fabs fraction 1

Percentage Accurate: 91.9% → 99.9%

Time: 8.3s

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: 91.9% 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

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-19)
   (fabs (/ (+ 4.0 (* x (- 1.0 z))) y_m))
   (fabs (- (/ (+ 4.0 x) 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-19) {
		tmp = fabs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = fabs((((4.0 + x) / 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-19) then
        tmp = abs(((4.0d0 + (x * (1.0d0 - z))) / y_m))
    else
        tmp = abs((((4.0d0 + x) / 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-19) {
		tmp = Math.abs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = Math.abs((((4.0 + x) / 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-19:
		tmp = math.fabs(((4.0 + (x * (1.0 - z))) / y_m))
	else:
		tmp = math.fabs((((4.0 + x) / 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-19)
		tmp = abs(Float64(Float64(4.0 + Float64(x * Float64(1.0 - z))) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / 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-19)
		tmp = abs(((4.0 + (x * (1.0 - z))) / y_m));
	else
		tmp = abs((((4.0 + x) / 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-19], N[Abs[N[(N[(4.0 + N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $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 < 2e-19

   1. Initial program 90.0%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in y around 0 96.9%

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

      1. associate-*r/ 96.9%

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

      2. associate--r+ 96.9%

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

      3. sub-neg 96.9%

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

      4. metadata-eval 96.9%

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

      5. associate-+r- 96.9%

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

      6. distribute-lft-in 96.9%

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

      7. neg-mul-1 96.9%

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

      8. neg-sub0 96.9%

         \[\leadsto \left|\frac{-1 \cdot \left(x \cdot z\right) + \color{blue}{\left(0 - \left(-4 - x\right)\right)}}{y}\right| \]

      9. associate--r- 96.9%

         \[\leadsto \left|\frac{-1 \cdot \left(x \cdot z\right) + \color{blue}{\left(\left(0 - -4\right) + x\right)}}{y}\right| \]

      10. metadata-eval 96.9%

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

      11. +-commutative 96.9%

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

      12. associate-+l+ 96.9%

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

      13. +-commutative 96.9%

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

      14. +-commutative 96.9%

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

      15. +-commutative 96.9%

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

      16. neg-mul-1 96.9%

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

      17. distribute-rgt-neg-in 96.9%

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

      18. *-rgt-identity 96.9%

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

      19. distribute-lft-in 96.9%

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

      20. +-commutative 96.9%

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

      21. unsub-neg 96.9%

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

   7. Simplified 96.9%

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

   if 2e-19 < 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/ 90.3%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.7%

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

      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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+67}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-52} \lor \neg \left(x \leq 0.02\right):\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.8e+67)
   (fabs (/ x y_m))
   (if (or (<= x -2.3e-52) (not (<= x 0.02)))
     (fabs (/ z (/ y_m x)))
     (fabs (/ 4.0 y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.8e+67) {
		tmp = fabs((x / y_m));
	} else if ((x <= -2.3e-52) || !(x <= 0.02)) {
		tmp = fabs((z / (y_m / x)));
	} 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 <= (-5.8d+67)) then
        tmp = abs((x / y_m))
    else if ((x <= (-2.3d-52)) .or. (.not. (x <= 0.02d0))) then
        tmp = abs((z / (y_m / x)))
    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 <= -5.8e+67) {
		tmp = Math.abs((x / y_m));
	} else if ((x <= -2.3e-52) || !(x <= 0.02)) {
		tmp = Math.abs((z / (y_m / x)));
	} 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 <= -5.8e+67:
		tmp = math.fabs((x / y_m))
	elif (x <= -2.3e-52) or not (x <= 0.02):
		tmp = math.fabs((z / (y_m / x)))
	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 <= -5.8e+67)
		tmp = abs(Float64(x / y_m));
	elseif ((x <= -2.3e-52) || !(x <= 0.02))
		tmp = abs(Float64(z / Float64(y_m / x)));
	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 <= -5.8e+67)
		tmp = abs((x / y_m));
	elseif ((x <= -2.3e-52) || ~((x <= 0.02)))
		tmp = abs((z / (y_m / x)));
	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[LessEqual[x, -5.8e+67], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, -2.3e-52], N[Not[LessEqual[x, 0.02]], $MachinePrecision]], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.80000000000000047e67

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0 71.3%

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

   4. Taylor expanded in x around inf 71.3%

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

   if -5.80000000000000047e67 < x < -2.29999999999999994e-52 or 0.0200000000000000004 < x

   1. Initial program 88.5%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in z around inf 52.0%

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

      1. associate-*r/ 52.0%

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

      2. neg-mul-1 52.0%

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

      3. distribute-lft-neg-in 52.0%

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

      4. *-commutative 52.0%

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

      5. associate-*r/ 73.9%

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

   7. Simplified 73.9%

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

      1. clear-num 73.9%

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

      2. un-div-inv 74.1%

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

      3. add-sqr-sqrt 19.4%

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

      4. sqrt-unprod 64.5%

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

      5. sqr-neg 64.5%

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

      6. sqrt-unprod 54.5%

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

      7. add-sqr-sqrt 74.1%

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

   9. Applied egg-rr 74.1%

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

   if -2.29999999999999994e-52 < x < 0.0200000000000000004

   1. Initial program 93.6%

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

   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 78.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+67}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-52} \lor \neg \left(x \leq 0.02\right):\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+67}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-51} \lor \neg \left(x \leq 0.02\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.1e+67)
   (fabs (/ x y_m))
   (if (or (<= x -3.1e-51) (not (<= x 0.02)))
     (fabs (* x (/ z 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 <= -5.1e+67) {
		tmp = fabs((x / y_m));
	} else if ((x <= -3.1e-51) || !(x <= 0.02)) {
		tmp = fabs((x * (z / 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 <= (-5.1d+67)) then
        tmp = abs((x / y_m))
    else if ((x <= (-3.1d-51)) .or. (.not. (x <= 0.02d0))) then
        tmp = abs((x * (z / 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 <= -5.1e+67) {
		tmp = Math.abs((x / y_m));
	} else if ((x <= -3.1e-51) || !(x <= 0.02)) {
		tmp = Math.abs((x * (z / 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 <= -5.1e+67:
		tmp = math.fabs((x / y_m))
	elif (x <= -3.1e-51) or not (x <= 0.02):
		tmp = math.fabs((x * (z / 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 <= -5.1e+67)
		tmp = abs(Float64(x / y_m));
	elseif ((x <= -3.1e-51) || !(x <= 0.02))
		tmp = abs(Float64(x * Float64(z / 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 <= -5.1e+67)
		tmp = abs((x / y_m));
	elseif ((x <= -3.1e-51) || ~((x <= 0.02)))
		tmp = abs((x * (z / 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[LessEqual[x, -5.1e+67], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, -3.1e-51], N[Not[LessEqual[x, 0.02]], $MachinePrecision]], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1000000000000002e67

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0 71.3%

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

   4. Taylor expanded in x around inf 71.3%

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

   if -5.1000000000000002e67 < x < -3.0999999999999997e-51 or 0.0200000000000000004 < x

   1. Initial program 88.5%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in z around inf 52.0%

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

      1. associate-*r/ 52.0%

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

      2. neg-mul-1 52.0%

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

      3. distribute-lft-neg-in 52.0%

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

      4. *-commutative 52.0%

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

      5. associate-*r/ 73.9%

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

   7. Simplified 73.9%

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

      1. associate-*r/ 52.0%

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

      2. add-cube-cbrt 51.6%

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

      3. times-frac 64.1%

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

      4. add-sqr-sqrt 19.1%

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

      5. sqrt-unprod 59.3%

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

      6. sqr-neg 59.3%

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

      7. sqrt-unprod 45.0%

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

      8. add-sqr-sqrt 64.1%

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

      9. times-frac 51.6%

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

      10. *-commutative 51.6%

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

      11. add-cube-cbrt 52.0%

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

      12. clear-num 52.0%

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

   9. Applied egg-rr 52.0%

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

       1. associate-/r/ 52.0%

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

       2. *-commutative 52.0%

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

       3. associate-*l* 62.3%

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

       4. *-commutative 62.3%

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

   11. Simplified 62.3%

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

   12. Taylor expanded in y around 0 62.4%

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

   if -3.0999999999999997e-51 < x < 0.0200000000000000004

   1. Initial program 93.6%

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

   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 78.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+67}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-51} \lor \neg \left(x \leq 0.02\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+67}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.8e+67)
   (fabs (/ x y_m))
   (if (<= x -1.52e-52)
     (fabs (/ x (/ y_m z)))
     (if (<= x 0.02) (fabs (/ 4.0 y_m)) (fabs (* z (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.8e+67) {
		tmp = fabs((x / y_m));
	} else if (x <= -1.52e-52) {
		tmp = fabs((x / (y_m / z)));
	} else if (x <= 0.02) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((z * (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 <= (-4.8d+67)) then
        tmp = abs((x / y_m))
    else if (x <= (-1.52d-52)) then
        tmp = abs((x / (y_m / z)))
    else if (x <= 0.02d0) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((z * (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 <= -4.8e+67) {
		tmp = Math.abs((x / y_m));
	} else if (x <= -1.52e-52) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (x <= 0.02) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.8e+67:
		tmp = math.fabs((x / y_m))
	elif x <= -1.52e-52:
		tmp = math.fabs((x / (y_m / z)))
	elif x <= 0.02:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.8e+67)
		tmp = abs(Float64(x / y_m));
	elseif (x <= -1.52e-52)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (x <= 0.02)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(z * Float64(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 <= -4.8e+67)
		tmp = abs((x / y_m));
	elseif (x <= -1.52e-52)
		tmp = abs((x / (y_m / z)));
	elseif (x <= 0.02)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -4.8e+67], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.52e-52], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.02], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.80000000000000004e67

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0 71.3%

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

   4. Taylor expanded in x around inf 71.3%

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

   if -4.80000000000000004e67 < x < -1.5199999999999999e-52

   1. Initial program 99.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in z around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. neg-mul-1 67.2%

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

      3. distribute-lft-neg-in 67.2%

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

      4. *-commutative 67.2%

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

      5. associate-*r/ 67.2%

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

   7. Simplified 67.2%

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

      1. associate-*r/ 67.2%

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

      2. add-cube-cbrt 66.5%

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

      3. times-frac 66.4%

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

      4. add-sqr-sqrt 66.5%

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

      5. sqrt-unprod 66.4%

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

      6. sqr-neg 66.4%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 66.4%

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

      9. times-frac 66.5%

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

      10. *-commutative 66.5%

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

      11. add-cube-cbrt 67.2%

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

      12. clear-num 67.2%

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

   9. Applied egg-rr 67.2%

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

       1. associate-/r/ 67.2%

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

       2. *-commutative 67.2%

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

       3. associate-*l* 67.2%

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

       4. *-commutative 67.2%

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

   11. Simplified 67.2%

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

   12. Taylor expanded in y around 0 67.3%

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

       1. clear-num 67.3%

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

       2. div-inv 67.4%

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

   14. Applied egg-rr 67.4%

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

   if -1.5199999999999999e-52 < x < 0.0200000000000000004

   1. Initial program 93.6%

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

   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 78.1%

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

   if 0.0200000000000000004 < x

   1. Initial program 84.0%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in z around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. neg-mul-1 45.8%

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

      3. distribute-lft-neg-in 45.8%

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

      4. *-commutative 45.8%

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

      5. associate-*r/ 76.7%

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

   7. Simplified 76.7%

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

      1. associate-*r/ 45.8%

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

      2. add-cube-cbrt 45.6%

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

      3. times-frac 63.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 56.4%

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

      6. sqr-neg 56.4%

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

      7. sqrt-unprod 63.2%

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

      8. add-sqr-sqrt 63.1%

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

      9. times-frac 45.6%

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

      10. *-commutative 45.6%

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

      11. add-cube-cbrt 45.8%

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

      12. clear-num 45.9%

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

   9. Applied egg-rr 45.9%

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

       1. associate-/r/ 45.8%

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

       2. associate-*l/ 45.8%

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

       3. *-lft-identity 45.8%

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

       4. *-commutative 45.8%

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

       5. associate-*r/ 76.7%

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

   11. Simplified 76.7%

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

Alternative 5: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+67}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-52}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.4e+67)
   (fabs (/ x y_m))
   (if (<= x -1.85e-52)
     (fabs (* x (/ z y_m)))
     (if (<= x 0.02) (fabs (/ 4.0 y_m)) (fabs (* z (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.4e+67) {
		tmp = fabs((x / y_m));
	} else if (x <= -1.85e-52) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 0.02) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((z * (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 <= (-5.4d+67)) then
        tmp = abs((x / y_m))
    else if (x <= (-1.85d-52)) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 0.02d0) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((z * (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 <= -5.4e+67) {
		tmp = Math.abs((x / y_m));
	} else if (x <= -1.85e-52) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 0.02) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -5.4e+67:
		tmp = math.fabs((x / y_m))
	elif x <= -1.85e-52:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 0.02:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5.4e+67)
		tmp = abs(Float64(x / y_m));
	elseif (x <= -1.85e-52)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 0.02)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(z * Float64(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 <= -5.4e+67)
		tmp = abs((x / y_m));
	elseif (x <= -1.85e-52)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 0.02)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -5.4e+67], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.85e-52], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.02], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.3999999999999998e67

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0 71.3%

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

   4. Taylor expanded in x around inf 71.3%

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

   if -5.3999999999999998e67 < x < -1.8499999999999999e-52

   1. Initial program 99.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in z around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. neg-mul-1 67.2%

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

      3. distribute-lft-neg-in 67.2%

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

      4. *-commutative 67.2%

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

      5. associate-*r/ 67.2%

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

   7. Simplified 67.2%

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

      1. associate-*r/ 67.2%

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

      2. add-cube-cbrt 66.5%

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

      3. times-frac 66.4%

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

      4. add-sqr-sqrt 66.5%

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

      5. sqrt-unprod 66.4%

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

      6. sqr-neg 66.4%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 66.4%

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

      9. times-frac 66.5%

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

      10. *-commutative 66.5%

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

      11. add-cube-cbrt 67.2%

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

      12. clear-num 67.2%

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

   9. Applied egg-rr 67.2%

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

       1. associate-/r/ 67.2%

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

       2. *-commutative 67.2%

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

       3. associate-*l* 67.2%

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

       4. *-commutative 67.2%

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

   11. Simplified 67.2%

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

   12. Taylor expanded in y around 0 67.3%

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

   if -1.8499999999999999e-52 < x < 0.0200000000000000004

   1. Initial program 93.6%

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

   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 78.1%

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

   if 0.0200000000000000004 < x

   1. Initial program 84.0%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in z around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. neg-mul-1 45.8%

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

      3. distribute-lft-neg-in 45.8%

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

      4. *-commutative 45.8%

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

      5. associate-*r/ 76.7%

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

   7. Simplified 76.7%

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

      1. associate-*r/ 45.8%

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

      2. add-cube-cbrt 45.6%

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

      3. times-frac 63.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 56.4%

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

      6. sqr-neg 56.4%

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

      7. sqrt-unprod 63.2%

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

      8. add-sqr-sqrt 63.1%

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

      9. times-frac 45.6%

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

      10. *-commutative 45.6%

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

      11. add-cube-cbrt 45.8%

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

      12. clear-num 45.9%

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

   9. Applied egg-rr 45.9%

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

       1. associate-/r/ 45.8%

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

       2. associate-*l/ 45.8%

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

       3. *-lft-identity 45.8%

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

       4. *-commutative 45.8%

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

       5. associate-*r/ 76.7%

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

   11. Simplified 76.7%

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

Alternative 6: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot \left(\frac{4}{z} - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 3.4)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (* z (- (/ 4.0 z) x)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 3.4)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs(((z * ((4.0 / z) - 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 <= (-10.5d0)) .or. (.not. (x <= 3.4d0))) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs(((z * ((4.0d0 / z) - 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 <= -10.5) || !(x <= 3.4)) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs(((z * ((4.0 / z) - x)) / 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 <= 3.4):
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = math.fabs(((z * ((4.0 / z) - x)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 3.4))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(z * Float64(Float64(4.0 / z) - 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 <= -10.5) || ~((x <= 3.4)))
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = abs(((z * ((4.0 / z) - 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, -10.5], N[Not[LessEqual[x, 3.4]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * N[(N[(4.0 / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 3.39999999999999991 < x

   1. Initial program 87.5%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. *-commutative 89.3%

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

      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%

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

   if -10.5 < x < 3.39999999999999991

   1. Initial program 93.9%

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

   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 y around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate--r+ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+r- 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. neg-mul-1 99.9%

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

      8. neg-sub0 99.9%

         \[\leadsto \left|\frac{-1 \cdot \left(x \cdot z\right) + \color{blue}{\left(0 - \left(-4 - x\right)\right)}}{y}\right| \]

      9. associate--r- 99.9%

         \[\leadsto \left|\frac{-1 \cdot \left(x \cdot z\right) + \color{blue}{\left(\left(0 - -4\right) + x\right)}}{y}\right| \]

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

      15. +-commutative 99.9%

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

      16. neg-mul-1 99.9%

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

      17. distribute-rgt-neg-in 99.9%

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

      18. *-rgt-identity 99.9%

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

      19. distribute-lft-in 99.9%

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

      20. +-commutative 99.9%

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

      21. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r/ 99.8%

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

      5. metadata-eval 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 98.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot \left(\frac{4}{z} - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+100}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 10^{+23}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.9e+100)
   (fabs (/ x (/ y_m z)))
   (if (<= z 1e+23) (fabs (/ (- -4.0 x) y_m)) (fabs (/ z (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.9e+100) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 1e+23) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	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 <= (-1.9d+100)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 1d+23) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z / (y_m / x)))
    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 <= -1.9e+100) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 1e+23) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.9e+100:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 1e+23:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.9e+100)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 1e+23)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.9e+100)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 1e+23)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.9e+100], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1e+23], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.89999999999999982e100

   1. Initial program 88.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in z around inf 78.3%

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

      1. associate-*r/ 78.3%

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

      2. neg-mul-1 78.3%

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

      3. distribute-lft-neg-in 78.3%

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

      4. *-commutative 78.3%

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

      5. associate-*r/ 80.9%

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

   7. Simplified 80.9%

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

      1. associate-*r/ 78.3%

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

      2. add-cube-cbrt 77.6%

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

      3. times-frac 84.6%

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

      4. add-sqr-sqrt 49.3%

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

      5. sqrt-unprod 69.5%

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

      6. sqr-neg 69.5%

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

      7. sqrt-unprod 35.4%

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

      8. add-sqr-sqrt 84.6%

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

      9. times-frac 77.6%

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

      10. *-commutative 77.6%

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

      11. add-cube-cbrt 78.3%

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

      12. clear-num 78.3%

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

   9. Applied egg-rr 78.3%

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

       1. associate-/r/ 78.2%

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

       2. *-commutative 78.2%

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

       3. associate-*l* 86.3%

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

       4. *-commutative 86.3%

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

   11. Simplified 86.3%

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

   12. Taylor expanded in y around 0 86.5%

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

       1. clear-num 86.4%

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

       2. div-inv 86.5%

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

   14. Applied egg-rr 86.5%

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

   if -1.89999999999999982e100 < z < 9.9999999999999992e22

   1. Initial program 92.4%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in z around 0 93.8%

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

      1. +-commutative 93.8%

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

      2. rem-square-sqrt 50.7%

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

      3. fabs-sqr 50.7%

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

      4. rem-square-sqrt 93.8%

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

      5. fabs-neg 93.8%

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

      6. distribute-neg-frac 93.8%

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

      7. distribute-neg-in 93.8%

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

      8. metadata-eval 93.8%

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

      9. +-commutative 93.8%

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

      10. sub-neg 93.8%

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

      11. rem-square-sqrt 42.4%

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

      12. fabs-sqr 42.4%

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

      13. rem-square-sqrt 93.8%

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

   7. Simplified 93.8%

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

   if 9.9999999999999992e22 < z

   1. Initial program 88.3%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in z around inf 70.9%

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

      1. associate-*r/ 70.9%

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

      2. neg-mul-1 70.9%

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

      3. distribute-lft-neg-in 70.9%

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

      4. *-commutative 70.9%

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

      5. associate-*r/ 78.5%

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

   7. Simplified 78.5%

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

      1. clear-num 78.5%

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

      2. un-div-inv 78.6%

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

      3. add-sqr-sqrt 37.9%

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

      4. sqrt-unprod 63.2%

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

      5. sqr-neg 63.2%

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

      6. sqrt-unprod 40.3%

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

      7. add-sqr-sqrt 78.6%

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

   9. Applied egg-rr 78.6%

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

Alternative 8: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 2e+16)
   (fabs (/ (+ 4.0 (* x (- 1.0 z))) y_m))
   (fabs (* (- 1.0 z) (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 2e+16) {
		tmp = fabs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (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 <= 2d+16) then
        tmp = abs(((4.0d0 + (x * (1.0d0 - z))) / y_m))
    else
        tmp = abs(((1.0d0 - z) * (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 <= 2e+16) {
		tmp = Math.abs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 2e+16:
		tmp = math.fabs(((4.0 + (x * (1.0 - z))) / y_m))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 2e+16)
		tmp = abs(Float64(Float64(4.0 + Float64(x * Float64(1.0 - z))) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(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 <= 2e+16)
		tmp = abs(((4.0 + (x * (1.0 - z))) / y_m));
	else
		tmp = abs(((1.0 - z) * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 2e+16], N[Abs[N[(N[(4.0 + N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e16

   1. Initial program 93.0%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. associate--r+ 98.0%

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

      3. sub-neg 98.0%

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

      4. metadata-eval 98.0%

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

      5. associate-+r- 98.0%

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

      6. distribute-lft-in 98.0%

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

      7. neg-mul-1 98.0%

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

      8. neg-sub0 98.0%

         \[\leadsto \left|\frac{-1 \cdot \left(x \cdot z\right) + \color{blue}{\left(0 - \left(-4 - x\right)\right)}}{y}\right| \]

      9. associate--r- 98.0%

         \[\leadsto \left|\frac{-1 \cdot \left(x \cdot z\right) + \color{blue}{\left(\left(0 - -4\right) + x\right)}}{y}\right| \]

      10. metadata-eval 98.0%

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

      11. +-commutative 98.0%

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

      12. associate-+l+ 98.0%

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

      13. +-commutative 98.0%

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

      14. +-commutative 98.0%

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

      15. +-commutative 98.0%

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

      16. neg-mul-1 98.0%

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

      17. distribute-rgt-neg-in 98.0%

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

      18. *-rgt-identity 98.0%

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

      19. distribute-lft-in 98.0%

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

      20. +-commutative 98.0%

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

      21. unsub-neg 98.0%

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

   7. Simplified 98.0%

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

   if 2e16 < x

   1. Initial program 82.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in x around inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. *-commutative 83.7%

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

      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%

      \[\leadsto \left|\color{blue}{\left(1 - z\right) \cdot \frac{x}{y}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.5) (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 <= -1.5) || !(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 <= (-1.5d0)) .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 <= -1.5) || !(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 <= -1.5) 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 <= -1.5) || !(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 <= -1.5) || ~((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, -1.5], 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 < -1.5 or 4 < x

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0 59.6%

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

   4. Taylor expanded in x around inf 59.1%

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

   if -1.5 < x < 4

   1. Initial program 93.9%

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

   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 75.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.4% 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 90.9%

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

3. Simplified 95.0%

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

5. Taylor expanded in x around 0 41.9%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))