fabs fraction 1

Percentage Accurate: 92.3% → 99.3%

Time: 11.1s

Alternatives: 17

Speedup: 1.0×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.3% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-94)
   (fabs (/ (- (* x z) (+ x 4.0)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-94) {
		tmp = fabs((((x * z) - (x + 4.0)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-94)
		tmp = abs(Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 4e-94], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-94

   1. Initial program 89.9%

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

      1. associate-*l/ 93.9%

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

      2. sub-div 98.9%

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

   4. Applied egg-rr 98.9%

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

   if 3.9999999999999998e-94 < y

   1. Initial program 96.2%

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

      1. fabs-sub 96.2%

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

      2. associate-*l/ 91.2%

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

      3. associate-*r/ 99.9%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37} \lor \neg \left(x \leq 1.2 \cdot 10^{-26}\right):\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.7e-37) (not (<= x 1.2e-26)))
   (fabs (* (/ x y_m) (- 1.0 z)))
   (/ (- 4.0 (* x z)) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.7e-37) || !(x <= 1.2e-26)) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = (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 <= (-1.7d-37)) .or. (.not. (x <= 1.2d-26))) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = (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 <= -1.7e-37) || !(x <= 1.2e-26)) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = (4.0 - (x * z)) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.7e-37) or not (x <= 1.2e-26):
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = (4.0 - (x * z)) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.7e-37) || !(x <= 1.2e-26))
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = Float64(Float64(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 <= -1.7e-37) || ~((x <= 1.2e-26)))
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = (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[Or[LessEqual[x, -1.7e-37], N[Not[LessEqual[x, 1.2e-26]], $MachinePrecision]], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000009e-37 or 1.2e-26 < x

   1. Initial program 88.8%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. associate-/l* 96.9%

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

      3. distribute-rgt-neg-in 96.9%

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

      4. mul-1-neg 96.9%

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

      5. associate-*r/ 96.9%

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

      6. sub-neg 96.9%

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

      7. metadata-eval 96.9%

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

      8. distribute-lft-in 96.9%

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

      9. neg-mul-1 96.9%

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

      10. metadata-eval 96.9%

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

      11. +-commutative 96.9%

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

      12. neg-mul-1 96.9%

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

      13. associate-/l* 91.1%

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

      14. *-commutative 91.1%

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

      15. associate-/l* 97.0%

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

      16. neg-mul-1 97.0%

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

      17. unsub-neg 97.0%

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

   7. Simplified 97.0%

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

   if -1.70000000000000009e-37 < x < 1.2e-26

   1. Initial program 95.7%

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

      1. add-sqr-sqrt 46.3%

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

      2. fabs-sqr 46.3%

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

      3. add-sqr-sqrt 47.6%

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

      4. sub-neg 47.6%

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

      5. +-commutative 47.6%

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

      6. associate-*l/ 50.1%

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

      7. associate-*r/ 45.8%

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

      8. frac-2neg 45.8%

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

      9. +-commutative 45.8%

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

      10. distribute-neg-in 45.8%

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

      11. metadata-eval 45.8%

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

      12. sub-neg 45.8%

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

      13. distribute-frac-neg2 45.8%

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

      14. distribute-neg-in 45.8%

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

      15. fma-undefine 45.8%

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

      16. neg-sub0 45.8%

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

      17. fma-undefine 45.8%

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

      18. associate-*r/ 50.1%

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

      19. associate-*l/ 47.6%

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

   4. Applied egg-rr 50.1%

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

   5. Taylor expanded in x around 0 50.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37} \lor \neg \left(x \leq 1.2 \cdot 10^{-26}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1e-52)
   (fabs (/ (- (* x z) (+ x 4.0)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1e-52) {
		tmp = fabs((((x * z) - (x + 4.0)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1d-52) then
        tmp = abs((((x * z) - (x + 4.0d0)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1e-52) {
		tmp = Math.abs((((x * z) - (x + 4.0)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 1e-52:
		tmp = math.fabs((((x * z) - (x + 4.0)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1e-52)
		tmp = abs(Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1e-52)
		tmp = abs((((x * z) - (x + 4.0)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1e-52], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e-52

   1. Initial program 90.3%

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

      1. associate-*l/ 94.1%

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

      2. sub-div 99.0%

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

   4. Applied egg-rr 99.0%

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

   if 1e-52 < y

   1. Initial program 95.8%

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

      1. associate-*l/ 90.5%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

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

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -16 \lor \neg \left(z \leq 950\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -16.0) (not (<= z 950.0)))
   (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 <= -16.0) || !(z <= 950.0)) {
		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 <= (-16.0d0)) .or. (.not. (z <= 950.0d0))) 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 <= -16.0) || !(z <= 950.0)) {
		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 <= -16.0) or not (z <= 950.0):
		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 <= -16.0) || !(z <= 950.0))
		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 <= -16.0) || ~((z <= 950.0)))
		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, -16.0], N[Not[LessEqual[z, 950.0]], $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 < -16 or 950 < z

   1. Initial program 90.7%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in z around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. distribute-frac-neg2 68.4%

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

      3. *-commutative 68.4%

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

      4. associate-*r/ 71.5%

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

   7. Simplified 71.5%

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

   if -16 < z < 950

   1. Initial program 92.7%

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

      1. fabs-sub 92.7%

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

      2. associate-*l/ 97.8%

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

      3. associate-*r/ 97.8%

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

      4. fmm-def 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 98.5%

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

      1. associate-*r/ 98.5%

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

      2. distribute-lft-in 98.5%

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

      3. metadata-eval 98.5%

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

      4. neg-mul-1 98.5%

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

      5. sub-neg 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -16 \lor \neg \left(z \leq 950\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -16:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 950:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -16.0)
   (fabs (/ (* x z) y_m))
   (if (<= z 950.0) (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 <= -16.0) {
		tmp = fabs(((x * z) / y_m));
	} else if (z <= 950.0) {
		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 <= (-16.0d0)) then
        tmp = abs(((x * z) / y_m))
    else if (z <= 950.0d0) 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 <= -16.0) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (z <= 950.0) {
		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 <= -16.0:
		tmp = math.fabs(((x * z) / y_m))
	elif z <= 950.0:
		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 <= -16.0)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (z <= 950.0)
		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 <= -16.0)
		tmp = abs(((x * z) / y_m));
	elseif (z <= 950.0)
		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, -16.0], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 950.0], 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 < -16

   1. Initial program 91.7%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in z around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. neg-mul-1 67.2%

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

      3. distribute-rgt-neg-in 67.2%

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

   7. Simplified 67.2%

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

   if -16 < z < 950

   1. Initial program 92.7%

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

      1. fabs-sub 92.7%

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

      2. associate-*l/ 97.8%

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

      3. associate-*r/ 97.8%

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

      4. fmm-def 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 98.5%

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

      1. associate-*r/ 98.5%

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

      2. distribute-lft-in 98.5%

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

      3. metadata-eval 98.5%

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

      4. neg-mul-1 98.5%

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

      5. sub-neg 98.5%

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

   7. Simplified 98.5%

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

   if 950 < z

   1. Initial program 89.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in z around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. distribute-frac-neg2 69.6%

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

      3. *-commutative 69.6%

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

      4. associate-*r/ 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -16:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 950:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1000000000000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 1000000000000.0)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (* (/ x y_m) (- 1.0 z)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1000000000000.0) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((x / y_m) * (1.0 - 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 <= 1000000000000.0d0) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((x / y_m) * (1.0d0 - 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 <= 1000000000000.0) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 1000000000000.0:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 1000000000000.0)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 1000000000000.0)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((x / y_m) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 1000000000000.0], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e12

   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. associate-*l/ 96.4%

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

      2. sub-div 98.0%

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

   4. Applied egg-rr 98.0%

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

   if 1e12 < x

   1. Initial program 86.2%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. associate-/l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. mul-1-neg 99.8%

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

      5. associate-*r/ 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. neg-mul-1 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

      12. neg-mul-1 99.8%

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

      13. associate-/l* 92.7%

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

      14. *-commutative 92.7%

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

      15. associate-/l* 99.8%

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

      16. neg-mul-1 99.8%

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

      17. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1000000000000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z + -1}{y\_m}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-18} \lor \neg \left(x \leq 6.8 \cdot 10^{+157}\right):\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.7e-37)
   (* x (/ (+ z -1.0) y_m))
   (if (or (<= x 1.4e-18) (not (<= x 6.8e+157)))
     (/ (+ x 4.0) y_m)
     (/ (* x (- 1.0 z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.7e-37) {
		tmp = x * ((z + -1.0) / y_m);
	} else if ((x <= 1.4e-18) || !(x <= 6.8e+157)) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = (x * (1.0 - 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 <= (-1.7d-37)) then
        tmp = x * ((z + (-1.0d0)) / y_m)
    else if ((x <= 1.4d-18) .or. (.not. (x <= 6.8d+157))) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = (x * (1.0d0 - 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 <= -1.7e-37) {
		tmp = x * ((z + -1.0) / y_m);
	} else if ((x <= 1.4e-18) || !(x <= 6.8e+157)) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = (x * (1.0 - z)) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.7e-37:
		tmp = x * ((z + -1.0) / y_m)
	elif (x <= 1.4e-18) or not (x <= 6.8e+157):
		tmp = (x + 4.0) / y_m
	else:
		tmp = (x * (1.0 - z)) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.7e-37)
		tmp = Float64(x * Float64(Float64(z + -1.0) / y_m));
	elseif ((x <= 1.4e-18) || !(x <= 6.8e+157))
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(Float64(x * Float64(1.0 - 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 <= -1.7e-37)
		tmp = x * ((z + -1.0) / y_m);
	elseif ((x <= 1.4e-18) || ~((x <= 6.8e+157)))
		tmp = (x + 4.0) / y_m;
	else
		tmp = (x * (1.0 - z)) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.7e-37], N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 1.4e-18], N[Not[LessEqual[x, 6.8e+157]], $MachinePrecision]], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. fabs-sub 89.2%

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

      2. div-inv 89.0%

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

      3. cancel-sign-sub-inv 89.0%

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

      4. associate-*l/ 89.4%

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

      5. associate-*r/ 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. sub-neg 95.1%

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

      10. div-inv 95.3%

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

      11. fma-undefine 96.8%

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

      12. add-sqr-sqrt 56.6%

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

      13. fabs-sqr 56.6%

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

      14. add-sqr-sqrt 57.2%

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

      15. fma-undefine 55.6%

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

      16. associate-*r/ 51.3%

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

      17. associate-*l/ 52.6%

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

      18. div-inv 52.5%

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

   4. Applied egg-rr 55.6%

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

      1. associate-*r/ 51.3%

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

      2. sub-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in x around inf 53.8%

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

      1. associate-/l* 58.1%

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

      2. sub-neg 58.1%

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

      3. metadata-eval 58.1%

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

   9. Simplified 58.1%

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

   if -1.70000000000000009e-37 < x < 1.40000000000000006e-18 or 6.79999999999999958e157 < x

   1. Initial program 91.3%

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

      1. fabs-sub 91.3%

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

      2. associate-*l/ 94.6%

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

      3. associate-*r/ 89.0%

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

      4. fmm-def 91.7%

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

      5. distribute-neg-frac 91.7%

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

      6. +-commutative 91.7%

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

      7. distribute-neg-in 91.7%

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

      8. unsub-neg 91.7%

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

      9. metadata-eval 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in z around 0 80.9%

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

      1. associate-*r/ 80.9%

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

      2. distribute-lft-in 80.9%

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

      3. metadata-eval 80.9%

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

      4. neg-mul-1 80.9%

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

      5. sub-neg 80.9%

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

   7. Simplified 80.9%

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

      1. add-sqr-sqrt 80.4%

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

      2. sqrt-unprod 53.5%

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

      3. neg-fabs 53.5%

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

      4. neg-fabs 53.5%

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

      5. sqr-abs 53.5%

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

      6. distribute-frac-neg2 53.5%

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

      7. sub-neg 53.5%

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

      8. metadata-eval 53.5%

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

      9. distribute-neg-in 53.5%

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

      10. +-commutative 53.5%

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

      11. frac-2neg 53.5%

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

      12. distribute-frac-neg2 53.5%

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

      13. sub-neg 53.5%

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

      14. metadata-eval 53.5%

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

      15. distribute-neg-in 53.5%

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

      16. +-commutative 53.5%

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

      17. frac-2neg 53.5%

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

      18. sqrt-unprod 37.6%

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

      19. add-sqr-sqrt 38.8%

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

   9. Applied egg-rr 38.8%

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

   if 1.40000000000000006e-18 < x < 6.79999999999999958e157

   1. Initial program 97.5%

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

      1. add-sqr-sqrt 37.1%

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

      2. fabs-sqr 37.1%

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

      3. add-sqr-sqrt 37.8%

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

      4. sub-neg 37.8%

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

      5. +-commutative 37.8%

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

      6. associate-*l/ 37.9%

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

      7. associate-*r/ 40.1%

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

      8. frac-2neg 40.1%

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

      9. +-commutative 40.1%

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

      10. distribute-neg-in 40.1%

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

      11. metadata-eval 40.1%

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

      12. sub-neg 40.1%

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

      13. distribute-frac-neg2 40.1%

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

      14. distribute-neg-in 40.1%

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

      15. fma-undefine 40.1%

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

      16. neg-sub0 40.1%

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

      17. fma-undefine 40.1%

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

      18. associate-*r/ 37.9%

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

      19. associate-*l/ 37.8%

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

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in x around inf 37.0%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z + -1}{y}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-18} \lor \neg \left(x \leq 6.8 \cdot 10^{+157}\right):\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z + -1}{y\_m}\\ \mathbf{elif}\;x \leq 1460000:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.7e-37)
   (* x (/ (+ z -1.0) y_m))
   (if (<= x 1460000.0)
     (/ (- 4.0 (* x z)) y_m)
     (if (<= x 1.32e+154) (/ (* x (- 1.0 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 <= -1.7e-37) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 1460000.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else if (x <= 1.32e+154) {
		tmp = (x * (1.0 - 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 <= (-1.7d-37)) then
        tmp = x * ((z + (-1.0d0)) / y_m)
    else if (x <= 1460000.0d0) then
        tmp = (4.0d0 - (x * z)) / y_m
    else if (x <= 1.32d+154) then
        tmp = (x * (1.0d0 - 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 <= -1.7e-37) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 1460000.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else if (x <= 1.32e+154) {
		tmp = (x * (1.0 - 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 <= -1.7e-37:
		tmp = x * ((z + -1.0) / y_m)
	elif x <= 1460000.0:
		tmp = (4.0 - (x * z)) / y_m
	elif x <= 1.32e+154:
		tmp = (x * (1.0 - 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 <= -1.7e-37)
		tmp = Float64(x * Float64(Float64(z + -1.0) / y_m));
	elseif (x <= 1460000.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	elseif (x <= 1.32e+154)
		tmp = Float64(Float64(x * Float64(1.0 - 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 <= -1.7e-37)
		tmp = x * ((z + -1.0) / y_m);
	elseif (x <= 1460000.0)
		tmp = (4.0 - (x * z)) / y_m;
	elseif (x <= 1.32e+154)
		tmp = (x * (1.0 - 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, -1.7e-37], N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1460000.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. fabs-sub 89.2%

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

      2. div-inv 89.0%

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

      3. cancel-sign-sub-inv 89.0%

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

      4. associate-*l/ 89.4%

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

      5. associate-*r/ 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. sub-neg 95.1%

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

      10. div-inv 95.3%

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

      11. fma-undefine 96.8%

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

      12. add-sqr-sqrt 56.6%

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

      13. fabs-sqr 56.6%

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

      14. add-sqr-sqrt 57.2%

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

      15. fma-undefine 55.6%

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

      16. associate-*r/ 51.3%

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

      17. associate-*l/ 52.6%

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

      18. div-inv 52.5%

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

   4. Applied egg-rr 55.6%

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

      1. associate-*r/ 51.3%

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

      2. sub-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in x around inf 53.8%

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

      1. associate-/l* 58.1%

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

      2. sub-neg 58.1%

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

      3. metadata-eval 58.1%

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

   9. Simplified 58.1%

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

   if -1.70000000000000009e-37 < x < 1.46e6

   1. Initial program 96.1%

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

      1. add-sqr-sqrt 45.0%

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

      2. fabs-sqr 45.0%

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

      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/ 48.6%

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

      7. associate-*r/ 44.7%

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

      8. frac-2neg 44.7%

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

      9. +-commutative 44.7%

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

      10. distribute-neg-in 44.7%

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

      11. metadata-eval 44.7%

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

      12. sub-neg 44.7%

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

      13. distribute-frac-neg2 44.7%

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

      14. distribute-neg-in 44.7%

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

      15. fma-undefine 44.7%

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

      16. neg-sub0 44.7%

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

      17. fma-undefine 44.7%

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

      18. associate-*r/ 48.6%

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

      19. associate-*l/ 46.4%

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

   4. Applied egg-rr 48.6%

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

   5. Taylor expanded in x around 0 48.4%

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

   if 1.46e6 < x < 1.31999999999999998e154

   1. Initial program 96.8%

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

      1. add-sqr-sqrt 38.1%

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

      2. fabs-sqr 38.1%

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

      3. add-sqr-sqrt 38.8%

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

      4. sub-neg 38.8%

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

      5. +-commutative 38.8%

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

      6. associate-*l/ 39.0%

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

      7. associate-*r/ 41.7%

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

      8. frac-2neg 41.7%

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

      9. +-commutative 41.7%

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

      10. distribute-neg-in 41.7%

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

      11. metadata-eval 41.7%

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

      12. sub-neg 41.7%

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

      13. distribute-frac-neg2 41.7%

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

      14. distribute-neg-in 41.7%

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

      15. fma-undefine 41.7%

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

      16. neg-sub0 41.7%

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

      17. fma-undefine 41.7%

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

      18. associate-*r/ 39.0%

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

      19. associate-*l/ 38.8%

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

   4. Applied egg-rr 39.0%

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

   5. Taylor expanded in x around inf 38.4%

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

   if 1.31999999999999998e154 < x

   1. Initial program 75.7%

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

      1. fabs-sub 75.7%

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

      2. associate-*l/ 75.9%

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

      3. associate-*r/ 81.8%

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

      4. fmm-def 93.9%

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

      5. distribute-neg-frac 93.9%

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

      6. +-commutative 93.9%

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

      7. distribute-neg-in 93.9%

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

      8. unsub-neg 93.9%

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

      9. metadata-eval 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. distribute-lft-in 73.9%

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

      3. metadata-eval 73.9%

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

      4. neg-mul-1 73.9%

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

      5. sub-neg 73.9%

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

   7. Simplified 73.9%

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

      1. add-sqr-sqrt 73.6%

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

      2. sqrt-unprod 74.7%

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

      3. neg-fabs 74.7%

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

      4. neg-fabs 74.7%

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

      5. sqr-abs 74.7%

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

      6. distribute-frac-neg2 74.7%

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

      7. sub-neg 74.7%

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

      8. metadata-eval 74.7%

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

      9. distribute-neg-in 74.7%

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

      10. +-commutative 74.7%

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

      11. frac-2neg 74.7%

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

      12. distribute-frac-neg2 74.7%

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

      13. sub-neg 74.7%

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

      14. metadata-eval 74.7%

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

      15. distribute-neg-in 74.7%

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

      16. +-commutative 74.7%

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

      17. frac-2neg 74.7%

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

      18. sqrt-unprod 38.0%

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

      19. add-sqr-sqrt 38.4%

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

   9. Applied egg-rr 38.4%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z + -1}{y}\\ \mathbf{elif}\;x \leq 1460000:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{x}{\frac{y\_m}{z}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)))
   (if (<= x -1.7e-37)
     (- (* x (/ z y_m)) t_0)
     (if (<= x 980.0) (/ (- 4.0 (* x z)) y_m) (+ t_0 (/ x (/ y_m z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -1.7e-37) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= 980.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    if (x <= (-1.7d-37)) then
        tmp = (x * (z / y_m)) - t_0
    else if (x <= 980.0d0) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = t_0 + (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 t_0 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -1.7e-37) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= 980.0) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = t_0 + (x / (y_m / z));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	tmp = 0
	if x <= -1.7e-37:
		tmp = (x * (z / y_m)) - t_0
	elif x <= 980.0:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = t_0 + (x / (y_m / z))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	tmp = 0.0
	if (x <= -1.7e-37)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (x <= 980.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = Float64(t_0 + Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	tmp = 0.0;
	if (x <= -1.7e-37)
		tmp = (x * (z / y_m)) - t_0;
	elseif (x <= 980.0)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = t_0 + (x / (y_m / z));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[x, -1.7e-37], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 980.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(t$95$0 + N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. fabs-sub 89.2%

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

      2. div-inv 89.0%

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

      3. cancel-sign-sub-inv 89.0%

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

      4. associate-*l/ 89.4%

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

      5. associate-*r/ 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. sub-neg 95.1%

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

      10. div-inv 95.3%

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

      11. fma-undefine 96.8%

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

      12. add-sqr-sqrt 56.6%

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

      13. fabs-sqr 56.6%

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

      14. add-sqr-sqrt 57.2%

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

      15. fma-undefine 55.6%

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

      16. associate-*r/ 51.3%

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

      17. associate-*l/ 52.6%

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

      18. div-inv 52.5%

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

   4. Applied egg-rr 55.6%

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

   if -1.70000000000000009e-37 < x < 980

   1. Initial program 96.1%

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

      1. add-sqr-sqrt 44.9%

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

      2. fabs-sqr 44.9%

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

      3. add-sqr-sqrt 46.3%

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

      4. sub-neg 46.3%

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

      5. +-commutative 46.3%

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

      6. associate-*l/ 48.6%

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

      7. associate-*r/ 44.6%

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

      8. frac-2neg 44.6%

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

      9. +-commutative 44.6%

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

      10. distribute-neg-in 44.6%

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

      11. metadata-eval 44.6%

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

      12. sub-neg 44.6%

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

      13. distribute-frac-neg2 44.6%

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

      14. distribute-neg-in 44.6%

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

      15. fma-undefine 44.6%

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

      16. neg-sub0 44.6%

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

      17. fma-undefine 44.6%

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

      18. associate-*r/ 48.6%

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

      19. associate-*l/ 46.3%

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

   4. Applied egg-rr 48.6%

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

   5. Taylor expanded in x around 0 48.4%

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

   if 980 < x

   1. Initial program 86.8%

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

      1. add-sqr-sqrt 31.7%

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

      2. fabs-sqr 31.7%

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

      3. add-sqr-sqrt 32.3%

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

      4. associate-*l/ 32.5%

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

      5. associate-*r/ 35.2%

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

   4. Applied egg-rr 35.2%

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

      1. add-sqr-sqrt 13.3%

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

      2. sqrt-unprod 32.5%

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

      3. sqr-neg 32.5%

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

      4. sqrt-unprod 20.4%

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

      5. add-sqr-sqrt 56.3%

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

      6. distribute-frac-neg 56.3%

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

      7. distribute-rgt-neg-in 56.3%

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

      8. distribute-lft-neg-in 56.3%

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

      9. clear-num 56.3%

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

      10. un-div-inv 56.2%

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

   6. Applied egg-rr 56.2%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{x}{\frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;x \leq 5.05 \cdot 10^{+157}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)))
   (if (<= x -1.7e-37)
     (- (* x (/ z y_m)) t_0)
     (if (<= x 5.05e+157) (/ (- (+ x 4.0) (* x z)) y_m) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -1.7e-37) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= 5.05e+157) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = t_0;
	}
	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 = (x + 4.0d0) / y_m
    if (x <= (-1.7d-37)) then
        tmp = (x * (z / y_m)) - t_0
    else if (x <= 5.05d+157) then
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    else
        tmp = t_0
    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 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -1.7e-37) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= 5.05e+157) {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	tmp = 0
	if x <= -1.7e-37:
		tmp = (x * (z / y_m)) - t_0
	elif x <= 5.05e+157:
		tmp = ((x + 4.0) - (x * z)) / y_m
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	tmp = 0.0
	if (x <= -1.7e-37)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (x <= 5.05e+157)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	tmp = 0.0;
	if (x <= -1.7e-37)
		tmp = (x * (z / y_m)) - t_0;
	elseif (x <= 5.05e+157)
		tmp = ((x + 4.0) - (x * z)) / y_m;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[x, -1.7e-37], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 5.05e+157], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. fabs-sub 89.2%

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

      2. div-inv 89.0%

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

      3. cancel-sign-sub-inv 89.0%

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

      4. associate-*l/ 89.4%

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

      5. associate-*r/ 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. sub-neg 95.1%

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

      10. div-inv 95.3%

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

      11. fma-undefine 96.8%

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

      12. add-sqr-sqrt 56.6%

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

      13. fabs-sqr 56.6%

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

      14. add-sqr-sqrt 57.2%

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

      15. fma-undefine 55.6%

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

      16. associate-*r/ 51.3%

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

      17. associate-*l/ 52.6%

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

      18. div-inv 52.5%

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

   4. Applied egg-rr 55.6%

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

   if -1.70000000000000009e-37 < x < 5.04999999999999988e157

   1. Initial program 96.3%

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

      1. add-sqr-sqrt 43.5%

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

      2. fabs-sqr 43.5%

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

      3. add-sqr-sqrt 44.7%

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

      4. sub-neg 44.7%

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

      5. +-commutative 44.7%

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

      6. associate-*l/ 46.5%

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

      7. associate-*r/ 44.1%

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

      8. frac-2neg 44.1%

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

      9. +-commutative 44.1%

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

      10. distribute-neg-in 44.1%

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

      11. metadata-eval 44.1%

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

      12. sub-neg 44.1%

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

      13. distribute-frac-neg2 44.1%

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

      14. distribute-neg-in 44.1%

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

      15. fma-undefine 44.1%

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

      16. neg-sub0 44.1%

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

      17. fma-undefine 44.1%

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

      18. associate-*r/ 46.5%

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

      19. associate-*l/ 44.7%

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

   4. Applied egg-rr 46.6%

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

   if 5.04999999999999988e157 < x

   1. Initial program 75.7%

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

      1. fabs-sub 75.7%

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

      2. associate-*l/ 75.9%

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

      3. associate-*r/ 81.8%

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

      4. fmm-def 93.9%

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

      5. distribute-neg-frac 93.9%

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

      6. +-commutative 93.9%

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

      7. distribute-neg-in 93.9%

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

      8. unsub-neg 93.9%

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

      9. metadata-eval 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. distribute-lft-in 73.9%

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

      3. metadata-eval 73.9%

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

      4. neg-mul-1 73.9%

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

      5. sub-neg 73.9%

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

   7. Simplified 73.9%

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

      1. add-sqr-sqrt 73.6%

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

      2. sqrt-unprod 74.7%

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

      3. neg-fabs 74.7%

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

      4. neg-fabs 74.7%

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

      5. sqr-abs 74.7%

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

      6. distribute-frac-neg2 74.7%

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

      7. sub-neg 74.7%

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

      8. metadata-eval 74.7%

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

      9. distribute-neg-in 74.7%

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

      10. +-commutative 74.7%

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

      11. frac-2neg 74.7%

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

      12. distribute-frac-neg2 74.7%

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

      13. sub-neg 74.7%

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

      14. metadata-eval 74.7%

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

      15. distribute-neg-in 74.7%

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

      16. +-commutative 74.7%

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

      17. frac-2neg 74.7%

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

      18. sqrt-unprod 38.0%

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

      19. add-sqr-sqrt 38.4%

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

   9. Applied egg-rr 38.4%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;x \leq 5.05 \cdot 10^{+157}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z + -1}{y\_m}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.7e-37)
   (* x (/ (+ z -1.0) y_m))
   (if (<= x 8e+156) (/ (- (+ x 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 <= -1.7e-37) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 8e+156) {
		tmp = ((x + 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 <= (-1.7d-37)) then
        tmp = x * ((z + (-1.0d0)) / y_m)
    else if (x <= 8d+156) then
        tmp = ((x + 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 <= -1.7e-37) {
		tmp = x * ((z + -1.0) / y_m);
	} else if (x <= 8e+156) {
		tmp = ((x + 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 <= -1.7e-37:
		tmp = x * ((z + -1.0) / y_m)
	elif x <= 8e+156:
		tmp = ((x + 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 <= -1.7e-37)
		tmp = Float64(x * Float64(Float64(z + -1.0) / y_m));
	elseif (x <= 8e+156)
		tmp = Float64(Float64(Float64(x + 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 <= -1.7e-37)
		tmp = x * ((z + -1.0) / y_m);
	elseif (x <= 8e+156)
		tmp = ((x + 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, -1.7e-37], N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+156], N[(N[(N[(x + 4.0), $MachinePrecision] - 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 < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. fabs-sub 89.2%

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

      2. div-inv 89.0%

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

      3. cancel-sign-sub-inv 89.0%

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

      4. associate-*l/ 89.4%

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

      5. associate-*r/ 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. sub-neg 95.1%

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

      10. div-inv 95.3%

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

      11. fma-undefine 96.8%

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

      12. add-sqr-sqrt 56.6%

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

      13. fabs-sqr 56.6%

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

      14. add-sqr-sqrt 57.2%

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

      15. fma-undefine 55.6%

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

      16. associate-*r/ 51.3%

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

      17. associate-*l/ 52.6%

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

      18. div-inv 52.5%

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

   4. Applied egg-rr 55.6%

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

      1. associate-*r/ 51.3%

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

      2. sub-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in x around inf 53.8%

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

      1. associate-/l* 58.1%

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

      2. sub-neg 58.1%

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

      3. metadata-eval 58.1%

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

   9. Simplified 58.1%

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

   if -1.70000000000000009e-37 < x < 7.9999999999999999e156

   1. Initial program 96.3%

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

      1. add-sqr-sqrt 43.5%

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

      2. fabs-sqr 43.5%

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

      3. add-sqr-sqrt 44.7%

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

      4. sub-neg 44.7%

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

      5. +-commutative 44.7%

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

      6. associate-*l/ 46.5%

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

      7. associate-*r/ 44.1%

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

      8. frac-2neg 44.1%

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

      9. +-commutative 44.1%

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

      10. distribute-neg-in 44.1%

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

      11. metadata-eval 44.1%

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

      12. sub-neg 44.1%

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

      13. distribute-frac-neg2 44.1%

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

      14. distribute-neg-in 44.1%

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

      15. fma-undefine 44.1%

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

      16. neg-sub0 44.1%

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

      17. fma-undefine 44.1%

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

      18. associate-*r/ 46.5%

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

      19. associate-*l/ 44.7%

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

   4. Applied egg-rr 46.6%

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

   if 7.9999999999999999e156 < x

   1. Initial program 75.7%

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

      1. fabs-sub 75.7%

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

      2. associate-*l/ 75.9%

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

      3. associate-*r/ 81.8%

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

      4. fmm-def 93.9%

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

      5. distribute-neg-frac 93.9%

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

      6. +-commutative 93.9%

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

      7. distribute-neg-in 93.9%

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

      8. unsub-neg 93.9%

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

      9. metadata-eval 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. distribute-lft-in 73.9%

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

      3. metadata-eval 73.9%

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

      4. neg-mul-1 73.9%

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

      5. sub-neg 73.9%

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

   7. Simplified 73.9%

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

      1. add-sqr-sqrt 73.6%

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

      2. sqrt-unprod 74.7%

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

      3. neg-fabs 74.7%

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

      4. neg-fabs 74.7%

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

      5. sqr-abs 74.7%

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

      6. distribute-frac-neg2 74.7%

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

      7. sub-neg 74.7%

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

      8. metadata-eval 74.7%

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

      9. distribute-neg-in 74.7%

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

      10. +-commutative 74.7%

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

      11. frac-2neg 74.7%

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

      12. distribute-frac-neg2 74.7%

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

      13. sub-neg 74.7%

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

      14. metadata-eval 74.7%

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

      15. distribute-neg-in 74.7%

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

      16. +-commutative 74.7%

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

      17. frac-2neg 74.7%

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

      18. sqrt-unprod 38.0%

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

      19. add-sqr-sqrt 38.4%

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

   9. Applied egg-rr 38.4%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z + -1}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.7e-37) (* 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 <= -1.7e-37) {
		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 <= (-1.7d-37)) 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 <= -1.7e-37) {
		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 <= -1.7e-37:
		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 <= -1.7e-37)
		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 <= -1.7e-37)
		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, -1.7e-37], 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 < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. add-sqr-sqrt 36.7%

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

      2. fabs-sqr 36.7%

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

      3. add-sqr-sqrt 37.6%

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

      4. associate-*l/ 39.2%

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

      5. associate-*r/ 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. add-sqr-sqrt 18.7%

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

      2. sqrt-unprod 41.1%

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

      3. sqr-neg 41.1%

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

      4. sqrt-unprod 25.3%

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

      5. add-sqr-sqrt 50.1%

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

      6. distribute-frac-neg 50.1%

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

      7. distribute-rgt-neg-in 50.1%

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

      8. distribute-lft-neg-in 50.1%

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

      9. clear-num 50.0%

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

      10. un-div-inv 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in z around inf 25.7%

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

      1. associate-*r/ 30.0%

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

   9. Simplified 30.0%

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

   if -1.70000000000000009e-37 < x < 4

   1. Initial program 96.0%

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

      1. add-sqr-sqrt 44.4%

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

      2. fabs-sqr 44.4%

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

      3. add-sqr-sqrt 45.8%

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

      4. associate-*l/ 48.1%

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

      5. associate-*r/ 44.0%

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

   4. Applied egg-rr 44.0%

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

      1. add-sqr-sqrt 20.7%

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

      2. sqrt-unprod 38.3%

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

      3. sqr-neg 38.3%

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

      4. sqrt-unprod 22.5%

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

      5. add-sqr-sqrt 42.2%

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

      6. distribute-frac-neg 42.2%

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

      7. distribute-rgt-neg-in 42.2%

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

      8. distribute-lft-neg-in 42.2%

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

      9. clear-num 42.2%

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

      10. un-div-inv 42.6%

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

   6. Applied egg-rr 42.6%

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

   7. Taylor expanded in x around 0 37.7%

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

   if 4 < x

   1. Initial program 87.3%

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

      1. fabs-sub 87.3%

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

      2. associate-*l/ 84.9%

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

      3. associate-*r/ 91.5%

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

      4. fmm-def 97.1%

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

      5. distribute-neg-frac 97.1%

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

      6. +-commutative 97.1%

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

      7. distribute-neg-in 97.1%

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

      8. unsub-neg 97.1%

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

      9. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in z around 0 58.6%

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

      1. associate-*r/ 58.6%

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

      2. distribute-lft-in 58.6%

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

      3. metadata-eval 58.6%

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

      4. neg-mul-1 58.6%

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

      5. sub-neg 58.6%

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

   7. Simplified 58.6%

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

      1. add-sqr-sqrt 58.3%

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

      2. sqrt-unprod 66.7%

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

      3. neg-fabs 66.7%

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

      4. neg-fabs 66.7%

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

      5. sqr-abs 66.7%

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

      6. distribute-frac-neg2 66.7%

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

      7. sub-neg 66.7%

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

      8. metadata-eval 66.7%

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

      9. distribute-neg-in 66.7%

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

      10. +-commutative 66.7%

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

      11. frac-2neg 66.7%

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

      12. distribute-frac-neg2 66.7%

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

      13. sub-neg 66.7%

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

      14. metadata-eval 66.7%

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

      15. distribute-neg-in 66.7%

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

      16. +-commutative 66.7%

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

      17. frac-2neg 66.7%

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

      18. sqrt-unprod 30.8%

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

      19. add-sqr-sqrt 31.4%

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

      20. clear-num 31.3%

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

   9. Applied egg-rr 31.3%

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

   10. Taylor expanded in x around inf 31.1%

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

Alternative 13: 71.6% accurate, 9.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z + -1}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.7e-37) (* x (/ (+ z -1.0) 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 <= -1.7e-37) {
		tmp = x * ((z + -1.0) / 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 <= (-1.7d-37)) then
        tmp = x * ((z + (-1.0d0)) / 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 <= -1.7e-37) {
		tmp = x * ((z + -1.0) / 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 <= -1.7e-37:
		tmp = x * ((z + -1.0) / 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 <= -1.7e-37)
		tmp = Float64(x * Float64(Float64(z + -1.0) / 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 <= -1.7e-37)
		tmp = x * ((z + -1.0) / 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, -1.7e-37], N[(x * N[(N[(z + -1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. fabs-sub 89.2%

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

      2. div-inv 89.0%

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

      3. cancel-sign-sub-inv 89.0%

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

      4. associate-*l/ 89.4%

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

      5. associate-*r/ 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. sub-neg 95.1%

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

      10. div-inv 95.3%

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

      11. fma-undefine 96.8%

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

      12. add-sqr-sqrt 56.6%

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

      13. fabs-sqr 56.6%

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

      14. add-sqr-sqrt 57.2%

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

      15. fma-undefine 55.6%

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

      16. associate-*r/ 51.3%

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

      17. associate-*l/ 52.6%

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

      18. div-inv 52.5%

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

   4. Applied egg-rr 55.6%

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

      1. associate-*r/ 51.3%

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

      2. sub-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in x around inf 53.8%

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

      1. associate-/l* 58.1%

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

      2. sub-neg 58.1%

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

      3. metadata-eval 58.1%

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

   9. Simplified 58.1%

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

   if -1.70000000000000009e-37 < x

   1. Initial program 92.7%

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

      1. fabs-sub 92.7%

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

      2. associate-*l/ 94.3%

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

      3. associate-*r/ 91.4%

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

      4. fmm-def 93.5%

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

      5. distribute-neg-frac 93.5%

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

      6. +-commutative 93.5%

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

      7. distribute-neg-in 93.5%

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

      8. unsub-neg 93.5%

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

      9. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in z around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. distribute-lft-in 72.2%

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

      3. metadata-eval 72.2%

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

      4. neg-mul-1 72.2%

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

      5. sub-neg 72.2%

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

   7. Simplified 72.2%

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

      1. add-sqr-sqrt 71.7%

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

      2. sqrt-unprod 54.8%

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

      3. neg-fabs 54.8%

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

      4. neg-fabs 54.8%

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

      5. sqr-abs 54.8%

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

      6. distribute-frac-neg2 54.8%

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

      7. sub-neg 54.8%

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

      8. metadata-eval 54.8%

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

      9. distribute-neg-in 54.8%

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

      10. +-commutative 54.8%

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

      11. frac-2neg 54.8%

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

      12. distribute-frac-neg2 54.8%

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

      13. sub-neg 54.8%

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

      14. metadata-eval 54.8%

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

      15. distribute-neg-in 54.8%

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

      16. +-commutative 54.8%

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

      17. frac-2neg 54.8%

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

      18. sqrt-unprod 34.2%

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

      19. add-sqr-sqrt 35.3%

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

   9. Applied egg-rr 35.3%

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

Alternative 14: 70.1% 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 88.3%

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

      1. fabs-sub 88.3%

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

      2. div-inv 88.1%

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

      3. cancel-sign-sub-inv 88.1%

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

      4. associate-*l/ 88.5%

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

      5. associate-*r/ 94.7%

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

      6. +-commutative 94.7%

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

      7. distribute-neg-in 94.7%

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

      8. metadata-eval 94.7%

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

      9. sub-neg 94.7%

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

      10. div-inv 94.9%

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

      11. fma-undefine 96.6%

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

      12. add-sqr-sqrt 58.0%

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

      13. fabs-sqr 58.0%

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

      14. add-sqr-sqrt 58.6%

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

      15. fma-undefine 56.9%

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

      16. associate-*r/ 52.3%

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

      17. associate-*l/ 53.6%

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

      18. div-inv 53.5%

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

   4. Applied egg-rr 56.9%

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

   5. Taylor expanded in z around 0 36.4%

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

      1. mul-1-neg 36.4%

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

      2. *-lft-identity 36.4%

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

      3. associate-*l/ 36.4%

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

      4. *-commutative 36.4%

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

      5. distribute-rgt-in 36.4%

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

      6. +-commutative 36.4%

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

      7. associate-*l/ 36.5%

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

      8. *-lft-identity 36.5%

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

      9. distribute-frac-neg 36.5%

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

      10. +-commutative 36.5%

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

      11. mul-1-neg 36.5%

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

      12. distribute-lft-in 36.5%

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

      13. metadata-eval 36.5%

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

      14. mul-1-neg 36.5%

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

      15. unsub-neg 36.5%

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

   7. Simplified 36.5%

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

   if -4 < x

   1. Initial program 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate-*l/ 94.4%

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

      3. associate-*r/ 91.6%

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

      4. fmm-def 93.7%

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

      5. distribute-neg-frac 93.7%

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

      6. +-commutative 93.7%

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

      7. distribute-neg-in 93.7%

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

      8. unsub-neg 93.7%

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

      9. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in z around 0 70.9%

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

      1. associate-*r/ 70.9%

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

      2. distribute-lft-in 70.9%

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

      3. metadata-eval 70.9%

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

      4. neg-mul-1 70.9%

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

      5. sub-neg 70.9%

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

   7. Simplified 70.9%

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

      1. add-sqr-sqrt 70.4%

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

      2. sqrt-unprod 54.5%

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

      3. neg-fabs 54.5%

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

      4. neg-fabs 54.5%

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

      5. sqr-abs 54.5%

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

      6. distribute-frac-neg2 54.5%

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

      7. sub-neg 54.5%

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

      8. metadata-eval 54.5%

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

      9. distribute-neg-in 54.5%

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

      10. +-commutative 54.5%

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

      11. frac-2neg 54.5%

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

      12. distribute-frac-neg2 54.5%

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

      13. sub-neg 54.5%

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

      14. metadata-eval 54.5%

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

      15. distribute-neg-in 54.5%

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

      16. +-commutative 54.5%

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

      17. frac-2neg 54.5%

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

      18. sqrt-unprod 33.9%

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

      19. add-sqr-sqrt 35.0%

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

   9. Applied egg-rr 35.0%

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

Alternative 15: 60.8% accurate, 11.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.7e-37) (* 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 <= -1.7e-37) {
		tmp = 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 <= (-1.7d-37)) then
        tmp = 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 <= -1.7e-37) {
		tmp = 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 <= -1.7e-37:
		tmp = 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 <= -1.7e-37)
		tmp = Float64(x * Float64(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 <= -1.7e-37)
		tmp = 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, -1.7e-37], N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000009e-37

   1. Initial program 89.2%

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

      1. add-sqr-sqrt 36.7%

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

      2. fabs-sqr 36.7%

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

      3. add-sqr-sqrt 37.6%

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

      4. associate-*l/ 39.2%

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

      5. associate-*r/ 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. add-sqr-sqrt 18.7%

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

      2. sqrt-unprod 41.1%

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

      3. sqr-neg 41.1%

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

      4. sqrt-unprod 25.3%

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

      5. add-sqr-sqrt 50.1%

         \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{-z}}{y} \]

      6. distribute-frac-neg 50.1%

         \[\leadsto \frac{x + 4}{y} - x \cdot \color{blue}{\left(-\frac{z}{y}\right)} \]

      7. distribute-rgt-neg-in 50.1%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x \cdot \frac{z}{y}\right)} \]

      8. distribute-lft-neg-in 50.1%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x\right) \cdot \frac{z}{y}} \]

      9. clear-num 50.0%

         \[\leadsto \frac{x + 4}{y} - \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}} \]

      10. un-div-inv 50.1%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   6. Applied egg-rr 50.1%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   7. Taylor expanded in z around inf 25.7%

      \[\leadsto \color{blue}{\frac{x \cdot z}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 30.0%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y}} \]

   9. Simplified 30.0%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y}} \]

   if -1.70000000000000009e-37 < x

   1. Initial program 92.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 92.7%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 94.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 91.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 93.5%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 93.5%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 72.2%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 72.2%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 72.2%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 72.2%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 72.2%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 72.2%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 72.2%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 71.7%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right|} \cdot \sqrt{\left|\frac{-4 - x}{y}\right|}} \]

      2. sqrt-unprod 54.8%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right| \cdot \left|\frac{-4 - x}{y}\right|}} \]

      3. neg-fabs 54.8%

         \[\leadsto \sqrt{\color{blue}{\left|-\frac{-4 - x}{y}\right|} \cdot \left|\frac{-4 - x}{y}\right|} \]

      4. neg-fabs 54.8%

         \[\leadsto \sqrt{\left|-\frac{-4 - x}{y}\right| \cdot \color{blue}{\left|-\frac{-4 - x}{y}\right|}} \]

      5. sqr-abs 54.8%

         \[\leadsto \sqrt{\color{blue}{\left(-\frac{-4 - x}{y}\right) \cdot \left(-\frac{-4 - x}{y}\right)}} \]

      6. distribute-frac-neg2 54.8%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 - x}{-y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      7. sub-neg 54.8%

         \[\leadsto \sqrt{\frac{\color{blue}{-4 + \left(-x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      8. metadata-eval 54.8%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      9. distribute-neg-in 54.8%

         \[\leadsto \sqrt{\frac{\color{blue}{-\left(4 + x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      10. +-commutative 54.8%

          \[\leadsto \sqrt{\frac{-\color{blue}{\left(x + 4\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      11. frac-2neg 54.8%

          \[\leadsto \sqrt{\color{blue}{\frac{x + 4}{y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      12. distribute-frac-neg2 54.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 54.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-4 + \left(-x\right)}}{-y}} \]

      14. metadata-eval 54.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y}} \]

      15. distribute-neg-in 54.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-\left(4 + x\right)}}{-y}} \]

      16. +-commutative 54.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{-\color{blue}{\left(x + 4\right)}}{-y}} \]

      17. frac-2neg 54.8%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{x + 4}{y}}} \]

      18. sqrt-unprod 34.2%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y}} \cdot \sqrt{\frac{x + 4}{y}}} \]

      19. add-sqr-sqrt 35.3%

          \[\leadsto \color{blue}{\frac{x + 4}{y}} \]

   9. Applied egg-rr 35.3%

      \[\leadsto \color{blue}{\frac{x + 4}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 55.4% 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 93.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 41.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      2. fabs-sqr 41.7%

         \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      3. add-sqr-sqrt 42.9%

         \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      4. associate-*l/ 45.0%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      5. associate-*r/ 42.8%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

   4. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 20.0%

         \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{y} \]

      2. sqrt-unprod 39.3%

         \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{\sqrt{z \cdot z}}}{y} \]

      3. sqr-neg 39.3%

         \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}{y} \]

      4. sqrt-unprod 23.5%

         \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}{y} \]

      5. add-sqr-sqrt 45.0%

         \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{-z}}{y} \]

      6. distribute-frac-neg 45.0%

         \[\leadsto \frac{x + 4}{y} - x \cdot \color{blue}{\left(-\frac{z}{y}\right)} \]

      7. distribute-rgt-neg-in 45.0%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x \cdot \frac{z}{y}\right)} \]

      8. distribute-lft-neg-in 45.0%

         \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x\right) \cdot \frac{z}{y}} \]

      9. clear-num 45.0%

         \[\leadsto \frac{x + 4}{y} - \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}} \]

      10. un-div-inv 45.2%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   6. Applied egg-rr 45.2%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

   7. Taylor expanded in x around 0 26.2%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 87.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 87.3%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 84.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 91.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 97.1%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 97.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 97.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 97.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 97.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 97.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 97.1%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 58.6%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
   6. Step-by-step derivation

      1. associate-*r/ 58.6%

         \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

      2. distribute-lft-in 58.6%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

      3. metadata-eval 58.6%

         \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

      4. neg-mul-1 58.6%

         \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

      5. sub-neg 58.6%

         \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

   7. Simplified 58.6%

      \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 58.3%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right|} \cdot \sqrt{\left|\frac{-4 - x}{y}\right|}} \]

      2. sqrt-unprod 66.7%

         \[\leadsto \color{blue}{\sqrt{\left|\frac{-4 - x}{y}\right| \cdot \left|\frac{-4 - x}{y}\right|}} \]

      3. neg-fabs 66.7%

         \[\leadsto \sqrt{\color{blue}{\left|-\frac{-4 - x}{y}\right|} \cdot \left|\frac{-4 - x}{y}\right|} \]

      4. neg-fabs 66.7%

         \[\leadsto \sqrt{\left|-\frac{-4 - x}{y}\right| \cdot \color{blue}{\left|-\frac{-4 - x}{y}\right|}} \]

      5. sqr-abs 66.7%

         \[\leadsto \sqrt{\color{blue}{\left(-\frac{-4 - x}{y}\right) \cdot \left(-\frac{-4 - x}{y}\right)}} \]

      6. distribute-frac-neg2 66.7%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 - x}{-y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      7. sub-neg 66.7%

         \[\leadsto \sqrt{\frac{\color{blue}{-4 + \left(-x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      8. metadata-eval 66.7%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      9. distribute-neg-in 66.7%

         \[\leadsto \sqrt{\frac{\color{blue}{-\left(4 + x\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      10. +-commutative 66.7%

          \[\leadsto \sqrt{\frac{-\color{blue}{\left(x + 4\right)}}{-y} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      11. frac-2neg 66.7%

          \[\leadsto \sqrt{\color{blue}{\frac{x + 4}{y}} \cdot \left(-\frac{-4 - x}{y}\right)} \]

      12. distribute-frac-neg2 66.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{-4 - x}{-y}}} \]

      13. sub-neg 66.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-4 + \left(-x\right)}}{-y}} \]

      14. metadata-eval 66.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{\left(-4\right)} + \left(-x\right)}{-y}} \]

      15. distribute-neg-in 66.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{\color{blue}{-\left(4 + x\right)}}{-y}} \]

      16. +-commutative 66.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \frac{-\color{blue}{\left(x + 4\right)}}{-y}} \]

      17. frac-2neg 66.7%

          \[\leadsto \sqrt{\frac{x + 4}{y} \cdot \color{blue}{\frac{x + 4}{y}}} \]

      18. sqrt-unprod 30.8%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y}} \cdot \sqrt{\frac{x + 4}{y}}} \]

      19. add-sqr-sqrt 31.4%

          \[\leadsto \color{blue}{\frac{x + 4}{y}} \]

      20. clear-num 31.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   9. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x + 4}}} \]

   10. Taylor expanded in x around inf 31.1%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.0% 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 91.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 39.3%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   2. fabs-sqr 39.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   3. add-sqr-sqrt 40.3%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   4. associate-*l/ 41.8%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   5. associate-*r/ 41.1%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{x \cdot \frac{z}{y}} \]

4. Applied egg-rr 41.1%

   \[\leadsto \color{blue}{\frac{x + 4}{y} - x \cdot \frac{z}{y}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 18.4%

      \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{y} \]

   2. sqrt-unprod 37.8%

      \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{\sqrt{z \cdot z}}}{y} \]

   3. sqr-neg 37.8%

      \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}{y} \]

   4. sqrt-unprod 22.8%

      \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}{y} \]

   5. add-sqr-sqrt 47.9%

      \[\leadsto \frac{x + 4}{y} - x \cdot \frac{\color{blue}{-z}}{y} \]

   6. distribute-frac-neg 47.9%

      \[\leadsto \frac{x + 4}{y} - x \cdot \color{blue}{\left(-\frac{z}{y}\right)} \]

   7. distribute-rgt-neg-in 47.9%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x \cdot \frac{z}{y}\right)} \]

   8. distribute-lft-neg-in 47.9%

      \[\leadsto \frac{x + 4}{y} - \color{blue}{\left(-x\right) \cdot \frac{z}{y}} \]

   9. clear-num 47.9%

      \[\leadsto \frac{x + 4}{y} - \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}} \]

   10. un-div-inv 48.1%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

6. Applied egg-rr 48.1%

   \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{-x}{\frac{y}{z}}} \]

7. Taylor expanded in x around 0 19.8%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))