate-*r/ 87.6%

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

      4. associate-*l/ 87.6%

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

      5. *-commutative 87.6%

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

   8. Simplified 87.6%

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

   if -6.3999999999999997e-43 < x < 4

   1. Initial program 94.5%

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

   2. Taylor expanded in x around 0 78.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9200000000000:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-43}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -320000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+76}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -320000.0)
   (fabs (/ z (/ y x)))
   (if (<= z 1.3e+76) (fabs (/ (- x -4.0) y)) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -320000.0) {
		tmp = fabs((z / (y / x)));
	} else if (z <= 1.3e+76) {
		tmp = fabs(((x - -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 <= (-320000.0d0)) then
        tmp = abs((z / (y / x)))
    else if (z <= 1.3d+76) then
        tmp = abs(((x - (-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 <= -320000.0) {
		tmp = Math.abs((z / (y / x)));
	} else if (z <= 1.3e+76) {
		tmp = Math.abs(((x - -4.0) / y));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -320000.0:
		tmp = math.fabs((z / (y / x)))
	elif z <= 1.3e+76:
		tmp = math.fabs(((x - -4.0) / y))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -320000.0)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (z <= 1.3e+76)
		tmp = abs(Float64(Float64(x - -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 <= -320000.0)
		tmp = abs((z / (y / x)));
	elseif (z <= 1.3e+76)
		tmp = abs(((x - -4.0) / y));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -320000.0], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.3e+76], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2e5

   1. Initial program 94.6%

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

   2. Taylor expanded in z around inf 71.5%

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

      1. mul-1-neg 71.5%

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

      2. associate-*r/ 78.4%

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

      3. distribute-rgt-neg-out 78.4%

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

      4. distribute-neg-frac 78.4%

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

   4. Simplified 78.4%

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

      1. associate-*r/ 71.5%

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

      2. add-cube-cbrt 70.9%

         \[\leadsto \left|\frac{x \cdot \left(-z\right)}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right| \]

      3. times-frac 79.3%

         \[\leadsto \left|\color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{-z}{\sqrt[3]{y}}}\right| \]

      4. add-sqr-sqrt 79.3%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}{\sqrt[3]{y}}\right| \]

      5. sqrt-unprod 57.5%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}{\sqrt[3]{y}}\right| \]

      6. sqr-neg 57.5%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt{\color{blue}{z \cdot z}}}{\sqrt[3]{y}}\right| \]

      7. sqrt-unprod 0.0%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt[3]{y}}\right| \]

      8. add-sqr-sqrt 79.3%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{z}}{\sqrt[3]{y}}\right| \]

      9. times-frac 70.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right| \]

      10. *-commutative 70.9%

          \[\leadsto \left|\frac{\color{blue}{z \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right| \]

      11. add-cube-cbrt 71.5%

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

      12. associate-/l* 79.9%

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

   6. Applied egg-rr 79.9%

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

   if -3.2e5 < z < 1.3e76

   1. Initial program 95.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. associate-*r/ 98.3%

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

      2. distribute-lft-in 98.3%

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

      3. metadata-eval 98.3%

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

      4. neg-mul-1 98.3%

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

      5. sub-neg 98.3%

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

   6. Simplified 98.3%

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

   if 1.3e76 < z

   1. Initial program 76.4%

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

   2. Taylor expanded in z around inf 78.8%

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

      1. mul-1-neg 78.8%

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

      2. associate-*r/ 85.5%

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

      3. distribute-rgt-neg-out 85.5%

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

      4. distribute-neg-frac 85.5%

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

   4. Simplified 85.5%

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

      1. associate-*r/ 78.8%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 48.2%

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

      4. sqr-neg 48.2%

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

      5. sqrt-unprod 78.6%

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

      6. add-sqr-sqrt 78.8%

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

      7. associate-/l* 85.5%

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

   6. Applied egg-rr 85.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -320000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+76}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 7: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -320000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+76}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -320000.0)
   (fabs (/ z (/ y x)))
   (if (<= z 3.05e+76) (fabs (/ (- x -4.0) y)) (fabs (* x (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -320000.0) {
		tmp = fabs((z / (y / x)));
	} else if (z <= 3.05e+76) {
		tmp = fabs(((x - -4.0) / y));
	} else {
		tmp = fabs((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 (z <= (-320000.0d0)) then
        tmp = abs((z / (y / x)))
    else if (z <= 3.05d+76) then
        tmp = abs(((x - (-4.0d0)) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -320000.0) {
		tmp = Math.abs((z / (y / x)));
	} else if (z <= 3.05e+76) {
		tmp = Math.abs(((x - -4.0) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -320000.0:
		tmp = math.fabs((z / (y / x)))
	elif z <= 3.05e+76:
		tmp = math.fabs(((x - -4.0) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -320000.0)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (z <= 3.05e+76)
		tmp = abs(Float64(Float64(x - -4.0) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -320000.0)
		tmp = abs((z / (y / x)));
	elseif (z <= 3.05e+76)
		tmp = abs(((x - -4.0) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -320000.0], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.05e+76], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2e5

   1. Initial program 94.6%

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

   2. Taylor expanded in z around inf 71.5%

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

      1. mul-1-neg 71.5%

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

      2. associate-*r/ 78.4%

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

      3. distribute-rgt-neg-out 78.4%

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

      4. distribute-neg-frac 78.4%

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

   4. Simplified 78.4%

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

      1. associate-*r/ 71.5%

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

      2. add-cube-cbrt 70.9%

         \[\leadsto \left|\frac{x \cdot \left(-z\right)}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right| \]

      3. times-frac 79.3%

         \[\leadsto \left|\color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{-z}{\sqrt[3]{y}}}\right| \]

      4. add-sqr-sqrt 79.3%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}{\sqrt[3]{y}}\right| \]

      5. sqrt-unprod 57.5%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}{\sqrt[3]{y}}\right| \]

      6. sqr-neg 57.5%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt{\color{blue}{z \cdot z}}}{\sqrt[3]{y}}\right| \]

      7. sqrt-unprod 0.0%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt[3]{y}}\right| \]

      8. add-sqr-sqrt 79.3%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{z}}{\sqrt[3]{y}}\right| \]

      9. times-frac 70.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right| \]

      10. *-commutative 70.9%

          \[\leadsto \left|\frac{\color{blue}{z \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right| \]

      11. add-cube-cbrt 71.5%

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

      12. associate-/l* 79.9%

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

   6. Applied egg-rr 79.9%

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

   if -3.2e5 < z < 3.05000000000000029e76

   1. Initial program 95.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. associate-*r/ 98.3%

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

      2. distribute-lft-in 98.3%

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

      3. metadata-eval 98.3%

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

      4. neg-mul-1 98.3%

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

      5. sub-neg 98.3%

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

   6. Simplified 98.3%

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

   if 3.05000000000000029e76 < z

   1. Initial program 76.4%

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

   2. Taylor expanded in z around inf 78.8%

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

      1. mul-1-neg 78.8%

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

      2. associate-*r/ 85.5%

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

      3. distribute-rgt-neg-out 85.5%

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

      4. distribute-neg-frac 85.5%

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

   4. Simplified 85.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -320000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+76}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 8: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0045:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+77}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.0045)
   (fabs (* (- 1.0 z) (/ x y)))
   (if (<= z 3.15e+77) (fabs (/ (- x -4.0) y)) (fabs (* x (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.0045) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else if (z <= 3.15e+77) {
		tmp = fabs(((x - -4.0) / y));
	} else {
		tmp = fabs((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 (z <= (-0.0045d0)) then
        tmp = abs(((1.0d0 - z) * (x / y)))
    else if (z <= 3.15d+77) then
        tmp = abs(((x - (-4.0d0)) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.0045) {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	} else if (z <= 3.15e+77) {
		tmp = Math.abs(((x - -4.0) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.0045:
		tmp = math.fabs(((1.0 - z) * (x / y)))
	elif z <= 3.15e+77:
		tmp = math.fabs(((x - -4.0) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.0045)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	elseif (z <= 3.15e+77)
		tmp = abs(Float64(Float64(x - -4.0) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.0045)
		tmp = abs(((1.0 - z) * (x / y)));
	elseif (z <= 3.15e+77)
		tmp = abs(((x - -4.0) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.0045], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.15e+77], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.00449999999999999966

   1. Initial program 94.6%

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

      1. associate-*l/ 86.2%

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

      2. sub-div 86.1%

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

   3. Applied egg-rr 86.1%

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

   4. Taylor expanded in x around inf 73.3%

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

      1. associate-/l* 80.2%

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

      2. associate-/r/ 81.7%

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

   6. Applied egg-rr 81.7%

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

   if -0.00449999999999999966 < z < 3.14999999999999981e77

   1. Initial program 95.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. associate-*r/ 98.3%

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

      2. distribute-lft-in 98.3%

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

      3. metadata-eval 98.3%

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

      4. neg-mul-1 98.3%

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

      5. sub-neg 98.3%

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

   6. Simplified 98.3%

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

   if 3.14999999999999981e77 < z

   1. Initial program 76.4%

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

   2. Taylor expanded in z around inf 78.8%

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

      1. mul-1-neg 78.8%

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

      2. associate-*r/ 85.5%

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

      3. distribute-rgt-neg-out 85.5%

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

      4. distribute-neg-frac 85.5%

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

   4. Simplified 85.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0045:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+77}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 9: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(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 <= (-10.5d0)) .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 <= -10.5) || !(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 <= -10.5) 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 <= -10.5) || !(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 <= -10.5) || ~((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, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 89.4%

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

      1. associate-*l/ 83.3%

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

      2. sub-div 90.8%

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

   3. Applied egg-rr 90.8%

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

   4. Taylor expanded in x around inf 89.5%

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

   5. Taylor expanded in z around 0 71.3%

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

   if -10.5 < x < 4

   1. Initial program 94.9%

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

   2. Taylor expanded in x around 0 72.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 10: 40.0% 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 92.0%

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

2. Taylor expanded in x around 0 37.6%

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

3. Final simplification 37.6%

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

Reproduce

herbie shell --seed 2023320 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))