fabs fraction 1

Percentage Accurate: 91.8% → 99.9%

Time: 7.0s

Alternatives: 9

Speedup: 1.0×

• 20.7% 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: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-10)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-10) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2d-10) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-10) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 2e-10:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e-10)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2e-10)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 2e-10], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000007e-10

   1. Initial program 93.1%

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

      1. associate-*l/ 93.7%

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

      2. sub-div 96.9%

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

   4. Applied egg-rr 96.9%

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

   if 2.00000000000000007e-10 < y

   1. Initial program 98.4%

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

      1. associate-*l/ 91.5%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.8%

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

      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. Add Preprocessing

Alternative 2: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-71} \lor \neg \left(x \leq 2.75 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -6.4e-71) (not (<= x 2.75e-21)))
   (fabs (* (/ x y_m) (- 1.0 z)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -6.4e-71) || !(x <= 2.75e-21)) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-6.4d-71)) .or. (.not. (x <= 2.75d-21))) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -6.4e-71) || !(x <= 2.75e-21)) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -6.4e-71) or not (x <= 2.75e-21):
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -6.4e-71) || !(x <= 2.75e-21))
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -6.4e-71) || ~((x <= 2.75e-21)))
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -6.4e-71], N[Not[LessEqual[x, 2.75e-21]], $MachinePrecision]], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.3999999999999998e-71 or 2.74999999999999989e-21 < x

   1. Initial program 92.7%

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

   3. Simplified 92.4%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   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* 95.2%

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

      4. distribute-lft-neg-in 95.2%

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

      5. neg-sub0 95.2%

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

      6. associate-+l- 95.2%

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

      7. neg-sub0 95.2%

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

      8. +-commutative 95.2%

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

      9. unsub-neg 95.2%

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

   7. Simplified 95.2%

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

   if -6.3999999999999998e-71 < x < 2.74999999999999989e-21

   1. Initial program 97.3%

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

   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 83.7%

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

      1. +-commutative 83.7%

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

      2. rem-square-sqrt 36.3%

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

      3. fabs-sqr 36.3%

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

      4. rem-square-sqrt 83.7%

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

      5. fabs-neg 83.7%

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

      6. distribute-neg-frac 83.7%

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

      7. distribute-neg-in 83.7%

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

      8. metadata-eval 83.7%

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

      9. +-commutative 83.7%

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

      10. sub-neg 83.7%

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

      11. rem-square-sqrt 46.8%

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

      12. fabs-sqr 46.8%

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

      13. rem-square-sqrt 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-71} \lor \neg \left(x \leq 2.75 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-71} \lor \neg \left(x \leq 1.28 \cdot 10^{-20}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -6.4e-71) (not (<= x 1.28e-20)))
   (fabs (* x (/ (- 1.0 z) y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -6.4e-71) || !(x <= 1.28e-20)) {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-6.4d-71)) .or. (.not. (x <= 1.28d-20))) then
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -6.4e-71) || !(x <= 1.28e-20)) {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -6.4e-71) or not (x <= 1.28e-20):
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -6.4e-71) || !(x <= 1.28e-20))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -6.4e-71) || ~((x <= 1.28e-20)))
		tmp = abs((x * ((1.0 - z) / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -6.4e-71], N[Not[LessEqual[x, 1.28e-20]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.3999999999999998e-71 or 1.2800000000000001e-20 < x

   1. Initial program 92.7%

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

   3. Simplified 92.4%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   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. *-commutative 87.9%

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

      2. associate-/l* 94.5%

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

      3. associate-*r* 94.5%

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

      4. *-commutative 94.5%

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

      5. associate-*r/ 94.5%

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

      6. mul-1-neg 94.5%

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

      7. neg-sub0 94.5%

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

      8. associate-+l- 94.5%

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

      9. neg-sub0 94.5%

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

      10. +-commutative 94.5%

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

      11. unsub-neg 94.5%

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

   7. Simplified 94.5%

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

   if -6.3999999999999998e-71 < x < 1.2800000000000001e-20

   1. Initial program 97.3%

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

   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 83.7%

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

      1. +-commutative 83.7%

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

      2. rem-square-sqrt 36.3%

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

      3. fabs-sqr 36.3%

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

      4. rem-square-sqrt 83.7%

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

      5. fabs-neg 83.7%

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

      6. distribute-neg-frac 83.7%

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

      7. distribute-neg-in 83.7%

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

      8. metadata-eval 83.7%

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

      9. +-commutative 83.7%

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

      10. sub-neg 83.7%

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

      11. rem-square-sqrt 46.8%

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

      12. fabs-sqr 46.8%

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

      13. rem-square-sqrt 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-71} \lor \neg \left(x \leq 1.28 \cdot 10^{-20}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m} - \frac{x + 4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-21)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (* z (/ x y_m)) (/ (+ x 4.0) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-21) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((z * (x / y_m)) - ((x + 4.0) / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2d-21) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((z * (x / y_m)) - ((x + 4.0d0) / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-21) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((z * (x / y_m)) - ((x + 4.0) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 2e-21:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((z * (x / y_m)) - ((x + 4.0) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e-21)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(z * Float64(x / y_m)) - Float64(Float64(x + 4.0) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2e-21)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((z * (x / y_m)) - ((x + 4.0) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 2e-21], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999982e-21

   1. Initial program 93.1%

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

      1. associate-*l/ 93.7%

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

      2. sub-div 96.9%

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

   4. Applied egg-rr 96.9%

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

   if 1.99999999999999982e-21 < y

   1. Initial program 98.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y} - \frac{x + 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-16} \lor \neg \left(x \leq 3.95 \cdot 10^{-20}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -2.5e-16) (not (<= x 3.95e-20)))
   (fabs (* z (/ x y_m)))
   (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -2.5e-16) || !(x <= 3.95e-20)) {
		tmp = fabs((z * (x / y_m)));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.5d-16)) .or. (.not. (x <= 3.95d-20))) then
        tmp = abs((z * (x / y_m)))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -2.5e-16) || !(x <= 3.95e-20)) {
		tmp = Math.abs((z * (x / y_m)));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -2.5e-16) or not (x <= 3.95e-20):
		tmp = math.fabs((z * (x / y_m)))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -2.5e-16) || !(x <= 3.95e-20))
		tmp = abs(Float64(z * Float64(x / y_m)));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -2.5e-16) || ~((x <= 3.95e-20)))
		tmp = abs((z * (x / y_m)));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -2.5e-16], N[Not[LessEqual[x, 3.95e-20]], $MachinePrecision]], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5000000000000002e-16 or 3.9499999999999999e-20 < x

   1. Initial program 92.0%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in z around inf 48.0%

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

      1. mul-1-neg 48.0%

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

      2. distribute-frac-neg2 48.0%

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

      3. associate-/l* 55.3%

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

   7. Simplified 55.3%

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

      1. associate-*r/ 48.0%

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

      2. add-sqr-sqrt 23.6%

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

      3. sqrt-unprod 46.9%

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

      4. sqr-neg 46.9%

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

      5. sqrt-unprod 24.3%

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

      6. add-sqr-sqrt 48.0%

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

      7. associate-*l/ 65.3%

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

      8. associate-/r/ 55.9%

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

   9. Applied egg-rr 55.9%

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

       1. associate-/r/ 65.3%

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

   11. Applied egg-rr 65.3%

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

   if -2.5000000000000002e-16 < x < 3.9499999999999999e-20

   1. Initial program 97.6%

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

   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 80.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-16} \lor \neg \left(x \leq 3.95 \cdot 10^{-20}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+55}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 900:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -2.3e+55)
   (fabs (* z (/ x y_m)))
   (if (<= z 900.0) (fabs (/ (- -4.0 x) y_m)) (fabs (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -2.3e+55) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 900.0) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x / (y_m / z)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.3d+55)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 900.0d0) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x / (y_m / z)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -2.3e+55) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 900.0) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -2.3e+55:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 900.0:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -2.3e+55)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 900.0)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -2.3e+55)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 900.0)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x / (y_m / z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -2.3e+55], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 900.0], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.29999999999999987e55

   1. Initial program 98.0%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in z around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. distribute-frac-neg2 74.8%

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

      3. associate-/l* 80.9%

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

   7. Simplified 80.9%

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

      1. associate-*r/ 74.8%

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

      2. add-sqr-sqrt 47.6%

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

      3. sqrt-unprod 58.6%

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

      4. sqr-neg 58.6%

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

      5. sqrt-unprod 27.2%

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

      6. add-sqr-sqrt 74.8%

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

      7. associate-*l/ 84.4%

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

      8. associate-/r/ 82.6%

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

   9. Applied egg-rr 82.6%

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

       1. associate-/r/ 84.4%

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

   11. Applied egg-rr 84.4%

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

   if -2.29999999999999987e55 < z < 900

   1. Initial program 95.3%

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

   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 z around 0 94.3%

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

      1. +-commutative 94.3%

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

      2. rem-square-sqrt 41.3%

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

      3. fabs-sqr 41.3%

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

      4. rem-square-sqrt 94.3%

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

      5. fabs-neg 94.3%

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

      6. distribute-neg-frac 94.3%

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

      7. distribute-neg-in 94.3%

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

      8. metadata-eval 94.3%

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

      9. +-commutative 94.3%

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

      10. sub-neg 94.3%

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

      11. rem-square-sqrt 52.6%

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

      12. fabs-sqr 52.6%

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

      13. rem-square-sqrt 94.3%

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

   7. Simplified 94.3%

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

   if 900 < z

   1. Initial program 89.7%

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

   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 z around inf 69.4%

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

      1. mul-1-neg 69.4%

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

      2. distribute-frac-neg2 69.4%

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

      3. associate-/l* 75.8%

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

   7. Simplified 75.8%

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

      1. associate-*r/ 69.4%

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

      2. add-sqr-sqrt 23.5%

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

      3. sqrt-unprod 49.2%

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

      4. sqr-neg 49.2%

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

      5. sqrt-unprod 45.6%

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

      6. add-sqr-sqrt 69.4%

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

      7. associate-*l/ 75.6%

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

      8. associate-/r/ 76.1%

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

   9. Applied egg-rr 76.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+55}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 900:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1e+27)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1e+27) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1d+27)) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1e+27) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1e+27:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1e+27)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1e+27)
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1e+27], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e27

   1. Initial program 95.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in x around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. *-commutative 88.3%

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

      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 -1e27 < x

   1. Initial program 94.3%

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

      1. associate-*l/ 95.3%

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

      2. sub-div 97.9%

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

   4. Applied egg-rr 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.8% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -10.5) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 91.8%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in x around inf 91.0%

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

      1. mul-1-neg 91.0%

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

      2. *-commutative 91.0%

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

      3. associate-/l* 99.3%

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

      4. distribute-lft-neg-in 99.3%

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

      5. neg-sub0 99.3%

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

      6. associate-+l- 99.3%

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

      7. neg-sub0 99.3%

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

      8. +-commutative 99.3%

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

      9. unsub-neg 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in z around 0 57.4%

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

   if -10.5 < x < 4

   1. Initial program 97.7%

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

   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 77.9%

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

4. Final simplification 67.0%

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

Alternative 9: 39.9% accurate, 1.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ 4.0 y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 94.5%

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

3. Simplified 95.4%

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

5. Taylor expanded in x around 0 39.3%

   \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024094 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))