fabs fraction 1

Percentage Accurate: 91.6% → 99.7%

Time: 14.9s

Alternatives: 19

Speedup: 1.0×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-38}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \left(\left(x \cdot z + -4\right) - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1e-38)
   (fabs (* (/ -1.0 y_m) (- (+ (* x z) -4.0) x)))
   (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 <= 1e-38) {
		tmp = fabs(((-1.0 / y_m) * (((x * z) + -4.0) - x)));
	} 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 <= 1d-38) then
        tmp = abs((((-1.0d0) / y_m) * (((x * z) + (-4.0d0)) - x)))
    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 <= 1e-38) {
		tmp = Math.abs(((-1.0 / y_m) * (((x * z) + -4.0) - x)));
	} 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 <= 1e-38:
		tmp = math.fabs(((-1.0 / y_m) * (((x * z) + -4.0) - x)))
	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 <= 1e-38)
		tmp = abs(Float64(Float64(-1.0 / y_m) * Float64(Float64(Float64(x * z) + -4.0) - x)));
	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 <= 1e-38)
		tmp = abs(((-1.0 / y_m) * (((x * z) + -4.0) - x)));
	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, 1e-38], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(N[(N[(x * z), $MachinePrecision] + -4.0), $MachinePrecision] - x), $MachinePrecision]), $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 < 9.9999999999999996e-39

   1. Initial program 92.6%

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

   3. Simplified 97.3%

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

      1. fma-undefine 97.3%

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

      2. associate-+r- 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 9.9999999999999996e-39 < y

   1. Initial program 97.1%

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

      1. associate-*l/ 95.9%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

Alternative 2: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 7.8 \cdot 10^{-57}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m} + \frac{-1}{\frac{y\_m}{x \cdot z}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -4.0) (not (<= x 7.8e-57)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (+ (/ (+ x 4.0) y_m) (/ -1.0 (/ y_m (* x z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -4.0) || !(x <= 7.8e-57)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * 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 ((x <= (-4.0d0)) .or. (.not. (x <= 7.8d-57))) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = ((x + 4.0d0) / y_m) + ((-1.0d0) / (y_m / (x * 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 ((x <= -4.0) || !(x <= 7.8e-57)) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -4.0) or not (x <= 7.8e-57):
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -4.0) || !(x <= 7.8e-57))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = Float64(Float64(Float64(x + 4.0) / y_m) + Float64(-1.0 / Float64(y_m / Float64(x * z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -4.0) || ~((x <= 7.8e-57)))
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -4.0], N[Not[LessEqual[x, 7.8e-57]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] + N[(-1.0 / N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4 or 7.80000000000000013e-57 < x

   1. Initial program 91.2%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in x around inf 91.4%

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

      1. mul-1-neg 91.4%

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

      2. associate-/l* 96.0%

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

      3. distribute-rgt-neg-in 96.0%

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

      4. mul-1-neg 96.0%

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

      5. associate-*r/ 96.0%

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

      6. sub-neg 96.0%

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

      7. metadata-eval 96.0%

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

      8. distribute-lft-in 96.0%

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

      9. neg-mul-1 96.0%

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

      10. metadata-eval 96.0%

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

      11. +-commutative 96.0%

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

      12. neg-mul-1 96.0%

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

      13. associate-/l* 91.4%

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

      14. *-commutative 91.4%

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

      15. associate-/l* 96.8%

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

      16. neg-mul-1 96.8%

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

      17. unsub-neg 96.8%

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

   7. Simplified 96.8%

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

   if -4 < x < 7.80000000000000013e-57

   1. Initial program 96.9%

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

      1. add-sqr-sqrt 49.7%

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

      2. fabs-sqr 49.7%

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

      3. add-sqr-sqrt 51.1%

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

      4. associate-*l/ 52.0%

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

      5. associate-*r/ 49.6%

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

   4. Applied egg-rr 49.6%

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

      1. associate-*r/ 52.0%

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

      2. clear-num 52.0%

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

   6. Applied egg-rr 52.0%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 7.8 \cdot 10^{-57}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{-1}{\frac{y}{x \cdot z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2e+20)
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (* (/ -1.0 y_m) (- (+ (* x z) -4.0) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2e+20) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs(((-1.0 / y_m) * (((x * z) + -4.0) - 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 (x <= (-2d+20)) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs((((-1.0d0) / y_m) * (((x * z) + (-4.0d0)) - 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 (x <= -2e+20) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs(((-1.0 / y_m) * (((x * z) + -4.0) - x)));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2e+20], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(N[(N[(x * z), $MachinePrecision] + -4.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e20

   1. Initial program 93.0%

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

   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 x around inf 91.9%

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

      1. mul-1-neg 91.9%

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

      2. associate-/l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. mul-1-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. distribute-lft-in 99.7%

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

      9. neg-mul-1 99.7%

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

      10. metadata-eval 99.7%

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

      11. +-commutative 99.7%

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

      12. neg-mul-1 99.7%

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

      13. associate-/l* 91.9%

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

      14. *-commutative 91.9%

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

      15. associate-/l* 99.9%

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

      16. neg-mul-1 99.9%

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

      17. 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 -2e20 < x

   1. Initial program 94.1%

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

   3. Simplified 98.4%

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

      1. fma-undefine 98.4%

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

      2. associate-+r- 98.4%

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

   6. Applied egg-rr 98.4%

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

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+23} \lor \neg \left(z \leq 9 \cdot 10^{+62}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.4e+23) (not (<= z 9e+62)))
   (fabs (* z (/ x 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 ((z <= -1.4e+23) || !(z <= 9e+62)) {
		tmp = fabs((z * (x / 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 ((z <= (-1.4d+23)) .or. (.not. (z <= 9d+62))) then
        tmp = abs((z * (x / 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 ((z <= -1.4e+23) || !(z <= 9e+62)) {
		tmp = Math.abs((z * (x / 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 (z <= -1.4e+23) or not (z <= 9e+62):
		tmp = math.fabs((z * (x / 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 ((z <= -1.4e+23) || !(z <= 9e+62))
		tmp = abs(Float64(z * Float64(x / 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 ((z <= -1.4e+23) || ~((z <= 9e+62)))
		tmp = abs((z * (x / 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[z, -1.4e+23], N[Not[LessEqual[z, 9e+62]], $MachinePrecision]], N[Abs[N[(z * N[(x / 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 z < -1.4e23 or 8.99999999999999997e62 < z

   1. Initial program 90.9%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in z around inf 73.0%

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

      1. mul-1-neg 73.0%

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

      2. distribute-frac-neg2 73.0%

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

      3. associate-*l/ 76.6%

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

      4. *-commutative 76.6%

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

   7. Simplified 76.6%

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

   if -1.4e23 < z < 8.99999999999999997e62

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 98.6%

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

      3. associate-*r/ 99.3%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.8%

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

      1. associate-*r/ 95.8%

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

      2. distribute-lft-in 95.8%

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

      3. metadata-eval 95.8%

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

      4. neg-mul-1 95.8%

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

      5. sub-neg 95.8%

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

   7. Simplified 95.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+23} \lor \neg \left(z \leq 9 \cdot 10^{+62}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+22}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+68}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -9.2e+22)
   (fabs (/ (* x z) y_m))
   (if (<= z 2.4e+68) (fabs (/ (- -4.0 x) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -9.2e+22) {
		tmp = fabs(((x * z) / y_m));
	} else if (z <= 2.4e+68) {
		tmp = fabs(((-4.0 - x) / 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 (z <= (-9.2d+22)) then
        tmp = abs(((x * z) / y_m))
    else if (z <= 2.4d+68) then
        tmp = abs((((-4.0d0) - x) / 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 (z <= -9.2e+22) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (z <= 2.4e+68) {
		tmp = Math.abs(((-4.0 - x) / 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 z <= -9.2e+22:
		tmp = math.fabs(((x * z) / y_m))
	elif z <= 2.4e+68:
		tmp = math.fabs(((-4.0 - x) / 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 (z <= -9.2e+22)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (z <= 2.4e+68)
		tmp = abs(Float64(Float64(-4.0 - x) / 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 (z <= -9.2e+22)
		tmp = abs(((x * z) / y_m));
	elseif (z <= 2.4e+68)
		tmp = abs(((-4.0 - x) / 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[z, -9.2e+22], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.4e+68], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.2000000000000008e22

   1. Initial program 96.3%

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

   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 z around inf 74.2%

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

      1. associate-*r/ 74.2%

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

      2. neg-mul-1 74.2%

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

      3. distribute-rgt-neg-in 74.2%

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

   7. Simplified 74.2%

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

   if -9.2000000000000008e22 < z < 2.40000000000000008e68

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 98.6%

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

      3. associate-*r/ 99.3%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.8%

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

      1. associate-*r/ 95.8%

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

      2. distribute-lft-in 95.8%

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

      3. metadata-eval 95.8%

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

      4. neg-mul-1 95.8%

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

      5. sub-neg 95.8%

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

   7. Simplified 95.8%

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

   if 2.40000000000000008e68 < z

   1. Initial program 85.9%

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

   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 z around inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. distribute-frac-neg2 72.0%

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

      3. associate-*l/ 79.9%

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

      4. *-commutative 79.9%

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

   7. Simplified 79.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+22}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+68}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{x + 4}{y\_m} + \frac{-1}{\frac{y\_m}{x \cdot z}}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{4}{y\_m} - x \cdot \frac{z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{y\_m}}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -4.3e+24)
   (+ (/ (+ x 4.0) y_m) (/ -1.0 (/ y_m (* x z))))
   (if (<= z 4.1e+21)
     (fabs (/ (- -4.0 x) y_m))
     (if (<= z 3.5e+182)
       (- (/ 4.0 y_m) (* x (/ z y_m)))
       (/ (/ z y_m) (/ 1.0 x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -4.3e+24) {
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * z)));
	} else if (z <= 4.1e+21) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else if (z <= 3.5e+182) {
		tmp = (4.0 / y_m) - (x * (z / y_m));
	} else {
		tmp = (z / y_m) / (1.0 / 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 <= (-4.3d+24)) then
        tmp = ((x + 4.0d0) / y_m) + ((-1.0d0) / (y_m / (x * z)))
    else if (z <= 4.1d+21) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else if (z <= 3.5d+182) then
        tmp = (4.0d0 / y_m) - (x * (z / y_m))
    else
        tmp = (z / y_m) / (1.0d0 / 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 <= -4.3e+24) {
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * z)));
	} else if (z <= 4.1e+21) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else if (z <= 3.5e+182) {
		tmp = (4.0 / y_m) - (x * (z / y_m));
	} else {
		tmp = (z / y_m) / (1.0 / x);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -4.3e+24:
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * z)))
	elif z <= 4.1e+21:
		tmp = math.fabs(((-4.0 - x) / y_m))
	elif z <= 3.5e+182:
		tmp = (4.0 / y_m) - (x * (z / y_m))
	else:
		tmp = (z / y_m) / (1.0 / x)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -4.3e+24)
		tmp = Float64(Float64(Float64(x + 4.0) / y_m) + Float64(-1.0 / Float64(y_m / Float64(x * z))));
	elseif (z <= 4.1e+21)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	elseif (z <= 3.5e+182)
		tmp = Float64(Float64(4.0 / y_m) - Float64(x * Float64(z / y_m)));
	else
		tmp = Float64(Float64(z / y_m) / Float64(1.0 / x));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -4.3e+24)
		tmp = ((x + 4.0) / y_m) + (-1.0 / (y_m / (x * z)));
	elseif (z <= 4.1e+21)
		tmp = abs(((-4.0 - x) / y_m));
	elseif (z <= 3.5e+182)
		tmp = (4.0 / y_m) - (x * (z / y_m));
	else
		tmp = (z / y_m) / (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -4.3e+24], N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] + N[(-1.0 / N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.1e+21], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.5e+182], N[(N[(4.0 / y$95$m), $MachinePrecision] - N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / y$95$m), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.29999999999999987e24

   1. Initial program 96.2%

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

      1. add-sqr-sqrt 57.4%

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

      2. fabs-sqr 57.4%

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

      3. add-sqr-sqrt 58.0%

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

      4. associate-*l/ 58.4%

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

      5. associate-*r/ 54.5%

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

   4. Applied egg-rr 54.5%

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

      1. associate-*r/ 58.4%

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

      2. clear-num 58.4%

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

   6. Applied egg-rr 58.4%

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

   if -4.29999999999999987e24 < z < 4.1e21

   1. Initial program 95.7%

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

      1. fabs-sub 95.7%

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

      2. associate-*l/ 99.2%

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

      3. associate-*r/ 99.3%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 96.2%

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

      1. associate-*r/ 96.2%

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

      2. distribute-lft-in 96.2%

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

      3. metadata-eval 96.2%

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

      4. neg-mul-1 96.2%

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

      5. sub-neg 96.2%

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

   7. Simplified 96.2%

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

   if 4.1e21 < z < 3.50000000000000023e182

   1. Initial program 94.2%

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

      1. add-sqr-sqrt 30.5%

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

      2. fabs-sqr 30.5%

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

      3. add-sqr-sqrt 31.4%

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

      4. associate-*l/ 31.2%

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

      5. associate-*r/ 31.2%

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

   4. Applied egg-rr 31.2%

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

   5. Taylor expanded in x around 0 34.2%

      \[\leadsto \frac{\color{blue}{4}}{y} - x \cdot \frac{z}{y} \]

   if 3.50000000000000023e182 < z

   1. Initial program 81.2%

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

      1. add-sqr-sqrt 49.9%

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

      2. fabs-sqr 49.9%

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

      3. add-sqr-sqrt 50.2%

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

      4. sub-neg 50.2%

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

      5. +-commutative 50.2%

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

      6. associate-*l/ 38.4%

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

      7. associate-*r/ 50.3%

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

      8. frac-2neg 50.3%

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

      9. +-commutative 50.3%

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

      10. distribute-neg-in 50.3%

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

      11. metadata-eval 50.3%

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

      12. sub-neg 50.3%

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

      13. distribute-frac-neg2 50.3%

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

      14. distribute-neg-in 50.3%

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

      15. fma-undefine 50.3%

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

      16. neg-sub0 50.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 50.3%

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

      18. associate-*r/ 38.4%

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

      19. associate-*l/ 50.2%

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in z around inf 54.2%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 53.9%

         \[\leadsto \color{blue}{\sqrt{0 - \frac{x \cdot z}{y}} \cdot \sqrt{0 - \frac{x \cdot z}{y}}} \]

      2. sqrt-unprod 76.4%

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

      3. associate-*r/ 76.3%

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

      4. neg-sub0 76.3%

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

      5. associate-*r/ 82.2%

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

      6. neg-sub0 82.2%

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

      7. sqr-neg 82.2%

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

      8. sqrt-unprod 28.2%

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

      9. add-sqr-sqrt 28.5%

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

      10. *-commutative 28.5%

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

      11. associate-/r/ 25.7%

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

      12. div-inv 25.6%

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

      13. associate-/r* 28.5%

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

   7. Applied egg-rr 28.5%

      \[\leadsto \color{blue}{\frac{\frac{z}{y}}{\frac{1}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{-1}{\frac{y}{x \cdot z}}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{4}{y} - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{y}}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{z}{\frac{y\_m}{x}}\\ \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ z (/ y_m x))))
   (if (<= x -0.66)
     t_0
     (if (<= x 6.8e-18) (/ 4.0 y_m) (if (<= x 2.95e+32) t_0 (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = z / (y_m / x);
	double tmp;
	if (x <= -0.66) {
		tmp = t_0;
	} else if (x <= 6.8e-18) {
		tmp = 4.0 / y_m;
	} else if (x <= 2.95e+32) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = z / (y_m / x)
    if (x <= (-0.66d0)) then
        tmp = t_0
    else if (x <= 6.8d-18) then
        tmp = 4.0d0 / y_m
    else if (x <= 2.95d+32) then
        tmp = t_0
    else
        tmp = 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 t_0 = z / (y_m / x);
	double tmp;
	if (x <= -0.66) {
		tmp = t_0;
	} else if (x <= 6.8e-18) {
		tmp = 4.0 / y_m;
	} else if (x <= 2.95e+32) {
		tmp = t_0;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = z / (y_m / x)
	tmp = 0
	if x <= -0.66:
		tmp = t_0
	elif x <= 6.8e-18:
		tmp = 4.0 / y_m
	elif x <= 2.95e+32:
		tmp = t_0
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(z / Float64(y_m / x))
	tmp = 0.0
	if (x <= -0.66)
		tmp = t_0;
	elseif (x <= 6.8e-18)
		tmp = Float64(4.0 / y_m);
	elseif (x <= 2.95e+32)
		tmp = t_0;
	else
		tmp = Float64(x / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = z / (y_m / x);
	tmp = 0.0;
	if (x <= -0.66)
		tmp = t_0;
	elseif (x <= 6.8e-18)
		tmp = 4.0 / y_m;
	elseif (x <= 2.95e+32)
		tmp = t_0;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.66], t$95$0, If[LessEqual[x, 6.8e-18], N[(4.0 / y$95$m), $MachinePrecision], If[LessEqual[x, 2.95e+32], t$95$0, N[(x / y$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.660000000000000031 or 6.80000000000000002e-18 < x < 2.94999999999999983e32

   1. Initial program 94.5%

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

      1. add-sqr-sqrt 38.4%

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

      2. fabs-sqr 38.4%

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

      3. add-sqr-sqrt 39.0%

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

      4. sub-neg 39.0%

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

      5. +-commutative 39.0%

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

      6. associate-*l/ 36.5%

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

      7. associate-*r/ 39.0%

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

      8. frac-2neg 39.0%

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

      9. +-commutative 39.0%

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

      10. distribute-neg-in 39.0%

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

      11. metadata-eval 39.0%

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

      12. sub-neg 39.0%

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

      13. distribute-frac-neg2 39.0%

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

      14. distribute-neg-in 39.0%

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

      15. fma-undefine 39.0%

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

      16. neg-sub0 39.0%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 39.0%

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

      18. associate-*r/ 36.5%

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

      19. associate-*l/ 39.0%

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

   4. Applied egg-rr 40.5%

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

   5. Taylor expanded in z around inf 25.3%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 24.8%

         \[\leadsto \color{blue}{\sqrt{0 - \frac{x \cdot z}{y}} \cdot \sqrt{0 - \frac{x \cdot z}{y}}} \]

      2. sqrt-unprod 42.5%

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

      3. associate-*r/ 42.5%

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

      4. neg-sub0 42.5%

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

      5. associate-*r/ 46.4%

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

      6. neg-sub0 46.4%

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

      7. sqr-neg 46.4%

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

      8. sqrt-unprod 28.0%

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

      9. add-sqr-sqrt 28.4%

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

      10. *-commutative 28.4%

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

      11. associate-/r/ 32.1%

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

   7. Applied egg-rr 32.1%

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

   if -0.660000000000000031 < x < 6.80000000000000002e-18

   1. Initial program 97.0%

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

      1. add-sqr-sqrt 47.7%

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

      2. fabs-sqr 47.7%

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

      3. add-sqr-sqrt 49.1%

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

      4. associate-*l/ 50.0%

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

      5. associate-*r/ 47.7%

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

   4. Applied egg-rr 47.7%

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

      1. associate-*r/ 50.0%

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

      2. clear-num 50.0%

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

   6. Applied egg-rr 50.0%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]
   7. Step-by-step derivation

      1. div-inv 50.0%

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

      2. *-commutative 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l/ 50.0%

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

      3. times-frac 49.1%

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

      4. associate-*l/ 49.1%

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

      5. *-lft-identity 49.1%

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

   10. Simplified 49.1%

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

   11. Taylor expanded in x around 0 42.3%

       \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 2.94999999999999983e32 < x

   1. Initial program 86.1%

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

      1. add-sqr-sqrt 49.7%

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

      2. fabs-sqr 49.7%

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

      3. add-sqr-sqrt 50.2%

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

      4. sub-neg 50.2%

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

      5. +-commutative 50.2%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 53.6%

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

      8. frac-2neg 53.6%

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

      9. +-commutative 53.6%

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

      10. distribute-neg-in 53.6%

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

      11. metadata-eval 53.6%

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

      12. sub-neg 53.6%

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

      13. distribute-frac-neg2 53.6%

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

      14. distribute-neg-in 53.6%

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

      15. fma-undefine 55.3%

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

      16. neg-sub0 55.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 53.6%

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

      18. associate-*r/ 48.8%

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

      19. associate-*l/ 50.2%

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

   4. Applied egg-rr 55.7%

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

   5. Taylor expanded in z around 0 30.0%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 30.0%

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

      2. distribute-lft-in 30.0%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 30.0%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 30.0%

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

      5. sub-neg 30.0%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 30.0%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in x around inf 30.0%

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

Alternative 8: 59.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;x \cdot \frac{z}{y\_m}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-19}:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -0.66)
   (* x (/ z y_m))
   (if (<= x 6e-19)
     (/ 4.0 y_m)
     (if (<= x 7.4e+31) (/ x (/ y_m z)) (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -0.66) {
		tmp = x * (z / y_m);
	} else if (x <= 6e-19) {
		tmp = 4.0 / y_m;
	} else if (x <= 7.4e+31) {
		tmp = x / (y_m / z);
	} else {
		tmp = 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 <= (-0.66d0)) then
        tmp = x * (z / y_m)
    else if (x <= 6d-19) then
        tmp = 4.0d0 / y_m
    else if (x <= 7.4d+31) then
        tmp = x / (y_m / z)
    else
        tmp = 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 <= -0.66) {
		tmp = x * (z / y_m);
	} else if (x <= 6e-19) {
		tmp = 4.0 / y_m;
	} else if (x <= 7.4e+31) {
		tmp = x / (y_m / z);
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -0.66:
		tmp = x * (z / y_m)
	elif x <= 6e-19:
		tmp = 4.0 / y_m
	elif x <= 7.4e+31:
		tmp = x / (y_m / z)
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -0.66)
		tmp = Float64(x * Float64(z / y_m));
	elseif (x <= 6e-19)
		tmp = Float64(4.0 / y_m);
	elseif (x <= 7.4e+31)
		tmp = Float64(x / Float64(y_m / z));
	else
		tmp = 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 <= -0.66)
		tmp = x * (z / y_m);
	elseif (x <= 6e-19)
		tmp = 4.0 / y_m;
	elseif (x <= 7.4e+31)
		tmp = x / (y_m / z);
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -0.66], N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-19], N[(4.0 / y$95$m), $MachinePrecision], If[LessEqual[x, 7.4e+31], N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.660000000000000031

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 66.0%

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

      3. sub0-neg 66.0%

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

      4. sub0-neg 66.0%

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

      5. sqr-neg 66.0%

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

      6. sqrt-unprod 52.4%

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

      7. add-sqr-sqrt 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in z around inf 22.7%

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

      1. associate-*r/ 27.1%

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

   9. Simplified 27.1%

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

   if -0.660000000000000031 < x < 5.99999999999999985e-19

   1. Initial program 97.0%

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

      1. add-sqr-sqrt 47.7%

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

      2. fabs-sqr 47.7%

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

      3. add-sqr-sqrt 49.1%

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

      4. associate-*l/ 50.0%

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

      5. associate-*r/ 47.7%

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

   4. Applied egg-rr 47.7%

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

      1. associate-*r/ 50.0%

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

      2. clear-num 50.0%

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

   6. Applied egg-rr 50.0%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]
   7. Step-by-step derivation

      1. div-inv 50.0%

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

      2. *-commutative 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l/ 50.0%

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

      3. times-frac 49.1%

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

      4. associate-*l/ 49.1%

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

      5. *-lft-identity 49.1%

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

   10. Simplified 49.1%

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

   11. Taylor expanded in x around 0 42.3%

       \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 5.99999999999999985e-19 < x < 7.3999999999999996e31

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 41.3%

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

      2. fabs-sqr 41.3%

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

      3. add-sqr-sqrt 42.1%

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

      4. sub-neg 42.1%

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

      5. +-commutative 42.1%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. frac-2neg 42.3%

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

      9. +-commutative 42.3%

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

      10. distribute-neg-in 42.3%

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

      11. metadata-eval 42.3%

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

      12. sub-neg 42.3%

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

      13. distribute-frac-neg2 42.3%

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

      14. distribute-neg-in 42.3%

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

      15. fma-undefine 42.3%

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

      16. neg-sub0 42.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 42.3%

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

      18. associate-*r/ 42.3%

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

      19. associate-*l/ 42.1%

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

   4. Applied egg-rr 42.3%

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

   5. Taylor expanded in z around inf 42.4%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 41.3%

         \[\leadsto \color{blue}{\sqrt{0 - \frac{x \cdot z}{y}} \cdot \sqrt{0 - \frac{x \cdot z}{y}}} \]

      2. sqrt-unprod 53.6%

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

      3. associate-*r/ 53.6%

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

      4. neg-sub0 53.6%

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

      5. associate-*r/ 53.6%

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

      6. neg-sub0 53.6%

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

      7. sqr-neg 53.6%

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

      8. sqrt-unprod 34.8%

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

      9. add-sqr-sqrt 35.5%

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

      10. clear-num 35.5%

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

      11. un-div-inv 42.6%

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

   7. Applied egg-rr 42.6%

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

   if 7.3999999999999996e31 < x

   1. Initial program 86.1%

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

      1. add-sqr-sqrt 49.7%

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

      2. fabs-sqr 49.7%

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

      3. add-sqr-sqrt 50.2%

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

      4. sub-neg 50.2%

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

      5. +-commutative 50.2%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 53.6%

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

      8. frac-2neg 53.6%

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

      9. +-commutative 53.6%

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

      10. distribute-neg-in 53.6%

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

      11. metadata-eval 53.6%

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

      12. sub-neg 53.6%

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

      13. distribute-frac-neg2 53.6%

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

      14. distribute-neg-in 53.6%

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

      15. fma-undefine 55.3%

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

      16. neg-sub0 55.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 53.6%

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

      18. associate-*r/ 48.8%

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

      19. associate-*l/ 50.2%

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

   4. Applied egg-rr 55.7%

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

   5. Taylor expanded in z around 0 30.0%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 30.0%

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

      2. distribute-lft-in 30.0%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 30.0%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 30.0%

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

      5. sub-neg 30.0%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 30.0%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in x around inf 30.0%

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

Alternative 9: 77.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{x \cdot z - \left(x + 4\right)}{y\_m}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -3.7)
   (/ (- (* x z) (+ x 4.0)) y_m)
   (if (<= x 2.3e+23) (/ (- 4.0 (* x z)) y_m) (/ (+ x 4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -3.7) {
		tmp = ((x * z) - (x + 4.0)) / y_m;
	} else if (x <= 2.3e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (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 (x <= (-3.7d0)) then
        tmp = ((x * z) - (x + 4.0d0)) / y_m
    else if (x <= 2.3d+23) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = (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 (x <= -3.7) {
		tmp = ((x * z) - (x + 4.0)) / y_m;
	} else if (x <= 2.3e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -3.7:
		tmp = ((x * z) - (x + 4.0)) / y_m
	elif x <= 2.3e+23:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -3.7)
		tmp = Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y_m);
	elseif (x <= 2.3e+23)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = 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 (x <= -3.7)
		tmp = ((x * z) - (x + 4.0)) / y_m;
	elseif (x <= 2.3e+23)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = (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[x, -3.7], N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 2.3e+23], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 66.0%

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

      3. sub0-neg 66.0%

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

      4. sub0-neg 66.0%

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

      5. sqr-neg 66.0%

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

      6. sqrt-unprod 52.4%

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

      7. add-sqr-sqrt 53.0%

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

   6. Applied egg-rr 53.0%

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

   if -3.7000000000000002 < x < 2.3e23

   1. Initial program 97.2%

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

      1. add-sqr-sqrt 47.5%

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

      2. fabs-sqr 47.5%

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

      3. add-sqr-sqrt 48.9%

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

      4. associate-*l/ 49.7%

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

      5. associate-*r/ 47.6%

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

   4. Applied egg-rr 47.6%

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

   5. Taylor expanded in x around 0 46.5%

      \[\leadsto \frac{\color{blue}{4}}{y} - x \cdot \frac{z}{y} \]

   6. Taylor expanded in y around 0 48.6%

      \[\leadsto \color{blue}{\frac{4 - x \cdot z}{y}} \]

   if 2.3e23 < x

   1. Initial program 86.3%

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

      1. add-sqr-sqrt 48.9%

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

      2. fabs-sqr 48.9%

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

      3. add-sqr-sqrt 49.4%

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

      4. sub-neg 49.4%

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

      5. +-commutative 49.4%

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

      6. associate-*l/ 48.0%

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

      7. associate-*r/ 52.7%

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

      8. frac-2neg 52.7%

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

      9. +-commutative 52.7%

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

      10. distribute-neg-in 52.7%

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

      11. metadata-eval 52.7%

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

      12. sub-neg 52.7%

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

      13. distribute-frac-neg2 52.7%

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

      14. distribute-neg-in 52.7%

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

      15. fma-undefine 54.4%

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

      16. neg-sub0 54.4%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 52.7%

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

      18. associate-*r/ 48.0%

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

      19. associate-*l/ 49.4%

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

   4. Applied egg-rr 54.7%

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

   5. Taylor expanded in z around 0 29.6%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 29.6%

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

      2. distribute-lft-in 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 29.6%

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

      5. sub-neg 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 29.6%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in y around 0 29.6%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{x \cdot z - \left(x + 4\right)}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\frac{x \cdot z - x}{y\_m}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -0.8)
   (/ (- (* x z) x) y_m)
   (if (<= x 2.3e+23) (/ (- 4.0 (* x z)) y_m) (/ (+ x 4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -0.8) {
		tmp = ((x * z) - x) / y_m;
	} else if (x <= 2.3e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (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 (x <= (-0.8d0)) then
        tmp = ((x * z) - x) / y_m
    else if (x <= 2.3d+23) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = (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 (x <= -0.8) {
		tmp = ((x * z) - x) / y_m;
	} else if (x <= 2.3e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -0.8:
		tmp = ((x * z) - x) / y_m
	elif x <= 2.3e+23:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -0.8)
		tmp = Float64(Float64(Float64(x * z) - x) / y_m);
	elseif (x <= 2.3e+23)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = 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 (x <= -0.8)
		tmp = ((x * z) - x) / y_m;
	elseif (x <= 2.3e+23)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = (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[x, -0.8], N[(N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 2.3e+23], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.80000000000000004

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 66.0%

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

      3. sub0-neg 66.0%

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

      4. sub0-neg 66.0%

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

      5. sqr-neg 66.0%

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

      6. sqrt-unprod 52.4%

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

      7. add-sqr-sqrt 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in x around inf 52.8%

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

   if -0.80000000000000004 < x < 2.3e23

   1. Initial program 97.2%

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

      1. add-sqr-sqrt 47.5%

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

      2. fabs-sqr 47.5%

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

      3. add-sqr-sqrt 48.9%

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

      4. associate-*l/ 49.7%

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

      5. associate-*r/ 47.6%

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

   4. Applied egg-rr 47.6%

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

   5. Taylor expanded in x around 0 46.5%

      \[\leadsto \frac{\color{blue}{4}}{y} - x \cdot \frac{z}{y} \]

   6. Taylor expanded in y around 0 48.6%

      \[\leadsto \color{blue}{\frac{4 - x \cdot z}{y}} \]

   if 2.3e23 < x

   1. Initial program 86.3%

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

      1. add-sqr-sqrt 48.9%

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

      2. fabs-sqr 48.9%

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

      3. add-sqr-sqrt 49.4%

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

      4. sub-neg 49.4%

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

      5. +-commutative 49.4%

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

      6. associate-*l/ 48.0%

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

      7. associate-*r/ 52.7%

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

      8. frac-2neg 52.7%

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

      9. +-commutative 52.7%

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

      10. distribute-neg-in 52.7%

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

      11. metadata-eval 52.7%

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

      12. sub-neg 52.7%

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

      13. distribute-frac-neg2 52.7%

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

      14. distribute-neg-in 52.7%

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

      15. fma-undefine 54.4%

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

      16. neg-sub0 54.4%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 52.7%

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

      18. associate-*r/ 48.0%

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

      19. associate-*l/ 49.4%

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

   4. Applied egg-rr 54.7%

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

   5. Taylor expanded in z around 0 29.6%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 29.6%

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

      2. distribute-lft-in 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 29.6%

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

      5. sub-neg 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 29.6%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in y around 0 29.6%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\frac{x \cdot z - x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\frac{-4 - x}{y\_m}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.8)
   (/ (- -4.0 x) y_m)
   (if (<= x 2.3e+23) (/ (- 4.0 (* x z)) y_m) (/ (+ x 4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.8) {
		tmp = (-4.0 - x) / y_m;
	} else if (x <= 2.3e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (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 (x <= (-4.8d0)) then
        tmp = ((-4.0d0) - x) / y_m
    else if (x <= 2.3d+23) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = (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 (x <= -4.8) {
		tmp = (-4.0 - x) / y_m;
	} else if (x <= 2.3e+23) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.8:
		tmp = (-4.0 - x) / y_m
	elif x <= 2.3e+23:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.8)
		tmp = Float64(Float64(-4.0 - x) / y_m);
	elseif (x <= 2.3e+23)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = 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 (x <= -4.8)
		tmp = (-4.0 - x) / y_m;
	elseif (x <= 2.3e+23)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = (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[x, -4.8], N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 2.3e+23], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999982

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 66.0%

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

      3. sub0-neg 66.0%

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

      4. sub0-neg 66.0%

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

      5. sqr-neg 66.0%

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

      6. sqrt-unprod 52.4%

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

      7. add-sqr-sqrt 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in z around 0 40.9%

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

      1. distribute-lft-in 40.9%

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

      2. metadata-eval 40.9%

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

      3. neg-mul-1 40.9%

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

      4. sub-neg 40.9%

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

   9. Simplified 40.9%

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

   if -4.79999999999999982 < x < 2.3e23

   1. Initial program 97.2%

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

      1. add-sqr-sqrt 47.5%

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

      2. fabs-sqr 47.5%

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

      3. add-sqr-sqrt 48.9%

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

      4. associate-*l/ 49.7%

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

      5. associate-*r/ 47.6%

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

   4. Applied egg-rr 47.6%

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

   5. Taylor expanded in x around 0 46.5%

      \[\leadsto \frac{\color{blue}{4}}{y} - x \cdot \frac{z}{y} \]

   6. Taylor expanded in y around 0 48.6%

      \[\leadsto \color{blue}{\frac{4 - x \cdot z}{y}} \]

   if 2.3e23 < x

   1. Initial program 86.3%

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

      1. add-sqr-sqrt 48.9%

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

      2. fabs-sqr 48.9%

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

      3. add-sqr-sqrt 49.4%

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

      4. sub-neg 49.4%

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

      5. +-commutative 49.4%

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

      6. associate-*l/ 48.0%

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

      7. associate-*r/ 52.7%

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

      8. frac-2neg 52.7%

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

      9. +-commutative 52.7%

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

      10. distribute-neg-in 52.7%

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

      11. metadata-eval 52.7%

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

      12. sub-neg 52.7%

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

      13. distribute-frac-neg2 52.7%

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

      14. distribute-neg-in 52.7%

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

      15. fma-undefine 54.4%

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

      16. neg-sub0 54.4%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 52.7%

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

      18. associate-*r/ 48.0%

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

      19. associate-*l/ 49.4%

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

   4. Applied egg-rr 54.7%

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

   5. Taylor expanded in z around 0 29.6%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 29.6%

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

      2. distribute-lft-in 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 29.6%

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

      5. sub-neg 29.6%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 29.6%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in y around 0 29.6%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\frac{-4 - x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.5% accurate, 6.9× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 93.5%

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

      1. fabs-sub 93.5%

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

      2. div-inv 93.4%

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

      3. cancel-sign-sub-inv 93.4%

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

      4. associate-*l/ 87.5%

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

      5. associate-*r/ 95.0%

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

      6. +-commutative 95.0%

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

      7. distribute-neg-in 95.0%

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

      8. metadata-eval 95.0%

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

      9. sub-neg 95.0%

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

      10. div-inv 95.1%

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

      11. fma-undefine 96.7%

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

      12. add-sqr-sqrt 58.4%

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

      13. fabs-sqr 58.4%

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

      14. add-sqr-sqrt 59.1%

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

      15. fma-undefine 57.5%

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

      16. associate-*r/ 53.0%

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

      17. associate-*l/ 55.8%

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

      18. div-inv 55.7%

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

   4. Applied egg-rr 57.5%

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

   if -4 < x

   1. Initial program 93.9%

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

      1. add-sqr-sqrt 47.9%

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

      2. fabs-sqr 47.9%

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

      3. add-sqr-sqrt 49.0%

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

      4. sub-neg 49.0%

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

      5. +-commutative 49.0%

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

      6. associate-*l/ 49.1%

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

      7. associate-*r/ 49.1%

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

      8. frac-2neg 49.1%

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

      9. +-commutative 49.1%

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

      10. distribute-neg-in 49.1%

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

      11. metadata-eval 49.1%

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

      12. sub-neg 49.1%

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

      13. distribute-frac-neg2 49.1%

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

      14. distribute-neg-in 49.1%

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

      15. fma-undefine 49.6%

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

      16. neg-sub0 49.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 49.1%

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

      18. associate-*r/ 49.1%

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

      19. associate-*l/ 49.0%

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

   4. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.7%

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

Alternative 13: 80.1% accurate, 7.9× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 66.0%

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

      3. sub0-neg 66.0%

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

      4. sub0-neg 66.0%

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

      5. sqr-neg 66.0%

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

      6. sqrt-unprod 52.4%

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

      7. add-sqr-sqrt 53.0%

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

   6. Applied egg-rr 53.0%

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

   if -4 < x

   1. Initial program 93.9%

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

      1. add-sqr-sqrt 47.9%

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

      2. fabs-sqr 47.9%

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

      3. add-sqr-sqrt 49.0%

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

      4. sub-neg 49.0%

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

      5. +-commutative 49.0%

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

      6. associate-*l/ 49.1%

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

      7. associate-*r/ 49.1%

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

      8. frac-2neg 49.1%

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

      9. +-commutative 49.1%

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

      10. distribute-neg-in 49.1%

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

      11. metadata-eval 49.1%

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

      12. sub-neg 49.1%

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

      13. distribute-frac-neg2 49.1%

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

      14. distribute-neg-in 49.1%

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

      15. fma-undefine 49.6%

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

      16. neg-sub0 49.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 49.1%

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

      18. associate-*r/ 49.1%

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

      19. associate-*l/ 49.0%

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

   4. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

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

Alternative 14: 60.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.62:\\ \;\;\;\;x \cdot \frac{z}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -0.62) (* x (/ z y_m)) (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -0.62) {
		tmp = x * (z / y_m);
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = 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 <= (-0.62d0)) then
        tmp = x * (z / y_m)
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = 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 <= -0.62) {
		tmp = x * (z / y_m);
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -0.62:
		tmp = x * (z / y_m)
	elif x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -0.62)
		tmp = Float64(x * Float64(z / y_m));
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = 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 <= -0.62)
		tmp = x * (z / y_m);
	elseif (x <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -0.62], N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.619999999999999996

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 66.0%

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

      3. sub0-neg 66.0%

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

      4. sub0-neg 66.0%

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

      5. sqr-neg 66.0%

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

      6. sqrt-unprod 52.4%

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

      7. add-sqr-sqrt 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in z around inf 22.7%

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

      1. associate-*r/ 27.1%

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

   9. Simplified 27.1%

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

   if -0.619999999999999996 < x < 4

   1. Initial program 97.1%

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

      1. add-sqr-sqrt 47.4%

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

      2. fabs-sqr 47.4%

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

      3. add-sqr-sqrt 48.8%

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

      4. associate-*l/ 49.6%

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

      5. associate-*r/ 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. associate-*r/ 49.6%

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

      2. clear-num 49.6%

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

   6. Applied egg-rr 49.6%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]
   7. Step-by-step derivation

      1. div-inv 49.6%

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

      2. *-commutative 49.6%

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

   8. Applied egg-rr 49.6%

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

      1. *-commutative 49.6%

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

      2. associate-*l/ 49.6%

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

      3. times-frac 48.8%

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

      4. associate-*l/ 48.8%

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

      5. *-lft-identity 48.8%

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

   10. Simplified 48.8%

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

   11. Taylor expanded in x around 0 40.7%

       \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 87.5%

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

      1. add-sqr-sqrt 48.9%

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

      2. fabs-sqr 48.9%

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

      3. add-sqr-sqrt 49.5%

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

      4. sub-neg 49.5%

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

      5. +-commutative 49.5%

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

      6. associate-*l/ 48.2%

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

      7. associate-*r/ 52.5%

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

      8. frac-2neg 52.5%

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

      9. +-commutative 52.5%

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

      10. distribute-neg-in 52.5%

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

      11. metadata-eval 52.5%

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

      12. sub-neg 52.5%

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

      13. distribute-frac-neg2 52.5%

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

      14. distribute-neg-in 52.5%

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

      15. fma-undefine 54.0%

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

      16. neg-sub0 54.0%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 52.5%

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

      18. associate-*r/ 48.2%

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

      19. associate-*l/ 49.5%

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

   4. Applied egg-rr 54.4%

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

   5. Taylor expanded in z around 0 27.1%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 27.1%

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

      2. distribute-lft-in 27.1%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 27.1%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 27.1%

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

      5. sub-neg 27.1%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 27.1%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in x around inf 27.1%

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

Alternative 15: 70.3% accurate, 11.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.0) (/ (- -4.0 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 (x <= -4.0) {
		tmp = (-4.0 - x) / y_m;
	} else {
		tmp = (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 (x <= (-4.0d0)) then
        tmp = ((-4.0d0) - x) / y_m
    else
        tmp = (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 (x <= -4.0) {
		tmp = (-4.0 - x) / y_m;
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(-4.0 - x) / y_m);
	else
		tmp = 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 (x <= -4.0)
		tmp = (-4.0 - x) / y_m;
	else
		tmp = (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[x, -4.0], N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 66.0%

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

      3. sub0-neg 66.0%

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

      4. sub0-neg 66.0%

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

      5. sqr-neg 66.0%

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

      6. sqrt-unprod 52.4%

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

      7. add-sqr-sqrt 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in z around 0 40.9%

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

      1. distribute-lft-in 40.9%

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

      2. metadata-eval 40.9%

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

      3. neg-mul-1 40.9%

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

      4. sub-neg 40.9%

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

   9. Simplified 40.9%

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

   if -4 < x

   1. Initial program 93.9%

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

      1. add-sqr-sqrt 47.9%

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

      2. fabs-sqr 47.9%

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

      3. add-sqr-sqrt 49.0%

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

      4. sub-neg 49.0%

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

      5. +-commutative 49.0%

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

      6. associate-*l/ 49.1%

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

      7. associate-*r/ 49.1%

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

      8. frac-2neg 49.1%

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

      9. +-commutative 49.1%

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

      10. distribute-neg-in 49.1%

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

      11. metadata-eval 49.1%

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

      12. sub-neg 49.1%

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

      13. distribute-frac-neg2 49.1%

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

      14. distribute-neg-in 49.1%

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

      15. fma-undefine 49.6%

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

      16. neg-sub0 49.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 49.1%

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

      18. associate-*r/ 49.1%

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

      19. associate-*l/ 49.0%

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

   4. Applied egg-rr 51.2%

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

   5. Taylor expanded in z around 0 36.9%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 36.9%

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

      2. distribute-lft-in 36.9%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 36.9%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 36.9%

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

      5. sub-neg 36.9%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 36.9%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in y around 0 36.9%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.1% accurate, 11.1× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = z / (y_m / x);
	} else {
		tmp = (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 (x <= (-4.0d0)) then
        tmp = z / (y_m / x)
    else
        tmp = (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 (x <= -4.0) {
		tmp = z / (y_m / x);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(z / Float64(y_m / x));
	else
		tmp = 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 (x <= -4.0)
		tmp = z / (y_m / x);
	else
		tmp = (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[x, -4.0], N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 35.3%

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

      7. associate-*r/ 38.3%

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

      8. frac-2neg 38.3%

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

      9. +-commutative 38.3%

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

      10. distribute-neg-in 38.3%

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

      11. metadata-eval 38.3%

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

      12. sub-neg 38.3%

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

      13. distribute-frac-neg2 38.3%

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

      14. distribute-neg-in 38.3%

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

      15. fma-undefine 38.3%

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

      16. neg-sub0 38.3%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 38.3%

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

      18. associate-*r/ 35.3%

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

      19. associate-*l/ 38.4%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around inf 22.1%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 21.7%

         \[\leadsto \color{blue}{\sqrt{0 - \frac{x \cdot z}{y}} \cdot \sqrt{0 - \frac{x \cdot z}{y}}} \]

      2. sqrt-unprod 40.4%

         \[\leadsto \color{blue}{\sqrt{\left(0 - \frac{x \cdot z}{y}\right) \cdot \left(0 - \frac{x \cdot z}{y}\right)}} \]

      3. associate-*r/ 40.4%

         \[\leadsto \sqrt{\left(0 - \color{blue}{x \cdot \frac{z}{y}}\right) \cdot \left(0 - \frac{x \cdot z}{y}\right)} \]

      4. neg-sub0 40.4%

         \[\leadsto \sqrt{\color{blue}{\left(-x \cdot \frac{z}{y}\right)} \cdot \left(0 - \frac{x \cdot z}{y}\right)} \]

      5. associate-*r/ 45.0%

         \[\leadsto \sqrt{\left(-x \cdot \frac{z}{y}\right) \cdot \left(0 - \color{blue}{x \cdot \frac{z}{y}}\right)} \]

      6. neg-sub0 45.0%

         \[\leadsto \sqrt{\left(-x \cdot \frac{z}{y}\right) \cdot \color{blue}{\left(-x \cdot \frac{z}{y}\right)}} \]

      7. sqr-neg 45.0%

         \[\leadsto \sqrt{\color{blue}{\left(x \cdot \frac{z}{y}\right) \cdot \left(x \cdot \frac{z}{y}\right)}} \]

      8. sqrt-unprod 26.7%

         \[\leadsto \color{blue}{\sqrt{x \cdot \frac{z}{y}} \cdot \sqrt{x \cdot \frac{z}{y}}} \]

      9. add-sqr-sqrt 27.1%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y}} \]

      10. *-commutative 27.1%

          \[\leadsto \color{blue}{\frac{z}{y} \cdot x} \]

      11. associate-/r/ 30.1%

          \[\leadsto \color{blue}{\frac{z}{\frac{y}{x}}} \]

   7. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{\frac{z}{\frac{y}{x}}} \]

   if -4 < x

   1. Initial program 93.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 47.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 47.9%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 49.0%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. sub-neg 49.0%

         \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      5. +-commutative 49.0%

         \[\leadsto \color{blue}{\left(-\frac{x}{y} \cdot z\right) + \frac{x + 4}{y}} \]

      6. associate-*l/ 49.1%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \frac{x + 4}{y} \]

      7. associate-*r/ 49.1%

         \[\leadsto \left(-\color{blue}{x \cdot \frac{z}{y}}\right) + \frac{x + 4}{y} \]

      8. frac-2neg 49.1%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\frac{-\left(x + 4\right)}{-y}} \]

      9. +-commutative 49.1%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{-\color{blue}{\left(4 + x\right)}}{-y} \]

      10. distribute-neg-in 49.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{-y} \]

      11. metadata-eval 49.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4} + \left(-x\right)}{-y} \]

      12. sub-neg 49.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4 - x}}{-y} \]

      13. distribute-frac-neg2 49.1%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\left(-\frac{-4 - x}{y}\right)} \]

      14. distribute-neg-in 49.1%

          \[\leadsto \color{blue}{-\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      15. fma-undefine 49.6%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      16. neg-sub0 49.6%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 49.1%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 49.1%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 49.0%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around 0 36.9%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 36.9%

         \[\leadsto 0 - \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 36.9%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 36.9%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 36.9%

         \[\leadsto 0 - \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 36.9%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 36.9%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in y around 0 36.9%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.1% accurate, 13.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = 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 <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = 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 <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = 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 <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 95.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 44.2%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 44.2%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 45.4%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. associate-*l/ 44.9%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      5. associate-*r/ 44.4%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

   4. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
   5. Step-by-step derivation

      1. associate-*r/ 44.9%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      2. clear-num 44.9%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]

   6. Applied egg-rr 44.9%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]
   7. Step-by-step derivation

      1. div-inv 44.9%

         \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{y \cdot \frac{1}{x \cdot z}}} \]

      2. *-commutative 44.9%

         \[\leadsto \frac{x + 4}{y} - \frac{1}{y \cdot \frac{1}{\color{blue}{z \cdot x}}} \]

   8. Applied egg-rr 44.9%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{y \cdot \frac{1}{z \cdot x}}} \]
   9. Step-by-step derivation

      1. *-commutative 44.9%

         \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1}{z \cdot x} \cdot y}} \]

      2. associate-*l/ 44.9%

         \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1 \cdot y}{z \cdot x}}} \]

      3. times-frac 45.3%

         \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1}{z} \cdot \frac{y}{x}}} \]

      4. associate-*l/ 45.4%

         \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{z}}} \]

      5. *-lft-identity 45.4%

         \[\leadsto \frac{x + 4}{y} - \frac{1}{\frac{\color{blue}{\frac{y}{x}}}{z}} \]

   10. Simplified 45.4%

       \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{\frac{y}{x}}{z}}} \]

   11. Taylor expanded in x around 0 28.5%

       \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 87.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 48.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 48.9%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 49.5%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. sub-neg 49.5%

         \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      5. +-commutative 49.5%

         \[\leadsto \color{blue}{\left(-\frac{x}{y} \cdot z\right) + \frac{x + 4}{y}} \]

      6. associate-*l/ 48.2%

         \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \frac{x + 4}{y} \]

      7. associate-*r/ 52.5%

         \[\leadsto \left(-\color{blue}{x \cdot \frac{z}{y}}\right) + \frac{x + 4}{y} \]

      8. frac-2neg 52.5%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\frac{-\left(x + 4\right)}{-y}} \]

      9. +-commutative 52.5%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{-\color{blue}{\left(4 + x\right)}}{-y} \]

      10. distribute-neg-in 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{-y} \]

      11. metadata-eval 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4} + \left(-x\right)}{-y} \]

      12. sub-neg 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4 - x}}{-y} \]

      13. distribute-frac-neg2 52.5%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\left(-\frac{-4 - x}{y}\right)} \]

      14. distribute-neg-in 52.5%

          \[\leadsto \color{blue}{-\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      15. fma-undefine 54.0%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      16. neg-sub0 54.0%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 52.5%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 48.2%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 49.5%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in z around 0 27.1%

      \[\leadsto 0 - \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 27.1%

         \[\leadsto 0 - \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 27.1%

         \[\leadsto 0 - \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 27.1%

         \[\leadsto 0 - \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 27.1%

         \[\leadsto 0 - \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 27.1%

         \[\leadsto 0 - \frac{\color{blue}{-4 - x}}{y} \]

   7. Simplified 27.1%

      \[\leadsto 0 - \color{blue}{\frac{-4 - x}{y}} \]

   8. Taylor expanded in x around inf 27.1%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 40.9% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ 4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return 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 = 4.0d0 / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return 4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return 4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = 4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 45.4%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   2. fabs-sqr 45.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   3. add-sqr-sqrt 46.4%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   4. associate-*l/ 45.7%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   5. associate-*r/ 46.5%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

4. Applied egg-rr 46.5%

   \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
5. Step-by-step derivation

   1. associate-*r/ 45.7%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   2. clear-num 45.7%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]

6. Applied egg-rr 45.7%

   \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{1}{\frac{y}{x \cdot z}}} \]
7. Step-by-step derivation

   1. div-inv 45.7%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{y \cdot \frac{1}{x \cdot z}}} \]

   2. *-commutative 45.7%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{y \cdot \frac{1}{\color{blue}{z \cdot x}}} \]

8. Applied egg-rr 45.7%

   \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{y \cdot \frac{1}{z \cdot x}}} \]
9. Step-by-step derivation

   1. *-commutative 45.7%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1}{z \cdot x} \cdot y}} \]

   2. associate-*l/ 45.7%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1 \cdot y}{z \cdot x}}} \]

   3. times-frac 46.4%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1}{z} \cdot \frac{y}{x}}} \]

   4. associate-*l/ 46.4%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{z}}} \]

   5. *-lft-identity 46.4%

      \[\leadsto \frac{x + 4}{y} - \frac{1}{\frac{\color{blue}{\frac{y}{x}}}{z}} \]

10. Simplified 46.4%

    \[\leadsto \frac{x + 4}{y} - \frac{1}{\color{blue}{\frac{\frac{y}{x}}{z}}} \]

11. Taylor expanded in x around 0 22.1%

    \[\leadsto \color{blue}{\frac{4}{y}} \]
12. Add Preprocessing

Alternative 19: 1.7% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{-4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ -4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return -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 = (-4.0d0) / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return -4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return -4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(-4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = -4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(-4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 45.4%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   2. fabs-sqr 45.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   3. add-sqr-sqrt 46.4%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   4. sub-neg 46.4%

      \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

   5. +-commutative 46.4%

      \[\leadsto \color{blue}{\left(-\frac{x}{y} \cdot z\right) + \frac{x + 4}{y}} \]

   6. associate-*l/ 45.7%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \frac{x + 4}{y} \]

   7. associate-*r/ 46.5%

      \[\leadsto \left(-\color{blue}{x \cdot \frac{z}{y}}\right) + \frac{x + 4}{y} \]

   8. frac-2neg 46.5%

      \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\frac{-\left(x + 4\right)}{-y}} \]

   9. +-commutative 46.5%

      \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{-\color{blue}{\left(4 + x\right)}}{-y} \]

   10. distribute-neg-in 46.5%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{-y} \]

   11. metadata-eval 46.5%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4} + \left(-x\right)}{-y} \]

   12. sub-neg 46.5%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4 - x}}{-y} \]

   13. distribute-frac-neg2 46.5%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\left(-\frac{-4 - x}{y}\right)} \]

   14. distribute-neg-in 46.5%

       \[\leadsto \color{blue}{-\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

   15. fma-undefine 46.9%

       \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   16. neg-sub0 46.9%

       \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   17. fma-undefine 46.5%

       \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

   18. associate-*r/ 45.7%

       \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

   19. associate-*l/ 46.4%

       \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

4. Applied egg-rr 48.5%

   \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 47.5%

      \[\leadsto \color{blue}{\sqrt{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \cdot \sqrt{0 - \frac{x \cdot z - \left(x + 4\right)}{y}}} \]

   2. sqrt-unprod 65.3%

      \[\leadsto \color{blue}{\sqrt{\left(0 - \frac{x \cdot z - \left(x + 4\right)}{y}\right) \cdot \left(0 - \frac{x \cdot z - \left(x + 4\right)}{y}\right)}} \]

   3. sub0-neg 65.3%

      \[\leadsto \sqrt{\color{blue}{\left(-\frac{x \cdot z - \left(x + 4\right)}{y}\right)} \cdot \left(0 - \frac{x \cdot z - \left(x + 4\right)}{y}\right)} \]

   4. sub0-neg 65.3%

      \[\leadsto \sqrt{\left(-\frac{x \cdot z - \left(x + 4\right)}{y}\right) \cdot \color{blue}{\left(-\frac{x \cdot z - \left(x + 4\right)}{y}\right)}} \]

   5. sqr-neg 65.3%

      \[\leadsto \sqrt{\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y} \cdot \frac{x \cdot z - \left(x + 4\right)}{y}}} \]

   6. sqrt-unprod 49.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x \cdot z - \left(x + 4\right)}{y}} \cdot \sqrt{\frac{x \cdot z - \left(x + 4\right)}{y}}} \]

   7. add-sqr-sqrt 49.9%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

6. Applied egg-rr 49.9%

   \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

7. Taylor expanded in x around 0 22.6%

   \[\leadsto \frac{\color{blue}{-4}}{y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))