fabs fraction 1

Percentage Accurate: 91.6% → 99.7%

Time: 29.7s

Alternatives: 10

Speedup: 1.0×

• 20.5% 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: 91.6% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+78} \lor \neg \left(x \leq 3.6 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.2e+78) (not (<= x 3.6e+16)))
   (fabs (* (- 1.0 z) (/ x y)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.2e+78) || !(x <= 3.6e+16)) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} 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 <= (-6.2d+78)) .or. (.not. (x <= 3.6d+16))) then
        tmp = abs(((1.0d0 - z) * (x / y)))
    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 <= -6.2e+78) || !(x <= 3.6e+16)) {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.2e+78) or not (x <= 3.6e+16):
		tmp = math.fabs(((1.0 - z) * (x / y)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.2e+78) || !(x <= 3.6e+16))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	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 <= -6.2e+78) || ~((x <= 3.6e+16)))
		tmp = abs(((1.0 - z) * (x / y)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.2e+78], N[Not[LessEqual[x, 3.6e+16]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $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 < -6.2e78 or 3.6e16 < x

   1. Initial program 86.4%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in x around inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

      3. associate-/l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. neg-sub0 99.9%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.9%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -6.2e78 < x < 3.6e16

   1. Initial program 94.5%

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

      1. associate-*l/ 99.2%

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

      2. sub-div 99.9%

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

   4. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+78} \lor \neg \left(x \leq 3.6 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+46} \lor \neg \left(z \leq -1.3 \cdot 10^{+18}\right) \land z \leq 4.6 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5e+66)
   (fabs (/ x (/ y z)))
   (if (or (<= z -8e+46) (and (not (<= z -1.3e+18)) (<= z 4.6e+26)))
     (fabs (/ (- -4.0 x) y))
     (fabs (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e+66) {
		tmp = fabs((x / (y / z)));
	} else if ((z <= -8e+46) || (!(z <= -1.3e+18) && (z <= 4.6e+26))) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z * (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.5d+66)) then
        tmp = abs((x / (y / z)))
    else if ((z <= (-8d+46)) .or. (.not. (z <= (-1.3d+18))) .and. (z <= 4.6d+26)) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e+66) {
		tmp = Math.abs((x / (y / z)));
	} else if ((z <= -8e+46) || (!(z <= -1.3e+18) && (z <= 4.6e+26))) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.5e+66:
		tmp = math.fabs((x / (y / z)))
	elif (z <= -8e+46) or (not (z <= -1.3e+18) and (z <= 4.6e+26)):
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5e+66)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif ((z <= -8e+46) || (!(z <= -1.3e+18) && (z <= 4.6e+26)))
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.5e+66)
		tmp = abs((x / (y / z)));
	elseif ((z <= -8e+46) || (~((z <= -1.3e+18)) && (z <= 4.6e+26)))
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5e+66], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[z, -8e+46], And[N[Not[LessEqual[z, -1.3e+18]], $MachinePrecision], LessEqual[z, 4.6e+26]]], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5e66

   1. Initial program 91.5%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in z around inf 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

      3. distribute-lft-neg-in 82.5%

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

      4. *-commutative 82.5%

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

      5. associate-*r/ 83.8%

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

   7. Simplified 83.8%

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

      1. *-commutative 83.8%

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

      2. distribute-frac-neg 83.8%

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

      3. distribute-lft-neg-in 83.8%

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

      4. associate-/r/ 86.6%

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

      5. distribute-neg-frac 86.6%

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

      6. add-sqr-sqrt 47.2%

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

      7. sqrt-unprod 74.5%

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

      8. sqr-neg 74.5%

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

      9. sqrt-unprod 39.2%

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

      10. add-sqr-sqrt 86.6%

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

   9. Applied egg-rr 86.6%

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

   if -5.5e66 < z < -7.9999999999999999e46 or -1.3e18 < z < 4.6000000000000001e26

   1. Initial program 94.4%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in z around 0 96.3%

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

      1. +-commutative 96.3%

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

      2. rem-square-sqrt 50.8%

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

      3. fabs-sqr 50.8%

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

      4. rem-square-sqrt 96.3%

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

      5. fabs-neg 96.3%

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

      6. distribute-neg-frac 96.3%

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

      7. distribute-neg-in 96.3%

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

      8. metadata-eval 96.3%

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

      9. +-commutative 96.3%

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

      10. sub-neg 96.3%

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

      11. rem-square-sqrt 44.8%

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

      12. fabs-sqr 44.8%

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

      13. rem-square-sqrt 96.3%

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

   7. Simplified 96.3%

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

   if -7.9999999999999999e46 < z < -1.3e18 or 4.6000000000000001e26 < z

   1. Initial program 82.1%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in z around inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. neg-mul-1 69.3%

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

      3. distribute-lft-neg-in 69.3%

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

      4. *-commutative 69.3%

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

      5. associate-*r/ 81.2%

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

   7. Simplified 81.2%

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

      1. *-commutative 81.2%

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

      2. div-inv 81.2%

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

      3. associate-*l* 78.0%

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

      4. add-sqr-sqrt 44.2%

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

      5. sqrt-unprod 61.2%

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

      6. sqr-neg 61.2%

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

      7. sqrt-unprod 33.7%

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

      8. add-sqr-sqrt 78.0%

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

      9. associate-*l* 81.2%

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

      10. div-inv 81.2%

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

      11. clear-num 81.1%

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

      12. associate-*l/ 81.1%

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

      13. *-un-lft-identity 81.1%

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

   9. Applied egg-rr 81.1%

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

       1. clear-num 80.9%

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

       2. associate-/r/ 81.1%

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

       3. clear-num 81.2%

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

   11. Applied egg-rr 81.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+46} \lor \neg \left(z \leq -1.3 \cdot 10^{+18}\right) \land z \leq 4.6 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-34}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (* z (/ x y)))))
   (if (<= x -1.65e+120)
     t_0
     (if (<= x -7.5e-62)
       t_1
       (if (<= x 1.4e-34) (fabs (/ 4.0 y)) (if (<= x 5.4e+28) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((z * (x / y)));
	double tmp;
	if (x <= -1.65e+120) {
		tmp = t_0;
	} else if (x <= -7.5e-62) {
		tmp = t_1;
	} else if (x <= 1.4e-34) {
		tmp = fabs((4.0 / y));
	} else if (x <= 5.4e+28) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((z * (x / y)))
    if (x <= (-1.65d+120)) then
        tmp = t_0
    else if (x <= (-7.5d-62)) then
        tmp = t_1
    else if (x <= 1.4d-34) then
        tmp = abs((4.0d0 / y))
    else if (x <= 5.4d+28) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((z * (x / y)));
	double tmp;
	if (x <= -1.65e+120) {
		tmp = t_0;
	} else if (x <= -7.5e-62) {
		tmp = t_1;
	} else if (x <= 1.4e-34) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 5.4e+28) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((z * (x / y)))
	tmp = 0
	if x <= -1.65e+120:
		tmp = t_0
	elif x <= -7.5e-62:
		tmp = t_1
	elif x <= 1.4e-34:
		tmp = math.fabs((4.0 / y))
	elif x <= 5.4e+28:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(z * Float64(x / y)))
	tmp = 0.0
	if (x <= -1.65e+120)
		tmp = t_0;
	elseif (x <= -7.5e-62)
		tmp = t_1;
	elseif (x <= 1.4e-34)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 5.4e+28)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((z * (x / y)));
	tmp = 0.0;
	if (x <= -1.65e+120)
		tmp = t_0;
	elseif (x <= -7.5e-62)
		tmp = t_1;
	elseif (x <= 1.4e-34)
		tmp = abs((4.0 / y));
	elseif (x <= 5.4e+28)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.65e+120], t$95$0, If[LessEqual[x, -7.5e-62], t$95$1, If[LessEqual[x, 1.4e-34], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.4e+28], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.64999999999999995e120 or 5.4000000000000003e28 < x

   1. Initial program 85.4%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in x around inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. neg-sub0 99.9%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.9%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around 0 72.6%

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

   if -1.64999999999999995e120 < x < -7.5000000000000003e-62 or 1.39999999999999998e-34 < x < 5.4000000000000003e28

   1. Initial program 95.1%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. neg-mul-1 64.4%

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

      3. distribute-lft-neg-in 64.4%

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

      4. *-commutative 64.4%

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

      5. associate-*r/ 65.8%

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

   7. Simplified 65.8%

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

      1. *-commutative 65.8%

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

      2. div-inv 65.8%

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

      3. associate-*l* 66.4%

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

      4. add-sqr-sqrt 49.4%

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

      5. sqrt-unprod 66.4%

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

      6. sqr-neg 66.4%

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

      7. sqrt-unprod 16.9%

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

      8. add-sqr-sqrt 66.4%

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

      9. associate-*l* 65.8%

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

      10. div-inv 65.8%

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

      11. clear-num 64.4%

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

      12. associate-*l/ 64.4%

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

      13. *-un-lft-identity 64.4%

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

   9. Applied egg-rr 64.4%

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

       1. clear-num 64.2%

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

       2. associate-/r/ 64.4%

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

       3. clear-num 65.8%

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

   11. Applied egg-rr 65.8%

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

   if -7.5000000000000003e-62 < x < 1.39999999999999998e-34

   1. Initial program 94.7%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 77.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-62}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-34}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+28}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (* x (/ z y)))))
   (if (<= x -8.5e+114)
     t_0
     (if (<= x -1.4e-61)
       t_1
       (if (<= x 9.5e-33) (fabs (/ 4.0 y)) (if (<= x 1.95e+27) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((x * (z / y)));
	double tmp;
	if (x <= -8.5e+114) {
		tmp = t_0;
	} else if (x <= -1.4e-61) {
		tmp = t_1;
	} else if (x <= 9.5e-33) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.95e+27) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((x * (z / y)))
    if (x <= (-8.5d+114)) then
        tmp = t_0
    else if (x <= (-1.4d-61)) then
        tmp = t_1
    else if (x <= 9.5d-33) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.95d+27) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((x * (z / y)));
	double tmp;
	if (x <= -8.5e+114) {
		tmp = t_0;
	} else if (x <= -1.4e-61) {
		tmp = t_1;
	} else if (x <= 9.5e-33) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.95e+27) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -8.5e+114:
		tmp = t_0
	elif x <= -1.4e-61:
		tmp = t_1
	elif x <= 9.5e-33:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.95e+27:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -8.5e+114)
		tmp = t_0;
	elseif (x <= -1.4e-61)
		tmp = t_1;
	elseif (x <= 9.5e-33)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.95e+27)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -8.5e+114)
		tmp = t_0;
	elseif (x <= -1.4e-61)
		tmp = t_1;
	elseif (x <= 9.5e-33)
		tmp = abs((4.0 / y));
	elseif (x <= 1.95e+27)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8.5e+114], t$95$0, If[LessEqual[x, -1.4e-61], t$95$1, If[LessEqual[x, 9.5e-33], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.95e+27], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000001e114 or 1.9499999999999999e27 < x

   1. Initial program 85.4%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in x around inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. neg-sub0 99.9%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.9%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around 0 72.6%

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

   if -8.5000000000000001e114 < x < -1.4000000000000001e-61 or 9.50000000000000019e-33 < x < 1.9499999999999999e27

   1. Initial program 95.1%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. neg-mul-1 64.4%

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

      3. distribute-lft-neg-in 64.4%

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

      4. *-commutative 64.4%

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

      5. associate-*r/ 65.8%

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

   7. Simplified 65.8%

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

      1. *-commutative 65.8%

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

      2. div-inv 65.8%

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

      3. associate-*l* 66.4%

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

      4. add-sqr-sqrt 49.4%

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

      5. sqrt-unprod 66.4%

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

      6. sqr-neg 66.4%

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

      7. sqrt-unprod 16.9%

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

      8. add-sqr-sqrt 66.4%

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

      9. associate-*l* 65.8%

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

      10. div-inv 65.8%

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

      11. clear-num 64.4%

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

      12. associate-*l/ 64.4%

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

      13. *-un-lft-identity 64.4%

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

   9. Applied egg-rr 64.4%

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

       1. associate-/r/ 66.6%

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

   11. Applied egg-rr 66.6%

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

   if -1.4000000000000001e-61 < x < 9.50000000000000019e-33

   1. Initial program 94.7%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 77.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+114}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-61}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+27}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-61} \lor \neg \left(x \leq 9.5 \cdot 10^{-33}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4e-61) (not (<= x 9.5e-33)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (- -4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4e-61) || !(x <= 9.5e-33)) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = fabs(((-4.0 - x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d-61)) .or. (.not. (x <= 9.5d-33))) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4e-61) || !(x <= 9.5e-33)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4e-61) or not (x <= 9.5e-33):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4e-61) || !(x <= 9.5e-33))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4e-61) || ~((x <= 9.5e-33)))
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4e-61], N[Not[LessEqual[x, 9.5e-33]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4000000000000001e-61 or 9.50000000000000019e-33 < x

   1. Initial program 88.3%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around inf 85.1%

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

      1. mul-1-neg 85.1%

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

      2. associate-/l* 94.0%

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

      3. distribute-rgt-neg-in 94.0%

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

      4. mul-1-neg 94.0%

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

      5. associate-*r/ 94.0%

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

      6. mul-1-neg 94.0%

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

      7. neg-sub0 94.0%

         \[\leadsto \left|x \cdot \frac{\color{blue}{0 - \left(z - 1\right)}}{y}\right| \]

      8. associate-+l- 94.0%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      9. neg-sub0 94.0%

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

      10. +-commutative 94.0%

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

      11. unsub-neg 94.0%

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

   7. Simplified 94.0%

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

   if -1.4000000000000001e-61 < x < 9.50000000000000019e-33

   1. Initial program 94.7%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. rem-square-sqrt 40.1%

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

      3. fabs-sqr 40.1%

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

      4. rem-square-sqrt 77.3%

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

      5. fabs-neg 77.3%

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

      6. distribute-neg-frac 77.3%

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

      7. distribute-neg-in 77.3%

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

      8. metadata-eval 77.3%

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

      9. +-commutative 77.3%

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

      10. sub-neg 77.3%

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

      11. rem-square-sqrt 36.6%

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

      12. fabs-sqr 36.6%

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

      13. rem-square-sqrt 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-61} \lor \neg \left(x \leq 9.5 \cdot 10^{-33}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-61}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e-61)
   (fabs (* x (/ (- 1.0 z) y)))
   (if (<= x 8.2e-33) (fabs (/ (- -4.0 x) y)) (fabs (* (- 1.0 z) (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-61) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else if (x <= 8.2e-33) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs(((1.0 - z) * (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.4d-61)) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else if (x <= 8.2d-33) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs(((1.0d0 - z) * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-61) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else if (x <= 8.2e-33) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e-61:
		tmp = math.fabs((x * ((1.0 - z) / y)))
	elif x <= 8.2e-33:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e-61)
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	elseif (x <= 8.2e-33)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e-61)
		tmp = abs((x * ((1.0 - z) / y)));
	elseif (x <= 8.2e-33)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs(((1.0 - z) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e-61], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 8.2e-33], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4000000000000001e-61

   1. Initial program 90.8%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. associate-/l* 91.9%

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

      3. distribute-rgt-neg-in 91.9%

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

      4. mul-1-neg 91.9%

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

      5. associate-*r/ 91.9%

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

      6. mul-1-neg 91.9%

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

      7. neg-sub0 91.9%

         \[\leadsto \left|x \cdot \frac{\color{blue}{0 - \left(z - 1\right)}}{y}\right| \]

      8. associate-+l- 91.9%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      9. neg-sub0 91.9%

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

      10. +-commutative 91.9%

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

      11. unsub-neg 91.9%

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

   7. Simplified 91.9%

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

   if -1.4000000000000001e-61 < x < 8.2e-33

   1. Initial program 94.7%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. rem-square-sqrt 40.1%

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

      3. fabs-sqr 40.1%

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

      4. rem-square-sqrt 77.3%

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

      5. fabs-neg 77.3%

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

      6. distribute-neg-frac 77.3%

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

      7. distribute-neg-in 77.3%

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

      8. metadata-eval 77.3%

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

      9. +-commutative 77.3%

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

      10. sub-neg 77.3%

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

      11. rem-square-sqrt 36.6%

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

      12. fabs-sqr 36.6%

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

      13. rem-square-sqrt 77.3%

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

   7. Simplified 77.3%

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

   if 8.2e-33 < x

   1. Initial program 85.0%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-/l* 97.1%

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

      4. distribute-lft-neg-in 97.1%

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

      5. neg-sub0 97.1%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 97.1%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 97.1%

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

      8. +-commutative 97.1%

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

      9. unsub-neg 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-61}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+113)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ x (/ y z)) (/ (+ x 4.0) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+113) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x / (y / z)) - ((x + 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 (y <= 2d+113) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x / (y / z)) - ((x + 4.0d0) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+113) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x / (y / z)) - ((x + 4.0) / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+113:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x / (y / z)) - ((x + 4.0) / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+113)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x / Float64(y / z)) - Float64(Float64(x + 4.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+113)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x / (y / z)) - ((x + 4.0) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+113], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e113

   1. Initial program 90.2%

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

      1. associate-*l/ 93.3%

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

      2. sub-div 97.6%

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

   4. Applied egg-rr 97.6%

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

   if 2e113 < y

   1. Initial program 93.2%

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

      1. associate-*l/ 77.7%

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

      2. associate-*r/ 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+110)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+110) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5d+110) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x * (z / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+110) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e+110:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x * (z / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+110)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e+110)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+110], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999978e110

   1. Initial program 90.2%

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

      1. associate-*l/ 93.3%

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

      2. sub-div 97.6%

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

   4. Applied egg-rr 97.6%

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

   if 4.99999999999999978e110 < y

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0 77.7%

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

      1. associate-*r/ 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% 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 87.3%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in x around inf 88.6%

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

      1. mul-1-neg 88.6%

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

      2. *-commutative 88.6%

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

      3. associate-/l* 99.8%

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

      4. distribute-lft-neg-in 99.8%

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

      5. neg-sub0 99.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.8%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.8%

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

      8. +-commutative 99.8%

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

      9. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 65.2%

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

   if -10.5 < x < 4

   1. Initial program 94.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 68.8%

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

4. Final simplification 66.9%

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

Alternative 10: 39.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Simplified 93.9%

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

5. Taylor expanded in x around 0 35.1%

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

6. Final simplification 35.1%

   \[\leadsto \left|\frac{4}{y}\right| \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))