fabs fraction 1

Percentage Accurate: 91.9% → 99.7%

Time: 12.2s

Alternatives: 16

Speedup: 2.7×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.6 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}} - \frac{x + 4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 3.6e-43)
   (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x))))
   (fabs (- (/ x (/ y_m z)) (/ (+ x 4.0) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.6e-43) {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs(((x / (y_m / z)) - ((x + 4.0) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.6e-43)
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(Float64(Float64(x / Float64(y_m / z)) - Float64(Float64(x + 4.0) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 3.6e-43], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5999999999999999e-43

   1. Initial program 87.6%

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

   3. Simplified 98.3%

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

   if 3.5999999999999999e-43 < y

   1. Initial program 98.6%

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

      1. associate-*l/ 90.8%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := t\_0 - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{z}{\frac{y\_m}{x}} - t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (- t_0 (* z (/ x y_m)))))
   (if (<= t_1 -2e-243)
     (- (/ z (/ y_m x)) t_0)
     (if (<= t_1 INFINITY) t_1 (fabs (/ (- -4.0 x) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -2e-243) {
		tmp = (z / (y_m / x)) - t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

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 t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -2e-243) {
		tmp = (z / (y_m / x)) - t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = t_0 - (z * (x / y_m))
	tmp = 0
	if t_1 <= -2e-243:
		tmp = (z / (y_m / x)) - t_0
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(t_0 - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_1 <= -2e-243)
		tmp = Float64(Float64(z / Float64(y_m / x)) - t_0);
	elseif (t_1 <= Inf)
		tmp = t_1;
	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)
	t_0 = (x + 4.0) / y_m;
	t_1 = t_0 - (z * (x / y_m));
	tmp = 0.0;
	if (t_1 <= -2e-243)
		tmp = (z / (y_m / x)) - t_0;
	elseif (t_1 <= Inf)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-243], N[(N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -1.99999999999999999e-243

   1. Initial program 98.3%

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

      1. associate-*l/ 95.8%

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

      2. associate-*r/ 99.0%

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

      3. clear-num 99.0%

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

      4. un-div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. fabs-sub 99.1%

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

      2. div-inv 99.0%

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

      3. clear-num 99.0%

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

      4. associate-*r/ 95.8%

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

      5. div-sub 95.8%

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

      6. add-sqr-sqrt 93.5%

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

      7. fabs-sqr 93.5%

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

      8. add-sqr-sqrt 94.1%

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

      9. div-sub 94.1%

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

      10. associate-*r/ 97.3%

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

      11. clear-num 97.2%

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

      12. div-inv 97.3%

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

      13. associate-/r/ 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. clear-num 98.3%

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

      3. un-div-inv 98.3%

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

   8. Applied egg-rr 98.3%

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

   if -1.99999999999999999e-243 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.6%

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

      1. associate-*l/ 96.0%

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

      2. associate-*r/ 96.0%

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

      3. clear-num 96.0%

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

      4. un-div-inv 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. add-sqr-sqrt 93.0%

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

      2. fabs-sqr 93.0%

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

      3. add-sqr-sqrt 93.7%

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

      4. associate-/r/ 96.8%

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

   6. Applied egg-rr 96.8%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

      1. fabs-sub 0.0%

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

      2. associate-*l/ 50.0%

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

      3. associate-*r/ 50.0%

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

      4. fma-neg 70.0%

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

      5. distribute-neg-frac 70.0%

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

      6. +-commutative 70.0%

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

      7. distribute-neg-in 70.0%

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

      8. unsub-neg 70.0%

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

      9. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. sub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{z}{\frac{y}{x}} - \frac{x + 4}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq \infty:\\ \;\;\;\;\frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

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 ((t_0 - (z * (x / y_m))) <= ((double) INFINITY)) {
		tmp = fabs((t_0 - (z / (y_m / x))));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

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 ((t_0 - (z * (x / y_m))) <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs((t_0 - (z / (y_m / x))));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	tmp = 0
	if (t_0 - (z * (x / y_m))) <= math.inf:
		tmp = math.fabs((t_0 - (z / (y_m / x))))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	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 (Float64(t_0 - Float64(z * Float64(x / y_m))) <= Inf)
		tmp = abs(Float64(t_0 - Float64(z / Float64(y_m / x))));
	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)
	t_0 = (x + 4.0) / y_m;
	tmp = 0.0;
	if ((t_0 - (z * (x / y_m))) <= Inf)
		tmp = abs((t_0 - (z / (y_m / x))));
	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_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[Abs[N[(t$95$0 - N[(z / N[(y$95$m / x), $MachinePrecision]), $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 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

      2. clear-num 98.4%

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

      3. un-div-inv 98.5%

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

   4. Applied egg-rr 98.5%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

      1. fabs-sub 0.0%

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

      2. associate-*l/ 50.0%

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

      3. associate-*r/ 50.0%

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

      4. fma-neg 70.0%

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

      5. distribute-neg-frac 70.0%

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

      6. +-commutative 70.0%

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

      7. distribute-neg-in 70.0%

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

      8. unsub-neg 70.0%

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

      9. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. sub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.6%

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

Alternative 4: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - (z * (x / y_m));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

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) - (z * (x / y_m));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = abs(t_0);
	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)
	t_0 = ((x + 4.0) / y_m) - (z * (x / y_m));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = abs(t_0);
	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_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.5%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

      1. fabs-sub 0.0%

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

      2. associate-*l/ 50.0%

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

      3. associate-*r/ 50.0%

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

      4. fma-neg 70.0%

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

      5. distribute-neg-frac 70.0%

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

      6. +-commutative 70.0%

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

      7. distribute-neg-in 70.0%

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

      8. unsub-neg 70.0%

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

      9. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. sub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.6%

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

Alternative 5: 94.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := t\_0 - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{z}{\frac{y\_m}{x}} - t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (- t_0 (* z (/ x y_m)))))
   (if (<= t_1 -2e-243)
     (- (/ z (/ y_m x)) t_0)
     (if (<= t_1 INFINITY) t_1 (* x (/ (+ -1.0 z) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -2e-243) {
		tmp = (z / (y_m / x)) - t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x * ((-1.0 + z) / y_m);
	}
	return tmp;
}

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 t_1 = t_0 - (z * (x / y_m));
	double tmp;
	if (t_1 <= -2e-243) {
		tmp = (z / (y_m / x)) - t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = x * ((-1.0 + z) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = t_0 - (z * (x / y_m))
	tmp = 0
	if t_1 <= -2e-243:
		tmp = (z / (y_m / x)) - t_0
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = x * ((-1.0 + z) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(t_0 - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_1 <= -2e-243)
		tmp = Float64(Float64(z / Float64(y_m / x)) - t_0);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	t_1 = t_0 - (z * (x / y_m));
	tmp = 0.0;
	if (t_1 <= -2e-243)
		tmp = (z / (y_m / x)) - t_0;
	elseif (t_1 <= Inf)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-243], N[(N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -1.99999999999999999e-243

   1. Initial program 98.3%

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

      1. associate-*l/ 95.8%

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

      2. associate-*r/ 99.0%

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

      3. clear-num 99.0%

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

      4. un-div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. fabs-sub 99.1%

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

      2. div-inv 99.0%

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

      3. clear-num 99.0%

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

      4. associate-*r/ 95.8%

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

      5. div-sub 95.8%

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

      6. add-sqr-sqrt 93.5%

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

      7. fabs-sqr 93.5%

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

      8. add-sqr-sqrt 94.1%

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

      9. div-sub 94.1%

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

      10. associate-*r/ 97.3%

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

      11. clear-num 97.2%

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

      12. div-inv 97.3%

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

      13. associate-/r/ 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. clear-num 98.3%

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

      3. un-div-inv 98.3%

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

   8. Applied egg-rr 98.3%

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

   if -1.99999999999999999e-243 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.6%

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

      1. associate-*l/ 96.0%

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

      2. associate-*r/ 96.0%

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

      3. clear-num 96.0%

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

      4. un-div-inv 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. add-sqr-sqrt 93.0%

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

      2. fabs-sqr 93.0%

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

      3. add-sqr-sqrt 93.7%

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

      4. associate-/r/ 96.8%

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

   6. Applied egg-rr 96.8%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sub-neg 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-*l/ 20.0%

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

      7. associate-*r/ 20.0%

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

      8. frac-2neg 20.0%

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

      9. +-commutative 20.0%

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

      10. distribute-neg-in 20.0%

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

      11. metadata-eval 20.0%

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

      12. sub-neg 20.0%

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

      13. distribute-frac-neg2 20.0%

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

      14. distribute-neg-in 20.0%

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

      15. fma-undefine 25.0%

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

      16. neg-sub0 25.0%

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

      17. fma-undefine 20.0%

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

      18. associate-*r/ 20.0%

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

      19. associate-*l/ 0.0%

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

   4. Applied egg-rr 45.0%

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

   5. Taylor expanded in x around inf 45.0%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 45.0%

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

      2. sub-neg 45.0%

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

      3. metadata-eval 45.0%

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

   7. Simplified 45.0%

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

      1. add-sqr-sqrt 45.0%

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

      2. sqrt-unprod 100.0%

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

      3. sub0-neg 100.0%

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

      4. sub0-neg 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 55.0%

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

      7. add-sqr-sqrt 55.0%

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

      8. *-commutative 55.0%

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

   9. Applied egg-rr 55.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{z}{\frac{y}{x}} - \frac{x + 4}{y}\\ \mathbf{elif}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq \infty:\\ \;\;\;\;\frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m} - x \cdot \frac{z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{\left(-z\right) - -1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.6e-12)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 4.0)
     (- (/ 4.0 y_m) (* x (/ z y_m)))
     (/ x (/ y_m (- (- z) -1.0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 4.0) {
		tmp = (4.0 / y_m) - (x * (z / y_m));
	} else {
		tmp = x / (y_m / (-z - -1.0));
	}
	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) :: tmp
    if (x <= (-2.6d-12)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 4.0d0) then
        tmp = (4.0d0 / y_m) - (x * (z / y_m))
    else
        tmp = x / (y_m / (-z - (-1.0d0)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 4.0) {
		tmp = (4.0 / y_m) - (x * (z / y_m));
	} else {
		tmp = x / (y_m / (-z - -1.0));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.6e-12:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 4.0:
		tmp = (4.0 / y_m) - (x * (z / y_m))
	else:
		tmp = x / (y_m / (-z - -1.0))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.6e-12)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 4.0)
		tmp = Float64(Float64(4.0 / y_m) - Float64(x * Float64(z / y_m)));
	else
		tmp = Float64(x / Float64(y_m / Float64(Float64(-z) - -1.0)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.6e-12)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 4.0)
		tmp = (4.0 / y_m) - (x * (z / y_m));
	else
		tmp = x / (y_m / (-z - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.6e-12], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.0], N[(N[(4.0 / y$95$m), $MachinePrecision] - N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y$95$m / N[((-z) - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999983e-12

   1. Initial program 84.3%

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

      1. add-sqr-sqrt 37.4%

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

      2. fabs-sqr 37.4%

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

      3. add-sqr-sqrt 38.0%

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

      4. sub-neg 38.0%

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

      5. +-commutative 38.0%

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

      6. associate-*l/ 39.5%

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

      7. associate-*r/ 39.5%

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

      8. frac-2neg 39.5%

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

      9. +-commutative 39.5%

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

      10. distribute-neg-in 39.5%

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

      11. metadata-eval 39.5%

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

      12. sub-neg 39.5%

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

      13. distribute-frac-neg2 39.5%

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

      14. distribute-neg-in 39.5%

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

      15. fma-undefine 39.5%

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

      16. neg-sub0 39.5%

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

      17. fma-undefine 39.5%

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

      18. associate-*r/ 39.5%

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

      19. associate-*l/ 38.0%

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

   4. Applied egg-rr 42.6%

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

   5. Taylor expanded in x around inf 42.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.6%

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

      2. sub-neg 42.6%

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

      3. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-unprod 72.1%

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

      3. sub0-neg 72.1%

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

      4. sub0-neg 72.1%

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

      5. sqr-neg 72.1%

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

      6. sqrt-unprod 57.6%

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

      7. add-sqr-sqrt 57.9%

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

      8. *-commutative 57.9%

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

   9. Applied egg-rr 57.9%

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

   if -2.59999999999999983e-12 < x < 4

   1. Initial program 97.0%

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

      1. add-sqr-sqrt 53.8%

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

      2. fabs-sqr 53.8%

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

      3. add-sqr-sqrt 55.0%

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

      4. associate-*l/ 57.5%

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

      5. associate-*r/ 53.6%

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

   4. Applied egg-rr 53.6%

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

   5. Taylor expanded in x around 0 52.9%

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

   if 4 < x

   1. Initial program 86.4%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 39.5%

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

      4. sub-neg 39.5%

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

      5. +-commutative 39.5%

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

      6. associate-*l/ 37.2%

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

      7. associate-*r/ 43.6%

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

      8. frac-2neg 43.6%

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

      9. +-commutative 43.6%

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

      10. distribute-neg-in 43.6%

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

      11. metadata-eval 43.6%

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

      12. sub-neg 43.6%

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

      13. distribute-frac-neg2 43.6%

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

      14. distribute-neg-in 43.6%

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

      15. fma-undefine 44.9%

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

      16. neg-sub0 44.9%

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

      17. fma-undefine 43.6%

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

      18. associate-*r/ 37.2%

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

      19. associate-*l/ 39.5%

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

   4. Applied egg-rr 41.3%

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

   5. Taylor expanded in x around inf 39.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 46.0%

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

      2. sub-neg 46.0%

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

      3. metadata-eval 46.0%

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

   7. Simplified 46.0%

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

      1. clear-num 45.9%

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

      2. un-div-inv 45.9%

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

   9. Applied egg-rr 45.9%

      \[\leadsto 0 - \color{blue}{\frac{x}{\frac{y}{z + -1}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y} - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\left(-z\right) - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 1000000:\\ \;\;\;\;\frac{x - -4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{\left(-z\right) - -1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.6e-12)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 1000000.0) (/ (- x -4.0) y_m) (/ x (/ y_m (- (- z) -1.0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1000000.0) {
		tmp = (x - -4.0) / y_m;
	} else {
		tmp = x / (y_m / (-z - -1.0));
	}
	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) :: tmp
    if (x <= (-2.6d-12)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 1000000.0d0) then
        tmp = (x - (-4.0d0)) / y_m
    else
        tmp = x / (y_m / (-z - (-1.0d0)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1000000.0) {
		tmp = (x - -4.0) / y_m;
	} else {
		tmp = x / (y_m / (-z - -1.0));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.6e-12:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 1000000.0:
		tmp = (x - -4.0) / y_m
	else:
		tmp = x / (y_m / (-z - -1.0))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.6e-12)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 1000000.0)
		tmp = Float64(Float64(x - -4.0) / y_m);
	else
		tmp = Float64(x / Float64(y_m / Float64(Float64(-z) - -1.0)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.6e-12)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 1000000.0)
		tmp = (x - -4.0) / y_m;
	else
		tmp = x / (y_m / (-z - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.6e-12], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1000000.0], N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x / N[(y$95$m / N[((-z) - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999983e-12

   1. Initial program 84.3%

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

      1. add-sqr-sqrt 37.4%

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

      2. fabs-sqr 37.4%

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

      3. add-sqr-sqrt 38.0%

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

      4. sub-neg 38.0%

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

      5. +-commutative 38.0%

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

      6. associate-*l/ 39.5%

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

      7. associate-*r/ 39.5%

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

      8. frac-2neg 39.5%

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

      9. +-commutative 39.5%

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

      10. distribute-neg-in 39.5%

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

      11. metadata-eval 39.5%

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

      12. sub-neg 39.5%

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

      13. distribute-frac-neg2 39.5%

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

      14. distribute-neg-in 39.5%

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

      15. fma-undefine 39.5%

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

      16. neg-sub0 39.5%

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

      17. fma-undefine 39.5%

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

      18. associate-*r/ 39.5%

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

      19. associate-*l/ 38.0%

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

   4. Applied egg-rr 42.6%

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

   5. Taylor expanded in x around inf 42.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.6%

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

      2. sub-neg 42.6%

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

      3. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-unprod 72.1%

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

      3. sub0-neg 72.1%

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

      4. sub0-neg 72.1%

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

      5. sqr-neg 72.1%

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

      6. sqrt-unprod 57.6%

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

      7. add-sqr-sqrt 57.9%

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

      8. *-commutative 57.9%

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

   9. Applied egg-rr 57.9%

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

   if -2.59999999999999983e-12 < x < 1e6

   1. Initial program 97.1%

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

      1. fabs-sub 97.1%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.2%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in z around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. distribute-lft-in 75.4%

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

      3. metadata-eval 75.4%

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

      4. neg-mul-1 75.4%

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

      5. sub-neg 75.4%

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

   7. Simplified 75.4%

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

      1. add-sqr-sqrt 74.8%

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

      2. sqrt-unprod 45.2%

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

      3. neg-fabs 45.2%

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

      4. neg-fabs 45.2%

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

      5. sqr-abs 45.2%

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

      6. distribute-frac-neg2 45.2%

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

      7. sub-neg 45.2%

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

      8. metadata-eval 45.2%

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

      9. distribute-neg-in 45.2%

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

      10. +-commutative 45.2%

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

      11. frac-2neg 45.2%

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

      12. distribute-frac-neg2 45.2%

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

      13. sub-neg 45.2%

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

      14. metadata-eval 45.2%

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

      15. distribute-neg-in 45.2%

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

      16. +-commutative 45.2%

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

      17. frac-2neg 45.2%

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

      18. sqrt-unprod 41.6%

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

      19. add-sqr-sqrt 42.8%

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

      20. frac-2neg 42.8%

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

   9. Applied egg-rr 42.8%

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

   if 1e6 < x

   1. Initial program 85.8%

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

      1. add-sqr-sqrt 37.7%

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

      2. fabs-sqr 37.7%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 36.0%

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

      7. associate-*r/ 42.6%

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

      8. frac-2neg 42.6%

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

      9. +-commutative 42.6%

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

      10. distribute-neg-in 42.6%

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

      11. metadata-eval 42.6%

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

      12. sub-neg 42.6%

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

      13. distribute-frac-neg2 42.6%

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

      14. distribute-neg-in 42.6%

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

      15. fma-undefine 44.0%

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

      16. neg-sub0 44.0%

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

      17. fma-undefine 42.6%

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

      18. associate-*r/ 36.0%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.2%

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

   5. Taylor expanded in x around inf 40.1%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 46.8%

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

      2. sub-neg 46.8%

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

      3. metadata-eval 46.8%

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

   7. Simplified 46.8%

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

      1. clear-num 46.8%

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

      2. un-div-inv 46.8%

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

   9. Applied egg-rr 46.8%

      \[\leadsto 0 - \color{blue}{\frac{x}{\frac{y}{z + -1}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 1000000:\\ \;\;\;\;\frac{x - -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\left(-z\right) - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{-1 + z}{y\_m}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-15}:\\ \;\;\;\;x \cdot t\_0\\ \mathbf{elif}\;x \leq 490000:\\ \;\;\;\;\frac{x - -4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ -1.0 z) y_m)))
   (if (<= x -4e-15)
     (* x t_0)
     (if (<= x 490000.0) (/ (- x -4.0) y_m) (* t_0 (- x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (-1.0 + z) / y_m;
	double tmp;
	if (x <= -4e-15) {
		tmp = x * t_0;
	} else if (x <= 490000.0) {
		tmp = (x - -4.0) / y_m;
	} else {
		tmp = t_0 * -x;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) + z) / y_m
    if (x <= (-4d-15)) then
        tmp = x * t_0
    else if (x <= 490000.0d0) then
        tmp = (x - (-4.0d0)) / y_m
    else
        tmp = t_0 * -x
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = (-1.0 + z) / y_m;
	double tmp;
	if (x <= -4e-15) {
		tmp = x * t_0;
	} else if (x <= 490000.0) {
		tmp = (x - -4.0) / y_m;
	} else {
		tmp = t_0 * -x;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (-1.0 + z) / y_m
	tmp = 0
	if x <= -4e-15:
		tmp = x * t_0
	elif x <= 490000.0:
		tmp = (x - -4.0) / y_m
	else:
		tmp = t_0 * -x
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(-1.0 + z) / y_m)
	tmp = 0.0
	if (x <= -4e-15)
		tmp = Float64(x * t_0);
	elseif (x <= 490000.0)
		tmp = Float64(Float64(x - -4.0) / y_m);
	else
		tmp = Float64(t_0 * Float64(-x));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (-1.0 + z) / y_m;
	tmp = 0.0;
	if (x <= -4e-15)
		tmp = x * t_0;
	elseif (x <= 490000.0)
		tmp = (x - -4.0) / y_m;
	else
		tmp = t_0 * -x;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[x, -4e-15], N[(x * t$95$0), $MachinePrecision], If[LessEqual[x, 490000.0], N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(t$95$0 * (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000003e-15

   1. Initial program 84.3%

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

      1. add-sqr-sqrt 37.4%

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

      2. fabs-sqr 37.4%

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

      3. add-sqr-sqrt 38.0%

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

      4. sub-neg 38.0%

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

      5. +-commutative 38.0%

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

      6. associate-*l/ 39.5%

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

      7. associate-*r/ 39.5%

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

      8. frac-2neg 39.5%

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

      9. +-commutative 39.5%

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

      10. distribute-neg-in 39.5%

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

      11. metadata-eval 39.5%

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

      12. sub-neg 39.5%

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

      13. distribute-frac-neg2 39.5%

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

      14. distribute-neg-in 39.5%

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

      15. fma-undefine 39.5%

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

      16. neg-sub0 39.5%

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

      17. fma-undefine 39.5%

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

      18. associate-*r/ 39.5%

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

      19. associate-*l/ 38.0%

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

   4. Applied egg-rr 42.6%

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

   5. Taylor expanded in x around inf 42.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.6%

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

      2. sub-neg 42.6%

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

      3. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-unprod 72.1%

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

      3. sub0-neg 72.1%

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

      4. sub0-neg 72.1%

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

      5. sqr-neg 72.1%

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

      6. sqrt-unprod 57.6%

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

      7. add-sqr-sqrt 57.9%

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

      8. *-commutative 57.9%

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

   9. Applied egg-rr 57.9%

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

   if -4.0000000000000003e-15 < x < 4.9e5

   1. Initial program 97.1%

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

      1. fabs-sub 97.1%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.2%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in z around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. distribute-lft-in 75.4%

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

      3. metadata-eval 75.4%

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

      4. neg-mul-1 75.4%

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

      5. sub-neg 75.4%

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

   7. Simplified 75.4%

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

      1. add-sqr-sqrt 74.8%

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

      2. sqrt-unprod 45.2%

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

      3. neg-fabs 45.2%

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

      4. neg-fabs 45.2%

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

      5. sqr-abs 45.2%

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

      6. distribute-frac-neg2 45.2%

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

      7. sub-neg 45.2%

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

      8. metadata-eval 45.2%

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

      9. distribute-neg-in 45.2%

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

      10. +-commutative 45.2%

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

      11. frac-2neg 45.2%

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

      12. distribute-frac-neg2 45.2%

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

      13. sub-neg 45.2%

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

      14. metadata-eval 45.2%

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

      15. distribute-neg-in 45.2%

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

      16. +-commutative 45.2%

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

      17. frac-2neg 45.2%

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

      18. sqrt-unprod 41.6%

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

      19. add-sqr-sqrt 42.8%

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

      20. frac-2neg 42.8%

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

   9. Applied egg-rr 42.8%

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

   if 4.9e5 < x

   1. Initial program 85.8%

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

      1. add-sqr-sqrt 37.7%

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

      2. fabs-sqr 37.7%

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

      3. add-sqr-sqrt 38.4%

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

      4. sub-neg 38.4%

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

      5. +-commutative 38.4%

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

      6. associate-*l/ 36.0%

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

      7. associate-*r/ 42.6%

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

      8. frac-2neg 42.6%

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

      9. +-commutative 42.6%

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

      10. distribute-neg-in 42.6%

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

      11. metadata-eval 42.6%

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

      12. sub-neg 42.6%

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

      13. distribute-frac-neg2 42.6%

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

      14. distribute-neg-in 42.6%

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

      15. fma-undefine 44.0%

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

      16. neg-sub0 44.0%

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

      17. fma-undefine 42.6%

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

      18. associate-*r/ 36.0%

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

      19. associate-*l/ 38.4%

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

   4. Applied egg-rr 40.2%

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

   5. Taylor expanded in x around inf 40.1%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 46.8%

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

      2. sub-neg 46.8%

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

      3. metadata-eval 46.8%

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

   7. Simplified 46.8%

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

      1. sub0-neg 46.8%

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

      2. *-commutative 46.8%

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

      3. distribute-rgt-neg-in 46.8%

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

   9. Applied egg-rr 46.8%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 490000:\\ \;\;\;\;\frac{x - -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + z}{y} \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{-x}{y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{-4}{-y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.52) (/ (- x) 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 <= -1.52) {
		tmp = -x / 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 <= (-1.52d0)) then
        tmp = -x / 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 <= -1.52) {
		tmp = -x / 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 <= -1.52:
		tmp = -x / 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 <= -1.52)
		tmp = Float64(Float64(-x) / y_m);
	elseif (x <= 4.0)
		tmp = Float64(-4.0 / Float64(-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 <= -1.52)
		tmp = -x / 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, -1.52], N[((-x) / y$95$m), $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 < -1.52

   1. Initial program 84.0%

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

      1. fabs-sub 84.0%

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

      2. associate-*l/ 87.5%

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

      3. associate-*r/ 91.9%

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

      4. fma-neg 95.1%

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

      5. distribute-neg-frac 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. unsub-neg 95.1%

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

      9. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in z around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. distribute-lft-in 66.1%

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

      3. metadata-eval 66.1%

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

      4. neg-mul-1 66.1%

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

      5. sub-neg 66.1%

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

   7. Simplified 66.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. fabs-sqr 36.1%

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

      3. add-sqr-sqrt 36.5%

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

      4. div-inv 36.5%

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

   9. Applied egg-rr 36.5%

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

   10. Taylor expanded in x around inf 36.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   11. Step-by-step derivation

       1. neg-mul-1 36.5%

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

       2. distribute-neg-frac2 36.5%

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

   12. Simplified 36.5%

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

   if -1.52 < x < 4

   1. Initial program 97.0%

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

      1. add-sqr-sqrt 54.2%

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

      2. fabs-sqr 54.2%

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

      3. add-sqr-sqrt 55.4%

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

      4. sub-neg 55.4%

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

      5. +-commutative 55.4%

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

      6. associate-*l/ 57.9%

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

      7. associate-*r/ 54.0%

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

      8. frac-2neg 54.0%

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

      9. +-commutative 54.0%

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

      10. distribute-neg-in 54.0%

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

      11. metadata-eval 54.0%

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

      12. sub-neg 54.0%

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

      13. distribute-frac-neg2 54.0%

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

      14. distribute-neg-in 54.0%

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

      15. fma-undefine 53.9%

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

      16. neg-sub0 53.9%

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

      17. fma-undefine 54.0%

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

      18. associate-*r/ 57.9%

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

      19. associate-*l/ 55.4%

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

   4. Applied egg-rr 57.9%

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

   5. Taylor expanded in x around 0 41.1%

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

      1. sub0-neg 41.1%

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

      2. distribute-neg-frac2 41.1%

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

   7. Applied egg-rr 41.1%

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

   if 4 < x

   1. Initial program 86.4%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 39.5%

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

      4. sub-neg 39.5%

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

      5. +-commutative 39.5%

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

      6. associate-*l/ 37.2%

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

      7. associate-*r/ 43.6%

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

      8. frac-2neg 43.6%

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

      9. +-commutative 43.6%

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

      10. distribute-neg-in 43.6%

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

      11. metadata-eval 43.6%

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

      12. sub-neg 43.6%

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

      13. distribute-frac-neg2 43.6%

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

      14. distribute-neg-in 43.6%

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

      15. fma-undefine 44.9%

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

      16. neg-sub0 44.9%

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

      17. fma-undefine 43.6%

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

      18. associate-*r/ 37.2%

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

      19. associate-*l/ 39.5%

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

   4. Applied egg-rr 41.3%

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

   5. Taylor expanded in x around inf 39.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 46.0%

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

      2. sub-neg 46.0%

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

      3. metadata-eval 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in z around 0 22.8%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{-4}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{-1 + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.6e-12) (/ x (/ y_m (+ -1.0 z))) (/ (- (+ x 4.0) (* x z)) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x / (y_m / (-1.0 + z));
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.6d-12)) then
        tmp = x / (y_m / ((-1.0d0) + z))
    else
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x / (y_m / (-1.0 + z));
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.6e-12:
		tmp = x / (y_m / (-1.0 + z))
	else:
		tmp = ((x + 4.0) - (x * z)) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.6e-12)
		tmp = Float64(x / Float64(y_m / Float64(-1.0 + z)));
	else
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.6e-12)
		tmp = x / (y_m / (-1.0 + z));
	else
		tmp = ((x + 4.0) - (x * z)) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.6e-12], N[(x / N[(y$95$m / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999983e-12

   1. Initial program 84.3%

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

      1. add-sqr-sqrt 37.4%

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

      2. fabs-sqr 37.4%

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

      3. add-sqr-sqrt 38.0%

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

      4. sub-neg 38.0%

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

      5. +-commutative 38.0%

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

      6. associate-*l/ 39.5%

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

      7. associate-*r/ 39.5%

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

      8. frac-2neg 39.5%

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

      9. +-commutative 39.5%

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

      10. distribute-neg-in 39.5%

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

      11. metadata-eval 39.5%

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

      12. sub-neg 39.5%

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

      13. distribute-frac-neg2 39.5%

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

      14. distribute-neg-in 39.5%

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

      15. fma-undefine 39.5%

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

      16. neg-sub0 39.5%

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

      17. fma-undefine 39.5%

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

      18. associate-*r/ 39.5%

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

      19. associate-*l/ 38.0%

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

   4. Applied egg-rr 42.6%

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

   5. Taylor expanded in x around inf 42.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.6%

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

      2. sub-neg 42.6%

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

      3. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 0.8%

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

      3. sqr-neg 0.8%

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

      4. sub0-neg 0.8%

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

      5. sub0-neg 0.8%

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

      6. sqrt-unprod 0.3%

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

      7. add-sqr-sqrt 57.9%

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

      8. cancel-sign-sub-inv 57.9%

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

      9. +-lft-identity 57.9%

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

      10. clear-num 57.9%

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

      11. un-div-inv 58.0%

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

   9. Applied egg-rr 58.0%

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

   if -2.59999999999999983e-12 < x

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 94.4%

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

      3. clear-num 94.3%

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

      4. un-div-inv 94.4%

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

   4. Applied egg-rr 94.4%

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

      1. add-sqr-sqrt 48.8%

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

      2. fabs-sqr 48.8%

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

      3. add-sqr-sqrt 49.7%

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

      4. associate-/r/ 49.1%

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

   6. Applied egg-rr 49.1%

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

      1. associate-*l/ 49.7%

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

      2. sub-div 51.2%

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

   8. Applied egg-rr 51.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.6e-12) (* x (/ (+ -1.0 z) y_m)) (/ (- (+ x 4.0) (* x z)) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.6d-12)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else
        tmp = ((x + 4.0d0) - (x * z)) / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.6e-12:
		tmp = x * ((-1.0 + z) / y_m)
	else:
		tmp = ((x + 4.0) - (x * z)) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.6e-12)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	else
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.6e-12)
		tmp = x * ((-1.0 + z) / y_m);
	else
		tmp = ((x + 4.0) - (x * z)) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.6e-12], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999983e-12

   1. Initial program 84.3%

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

      1. add-sqr-sqrt 37.4%

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

      2. fabs-sqr 37.4%

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

      3. add-sqr-sqrt 38.0%

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

      4. sub-neg 38.0%

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

      5. +-commutative 38.0%

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

      6. associate-*l/ 39.5%

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

      7. associate-*r/ 39.5%

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

      8. frac-2neg 39.5%

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

      9. +-commutative 39.5%

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

      10. distribute-neg-in 39.5%

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

      11. metadata-eval 39.5%

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

      12. sub-neg 39.5%

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

      13. distribute-frac-neg2 39.5%

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

      14. distribute-neg-in 39.5%

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

      15. fma-undefine 39.5%

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

      16. neg-sub0 39.5%

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

      17. fma-undefine 39.5%

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

      18. associate-*r/ 39.5%

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

      19. associate-*l/ 38.0%

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

   4. Applied egg-rr 42.6%

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

   5. Taylor expanded in x around inf 42.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.6%

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

      2. sub-neg 42.6%

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

      3. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-unprod 72.1%

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

      3. sub0-neg 72.1%

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

      4. sub0-neg 72.1%

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

      5. sqr-neg 72.1%

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

      6. sqrt-unprod 57.6%

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

      7. add-sqr-sqrt 57.9%

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

      8. *-commutative 57.9%

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

   9. Applied egg-rr 57.9%

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

   if -2.59999999999999983e-12 < x

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 94.4%

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

      3. clear-num 94.3%

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

      4. un-div-inv 94.4%

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

   4. Applied egg-rr 94.4%

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

      1. add-sqr-sqrt 48.8%

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

      2. fabs-sqr 48.8%

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

      3. add-sqr-sqrt 49.7%

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

      4. associate-/r/ 49.1%

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

   6. Applied egg-rr 49.1%

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

      1. associate-*l/ 49.7%

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

      2. sub-div 51.2%

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

   8. Applied egg-rr 51.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.0% accurate, 9.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.7e-14) (* 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 <= -2.7e-14) {
		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 <= (-2.7d-14)) 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 <= -2.7e-14) {
		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 <= -2.7e-14:
		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 <= -2.7e-14)
		tmp = Float64(x * Float64(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 <= -2.7e-14)
		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, -2.7e-14], N[(x * N[(N[(-1.0 + z), $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 < -2.6999999999999999e-14

   1. Initial program 84.3%

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

      1. add-sqr-sqrt 37.4%

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

      2. fabs-sqr 37.4%

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

      3. add-sqr-sqrt 38.0%

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

      4. sub-neg 38.0%

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

      5. +-commutative 38.0%

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

      6. associate-*l/ 39.5%

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

      7. associate-*r/ 39.5%

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

      8. frac-2neg 39.5%

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

      9. +-commutative 39.5%

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

      10. distribute-neg-in 39.5%

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

      11. metadata-eval 39.5%

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

      12. sub-neg 39.5%

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

      13. distribute-frac-neg2 39.5%

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

      14. distribute-neg-in 39.5%

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

      15. fma-undefine 39.5%

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

      16. neg-sub0 39.5%

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

      17. fma-undefine 39.5%

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

      18. associate-*r/ 39.5%

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

      19. associate-*l/ 38.0%

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

   4. Applied egg-rr 42.6%

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

   5. Taylor expanded in x around inf 42.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.6%

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

      2. sub-neg 42.6%

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

      3. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-unprod 72.1%

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

      3. sub0-neg 72.1%

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

      4. sub0-neg 72.1%

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

      5. sqr-neg 72.1%

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

      6. sqrt-unprod 57.6%

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

      7. add-sqr-sqrt 57.9%

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

      8. *-commutative 57.9%

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

   9. Applied egg-rr 57.9%

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

   if -2.6999999999999999e-14 < 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/ 93.9%

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

      3. associate-*r/ 94.4%

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

      4. fma-neg 95.4%

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

      5. distribute-neg-frac 95.4%

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

      6. +-commutative 95.4%

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

      7. distribute-neg-in 95.4%

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

      8. unsub-neg 95.4%

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

      9. metadata-eval 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 71.3%

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

      1. associate-*r/ 71.3%

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

      2. distribute-lft-in 71.3%

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

      3. metadata-eval 71.3%

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

      4. neg-mul-1 71.3%

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

      5. sub-neg 71.3%

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

   7. Simplified 71.3%

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

      1. add-sqr-sqrt 70.8%

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

      2. sqrt-unprod 48.5%

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

      3. neg-fabs 48.5%

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

      4. neg-fabs 48.5%

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

      5. sqr-abs 48.5%

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

      6. distribute-frac-neg2 48.5%

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

      7. sub-neg 48.5%

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

      8. metadata-eval 48.5%

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

      9. distribute-neg-in 48.5%

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

      10. +-commutative 48.5%

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

      11. frac-2neg 48.5%

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

      12. distribute-frac-neg2 48.5%

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

      13. sub-neg 48.5%

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

      14. metadata-eval 48.5%

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

      15. distribute-neg-in 48.5%

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

      16. +-commutative 48.5%

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

      17. frac-2neg 48.5%

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

      18. sqrt-unprod 34.4%

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

      19. add-sqr-sqrt 35.4%

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

      20. frac-2neg 35.4%

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

   9. Applied egg-rr 35.4%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.7% accurate, 11.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.0) (/ (- 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 = -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 = -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 = -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 = -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(-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 = -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[((-x) / 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 84.0%

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

      1. fabs-sub 84.0%

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

      2. associate-*l/ 87.5%

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

      3. associate-*r/ 91.9%

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

      4. fma-neg 95.1%

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

      5. distribute-neg-frac 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. unsub-neg 95.1%

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

      9. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in z around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. distribute-lft-in 66.1%

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

      3. metadata-eval 66.1%

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

      4. neg-mul-1 66.1%

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

      5. sub-neg 66.1%

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

   7. Simplified 66.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. fabs-sqr 36.1%

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

      3. add-sqr-sqrt 36.5%

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

      4. div-inv 36.5%

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

   9. Applied egg-rr 36.5%

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

   10. Taylor expanded in x around inf 36.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   11. Step-by-step derivation

       1. neg-mul-1 36.5%

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

       2. distribute-neg-frac2 36.5%

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

   12. Simplified 36.5%

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

   if -4 < x

   1. Initial program 93.0%

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

      1. fabs-sub 93.0%

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

      2. associate-*l/ 93.9%

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

      3. associate-*r/ 94.4%

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

      4. fma-neg 95.4%

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

      5. distribute-neg-frac 95.4%

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

      6. +-commutative 95.4%

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

      7. distribute-neg-in 95.4%

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

      8. unsub-neg 95.4%

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

      9. metadata-eval 95.4%

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

   3. Simplified 95.4%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   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 48.2%

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

      3. neg-fabs 48.2%

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

      4. neg-fabs 48.2%

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

      5. sqr-abs 48.2%

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

      6. distribute-frac-neg2 48.2%

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

      7. sub-neg 48.2%

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

      8. metadata-eval 48.2%

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

      9. distribute-neg-in 48.2%

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

      10. +-commutative 48.2%

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

      11. frac-2neg 48.2%

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

      12. distribute-frac-neg2 48.2%

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

      13. sub-neg 48.2%

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

      14. metadata-eval 48.2%

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

      15. distribute-neg-in 48.2%

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

      16. +-commutative 48.2%

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

      17. frac-2neg 48.2%

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

      18. sqrt-unprod 34.2%

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

      19. add-sqr-sqrt 35.2%

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

      20. frac-2neg 35.2%

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

   9. Applied egg-rr 35.2%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.7% accurate, 12.3× 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 / Float64(-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 92.5%

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

      1. add-sqr-sqrt 48.1%

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

      2. fabs-sqr 48.1%

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

      3. add-sqr-sqrt 49.0%

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

      4. sub-neg 49.0%

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

      5. +-commutative 49.0%

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

      6. associate-*l/ 51.2%

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

      7. associate-*r/ 48.6%

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

      8. frac-2neg 48.6%

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

      9. +-commutative 48.6%

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

      10. distribute-neg-in 48.6%

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

      11. metadata-eval 48.6%

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

      12. sub-neg 48.6%

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

      13. distribute-frac-neg2 48.6%

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

      14. distribute-neg-in 48.6%

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

      15. fma-undefine 48.6%

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

      16. neg-sub0 48.6%

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

      17. fma-undefine 48.6%

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

      18. associate-*r/ 51.2%

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

      19. associate-*l/ 49.0%

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

   4. Applied egg-rr 52.3%

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

   5. Taylor expanded in x around 0 28.1%

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

      1. sub0-neg 28.1%

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

      2. distribute-neg-frac2 28.1%

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

   7. Applied egg-rr 28.1%

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

   if 4 < x

   1. Initial program 86.4%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 39.5%

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

      4. sub-neg 39.5%

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

      5. +-commutative 39.5%

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

      6. associate-*l/ 37.2%

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

      7. associate-*r/ 43.6%

         \[\leadsto \left(-\color{blue}{x \cdot \frac{z}{y}}\right) + \frac{x + 4}{y} \]

      8. frac-2neg 43.6%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\frac{-\left(x + 4\right)}{-y}} \]

      9. +-commutative 43.6%

         \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{-\color{blue}{\left(4 + x\right)}}{-y} \]

      10. distribute-neg-in 43.6%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{-y} \]

      11. metadata-eval 43.6%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4} + \left(-x\right)}{-y} \]

      12. sub-neg 43.6%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4 - x}}{-y} \]

      13. distribute-frac-neg2 43.6%

          \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\left(-\frac{-4 - x}{y}\right)} \]

      14. distribute-neg-in 43.6%

          \[\leadsto \color{blue}{-\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      15. fma-undefine 44.9%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      16. neg-sub0 44.9%

          \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      17. fma-undefine 43.6%

          \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

      18. associate-*r/ 37.2%

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

      19. associate-*l/ 39.5%

          \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

   4. Applied egg-rr 41.3%

      \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

   5. Taylor expanded in x around inf 39.6%

      \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   6. Step-by-step derivation

      1. associate-/l* 46.0%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 46.0%

         \[\leadsto 0 - x \cdot \frac{\color{blue}{z + \left(-1\right)}}{y} \]

      3. metadata-eval 46.0%

         \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

   7. Simplified 46.0%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]

   8. Taylor expanded in z around 0 22.8%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 18.0% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{x}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ x y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return x / 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 = x / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return x / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return x / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(x / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = x / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(x / y$95$m), $MachinePrecision]

Derivation

1. Initial program 90.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 45.4%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   2. fabs-sqr 45.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   3. add-sqr-sqrt 46.3%

      \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   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/ 47.1%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot z}{y}}\right) + \frac{x + 4}{y} \]

   7. associate-*r/ 47.2%

      \[\leadsto \left(-\color{blue}{x \cdot \frac{z}{y}}\right) + \frac{x + 4}{y} \]

   8. frac-2neg 47.2%

      \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\frac{-\left(x + 4\right)}{-y}} \]

   9. +-commutative 47.2%

      \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{-\color{blue}{\left(4 + x\right)}}{-y} \]

   10. distribute-neg-in 47.2%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{-y} \]

   11. metadata-eval 47.2%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4} + \left(-x\right)}{-y} \]

   12. sub-neg 47.2%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \frac{\color{blue}{-4 - x}}{-y} \]

   13. distribute-frac-neg2 47.2%

       \[\leadsto \left(-x \cdot \frac{z}{y}\right) + \color{blue}{\left(-\frac{-4 - x}{y}\right)} \]

   14. distribute-neg-in 47.2%

       \[\leadsto \color{blue}{-\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

   15. fma-undefine 47.6%

       \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   16. neg-sub0 47.6%

       \[\leadsto \color{blue}{0 - \mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   17. fma-undefine 47.2%

       \[\leadsto 0 - \color{blue}{\left(x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right)} \]

   18. associate-*r/ 47.1%

       \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right) \]

   19. associate-*l/ 46.3%

       \[\leadsto 0 - \left(\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right) \]

4. Applied egg-rr 49.1%

   \[\leadsto \color{blue}{0 - \frac{x \cdot z - \left(x + 4\right)}{y}} \]

5. Taylor expanded in x around inf 30.3%

   \[\leadsto 0 - \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
6. Step-by-step derivation

   1. associate-/l* 31.1%

      \[\leadsto 0 - \color{blue}{x \cdot \frac{z - 1}{y}} \]

   2. sub-neg 31.1%

      \[\leadsto 0 - x \cdot \frac{\color{blue}{z + \left(-1\right)}}{y} \]

   3. metadata-eval 31.1%

      \[\leadsto 0 - x \cdot \frac{z + \color{blue}{-1}}{y} \]

7. Simplified 31.1%

   \[\leadsto 0 - \color{blue}{x \cdot \frac{z + -1}{y}} \]

8. Taylor expanded in z around 0 16.0%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Add Preprocessing

Alternative 16: 1.7% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{-4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ -4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return -4.0 / y_m;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = (-4.0d0) / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return -4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return -4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(-4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = -4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(-4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 90.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 90.8%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 92.3%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 93.8%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fma-neg 95.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 95.3%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 69.7%

   \[\leadsto \left|\color{blue}{-1 \cdot \frac{4 + x}{y}}\right| \]
6. Step-by-step derivation

   1. associate-*r/ 69.7%

      \[\leadsto \left|\color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}}\right| \]

   2. distribute-lft-in 69.7%

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y}\right| \]

   3. metadata-eval 69.7%

      \[\leadsto \left|\frac{\color{blue}{-4} + -1 \cdot x}{y}\right| \]

   4. neg-mul-1 69.7%

      \[\leadsto \left|\frac{-4 + \color{blue}{\left(-x\right)}}{y}\right| \]

   5. sub-neg 69.7%

      \[\leadsto \left|\frac{\color{blue}{-4 - x}}{y}\right| \]

7. Simplified 69.7%

   \[\leadsto \left|\color{blue}{\frac{-4 - x}{y}}\right| \]
8. Step-by-step derivation

   1. add-sqr-sqrt 36.2%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{-4 - x}{y}} \cdot \sqrt{\frac{-4 - x}{y}}}\right| \]

   2. fabs-sqr 36.2%

      \[\leadsto \color{blue}{\sqrt{\frac{-4 - x}{y}} \cdot \sqrt{\frac{-4 - x}{y}}} \]

   3. add-sqr-sqrt 37.1%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   4. div-inv 37.1%

      \[\leadsto \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

9. Applied egg-rr 37.1%

   \[\leadsto \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

10. Taylor expanded in x around 0 17.6%

    \[\leadsto \color{blue}{\frac{-4}{y}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))