fabs fraction 1

Percentage Accurate: 92.0% → 99.7%

Time: 10.5s

Alternatives: 15

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.0% 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

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-52)
   (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x))))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-52) {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e-52)
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 2e-52], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-52

   1. Initial program 89.7%

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

   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 2e-52 < y

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 93.1%

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

      3. associate-*r/ 99.9%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.42 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.42e-52)
   (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x))))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.42e-52) {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.42e-52)
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.42e-52], 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[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4200000000000001e-52

   1. Initial program 89.7%

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

   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 1.4200000000000001e-52 < y

   1. Initial program 95.9%

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

      1. associate-*l/ 93.1%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

Alternative 3: 94.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 98.7%

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

      1. *-commutative 98.7%

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

      2. clear-num 98.7%

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

      3. un-div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

   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/ 21.1%

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

      3. associate-*r/ 21.1%

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

      4. fmm-def 52.6%

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

      5. distribute-neg-frac 52.6%

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

      6. +-commutative 52.6%

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

      7. distribute-neg-in 52.6%

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

      8. unsub-neg 52.6%

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

      9. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

      1. fma-undefine 21.1%

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

      2. associate-*r/ 21.1%

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

      3. associate-*l/ 0.0%

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

      4. div-inv 0.0%

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

      5. sub-neg 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-neg-in 0.0%

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

      8. +-commutative 0.0%

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

      9. cancel-sign-sub-inv 0.0%

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

      10. div-inv 0.0%

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

      11. fabs-sub 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. fabs-sqr 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. associate-*l/ 5.3%

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

      16. sub-div 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. Taylor expanded in z around 0 31.6%

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

4. Final simplification 93.8%

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

Alternative 4: 95.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= ((double) INFINITY)) {
		tmp = fabs((t_1 - t_0));
	} else {
		tmp = t_0;
	}
	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 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs((t_1 - t_0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	t_1 = z * (x / y_m);
	tmp = 0.0;
	if ((t_0 - t_1) <= Inf)
		tmp = abs((t_1 - t_0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - t$95$1), $MachinePrecision], Infinity], N[Abs[N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], t$95$0]]]

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 98.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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/ 21.1%

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

      3. associate-*r/ 21.1%

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

      4. fmm-def 52.6%

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

      5. distribute-neg-frac 52.6%

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

      6. +-commutative 52.6%

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

      7. distribute-neg-in 52.6%

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

      8. unsub-neg 52.6%

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

      9. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

      1. fma-undefine 21.1%

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

      2. associate-*r/ 21.1%

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

      3. associate-*l/ 0.0%

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

      4. div-inv 0.0%

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

      5. sub-neg 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-neg-in 0.0%

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

      8. +-commutative 0.0%

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

      9. cancel-sign-sub-inv 0.0%

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

      10. div-inv 0.0%

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

      11. fabs-sub 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. fabs-sqr 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. associate-*l/ 5.3%

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

      16. sub-div 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. Taylor expanded in z around 0 31.6%

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

4. Final simplification 93.7%

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

Alternative 5: 79.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.55e+36)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 2.7e+89) (/ (- (+ x 4.0) (* 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 <= -2.55e+36) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 2.7e+89) {
		tmp = ((x + 4.0) - (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 <= (-2.55d+36)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 2.7d+89) then
        tmp = ((x + 4.0d0) - (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 <= -2.55e+36) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 2.7e+89) {
		tmp = ((x + 4.0) - (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 <= -2.55e+36:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 2.7e+89:
		tmp = ((x + 4.0) - (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 <= -2.55e+36)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 2.7e+89)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	else
		tmp = Float64(x / Float64(y_m / 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 <= -2.55e+36)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 2.7e+89)
		tmp = ((x + 4.0) - (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, -2.55e+36], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e+89], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $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 < -2.54999999999999986e36

   1. Initial program 83.6%

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

      1. fabs-sub 83.6%

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

      2. associate-*l/ 83.9%

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

      3. associate-*r/ 87.7%

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

      4. fmm-def 91.7%

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

      5. distribute-neg-frac 91.7%

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

      6. +-commutative 91.7%

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

      7. distribute-neg-in 91.7%

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

      8. unsub-neg 91.7%

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

      9. metadata-eval 91.7%

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

   3. Simplified 91.7%

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

      1. add-sqr-sqrt 44.7%

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

      2. fabs-sqr 44.7%

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

      3. add-sqr-sqrt 45.2%

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

      4. fma-undefine 45.2%

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

      5. associate-*r/ 41.4%

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

      6. associate-*l/ 43.1%

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

      7. div-inv 43.0%

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

      8. sub-neg 43.0%

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

      9. metadata-eval 43.0%

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

      10. distribute-neg-in 43.0%

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

      11. +-commutative 43.0%

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

      12. cancel-sign-sub-inv 43.0%

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

      13. div-inv 43.1%

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

      14. associate-*l/ 41.4%

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

      15. sub-div 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 45.4%

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

      1. associate-/l* 49.2%

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

      2. sub-neg 49.2%

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

      3. metadata-eval 49.2%

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

   9. Simplified 49.2%

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

   if -2.54999999999999986e36 < x < 2.7e89

   1. Initial program 98.2%

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

      1. fabs-sub 98.2%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 92.6%

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

      4. fmm-def 92.6%

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

      5. distribute-neg-frac 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. unsub-neg 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

      1. fma-undefine 92.6%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 98.2%

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

      4. div-inv 98.1%

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

      5. sub-neg 98.1%

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

      6. metadata-eval 98.1%

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

      7. distribute-neg-in 98.1%

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

      8. +-commutative 98.1%

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

      9. cancel-sign-sub-inv 98.1%

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

      10. div-inv 98.2%

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

      11. fabs-sub 98.2%

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

      12. add-sqr-sqrt 38.8%

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

      13. fabs-sqr 38.8%

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

      14. add-sqr-sqrt 40.1%

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

      15. associate-*l/ 41.8%

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

      16. sub-div 41.8%

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

   6. Applied egg-rr 41.8%

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

   if 2.7e89 < x

   1. Initial program 74.4%

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

      1. fabs-sub 74.4%

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

      2. associate-*l/ 65.8%

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

      3. associate-*r/ 79.0%

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

      4. fmm-def 88.3%

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

      5. distribute-neg-frac 88.3%

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

      6. +-commutative 88.3%

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

      7. distribute-neg-in 88.3%

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

      8. unsub-neg 88.3%

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

      9. metadata-eval 88.3%

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

   3. Simplified 88.3%

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

      1. fma-undefine 79.0%

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

      2. associate-*r/ 65.8%

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

      3. associate-*l/ 74.4%

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

      4. div-inv 74.2%

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

      5. sub-neg 74.2%

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

      6. metadata-eval 74.2%

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

      7. distribute-neg-in 74.2%

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

      8. +-commutative 74.2%

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

      9. cancel-sign-sub-inv 74.2%

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

      10. div-inv 74.4%

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

      11. fabs-sub 74.4%

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

      12. add-sqr-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. add-sqr-sqrt 35.2%

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

      15. sub-neg 35.2%

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

      16. distribute-rgt-neg-in 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around -inf 51.3%

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

      1. mul-1-neg 51.3%

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

      2. div-sub 51.3%

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

      3. associate-/l* 49.2%

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

      4. sub-neg 49.2%

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

      5. metadata-eval 49.2%

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

      6. distribute-rgt-in 49.2%

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

      7. neg-mul-1 49.2%

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

      8. sub-neg 49.2%

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

      9. *-lft-identity 49.2%

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

      10. sub-neg 49.2%

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

      11. distribute-rgt-in 49.2%

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

      12. *-rgt-identity 49.2%

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

      13. remove-double-neg 49.2%

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

      14. *-commutative 49.2%

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

      15. mul-1-neg 49.2%

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

      16. associate-*r* 49.2%

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

      17. neg-mul-1 49.2%

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

      18. distribute-rgt-neg-in 49.2%

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

      19. mul-1-neg 49.2%

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

      20. distribute-lft-in 49.2%

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

      21. neg-mul-1 49.2%

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

      22. +-commutative 49.2%

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

      23. associate-*r* 49.2%

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

   9. Simplified 51.3%

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

       1. clear-num 51.3%

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

       2. un-div-inv 51.5%

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

       3. sub-neg 51.5%

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

       4. add-sqr-sqrt 18.7%

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

       5. sqrt-unprod 35.2%

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

       6. sqr-neg 35.2%

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

       7. sqrt-unprod 18.6%

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

       8. add-sqr-sqrt 37.4%

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

       9. +-commutative 37.4%

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

   11. Applied egg-rr 37.4%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.55e+36)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 8.5e-5) (/ (- 4.0 (* 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 <= -2.55e+36) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 8.5e-5) {
		tmp = (4.0 - (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 <= (-2.55d+36)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 8.5d-5) then
        tmp = (4.0d0 - (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 <= -2.55e+36) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 8.5e-5) {
		tmp = (4.0 - (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 <= -2.55e+36:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 8.5e-5:
		tmp = (4.0 - (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 <= -2.55e+36)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 8.5e-5)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = Float64(x / Float64(y_m / 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 <= -2.55e+36)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 8.5e-5)
		tmp = (4.0 - (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, -2.55e+36], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-5], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $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 < -2.54999999999999986e36

   1. Initial program 83.6%

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

      1. fabs-sub 83.6%

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

      2. associate-*l/ 83.9%

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

      3. associate-*r/ 87.7%

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

      4. fmm-def 91.7%

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

      5. distribute-neg-frac 91.7%

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

      6. +-commutative 91.7%

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

      7. distribute-neg-in 91.7%

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

      8. unsub-neg 91.7%

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

      9. metadata-eval 91.7%

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

   3. Simplified 91.7%

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

      1. add-sqr-sqrt 44.7%

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

      2. fabs-sqr 44.7%

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

      3. add-sqr-sqrt 45.2%

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

      4. fma-undefine 45.2%

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

      5. associate-*r/ 41.4%

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

      6. associate-*l/ 43.1%

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

      7. div-inv 43.0%

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

      8. sub-neg 43.0%

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

      9. metadata-eval 43.0%

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

      10. distribute-neg-in 43.0%

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

      11. +-commutative 43.0%

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

      12. cancel-sign-sub-inv 43.0%

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

      13. div-inv 43.1%

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

      14. associate-*l/ 41.4%

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

      15. sub-div 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 45.4%

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

      1. associate-/l* 49.2%

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

      2. sub-neg 49.2%

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

      3. metadata-eval 49.2%

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

   9. Simplified 49.2%

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

   if -2.54999999999999986e36 < x < 8.500000000000001e-5

   1. Initial program 97.9%

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

      1. fabs-sub 97.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 91.4%

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

      4. fmm-def 91.4%

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

      5. distribute-neg-frac 91.4%

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

      6. +-commutative 91.4%

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

      7. distribute-neg-in 91.4%

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

      8. unsub-neg 91.4%

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

      9. metadata-eval 91.4%

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

   3. Simplified 91.4%

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

      1. fma-undefine 91.4%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 97.9%

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

      4. div-inv 97.9%

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

      5. sub-neg 97.9%

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

      6. metadata-eval 97.9%

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

      7. distribute-neg-in 97.9%

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

      8. +-commutative 97.9%

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

      9. cancel-sign-sub-inv 97.9%

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

      10. div-inv 97.9%

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

      11. fabs-sub 97.9%

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

      12. add-sqr-sqrt 39.6%

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

      13. fabs-sqr 39.6%

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

      14. add-sqr-sqrt 40.9%

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

      15. associate-*l/ 42.8%

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

      16. sub-div 42.8%

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

   6. Applied egg-rr 42.8%

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

   7. Taylor expanded in x around 0 41.3%

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

   if 8.500000000000001e-5 < x

   1. Initial program 83.3%

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

      1. fabs-sub 83.3%

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

      2. associate-*l/ 77.7%

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

      3. associate-*r/ 86.2%

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

      4. fmm-def 92.3%

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

      5. distribute-neg-frac 92.3%

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

      6. +-commutative 92.3%

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

      7. distribute-neg-in 92.3%

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

      8. unsub-neg 92.3%

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

      9. metadata-eval 92.3%

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

   3. Simplified 92.3%

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

      1. fma-undefine 86.2%

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

      2. associate-*r/ 77.7%

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

      3. associate-*l/ 83.3%

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

      4. div-inv 83.0%

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

      5. sub-neg 83.0%

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

      6. metadata-eval 83.0%

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

      7. distribute-neg-in 83.0%

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

      8. +-commutative 83.0%

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

      9. cancel-sign-sub-inv 83.0%

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

      10. div-inv 83.3%

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

      11. fabs-sub 83.3%

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

      12. add-sqr-sqrt 34.6%

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

      13. fabs-sqr 34.6%

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

      14. add-sqr-sqrt 35.3%

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

      15. sub-neg 35.3%

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

      16. distribute-rgt-neg-in 35.3%

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

   6. Applied egg-rr 35.3%

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

   7. Taylor expanded in x around -inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. div-sub 45.4%

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

      3. associate-/l* 44.0%

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

      4. sub-neg 44.0%

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

      5. metadata-eval 44.0%

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

      6. distribute-rgt-in 44.0%

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

      7. neg-mul-1 44.0%

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

      8. sub-neg 44.0%

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

      9. *-lft-identity 44.0%

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

      10. sub-neg 44.0%

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

      11. distribute-rgt-in 44.0%

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

      12. *-rgt-identity 44.0%

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

      13. remove-double-neg 44.0%

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

      14. *-commutative 44.0%

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

      15. mul-1-neg 44.0%

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

      16. associate-*r* 44.0%

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

      17. neg-mul-1 44.0%

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

      18. distribute-rgt-neg-in 44.0%

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

      19. mul-1-neg 44.0%

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

      20. distribute-lft-in 44.0%

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

      21. neg-mul-1 44.0%

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

      22. +-commutative 44.0%

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

      23. associate-*r* 44.0%

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

   9. Simplified 45.4%

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

       1. clear-num 45.4%

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

       2. un-div-inv 45.5%

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

       3. sub-neg 45.5%

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

       4. add-sqr-sqrt 15.0%

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

       5. sqrt-unprod 31.9%

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

       6. sqr-neg 31.9%

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

       7. sqrt-unprod 18.4%

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

       8. add-sqr-sqrt 40.8%

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

       9. +-commutative 40.8%

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

   11. Applied egg-rr 40.8%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y\_m}{z + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -3.7)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 0.00045) (/ (+ 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 <= -3.7) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 0.00045) {
		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 <= (-3.7d0)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 0.00045d0) 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 <= -3.7) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 0.00045) {
		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 <= -3.7:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 0.00045:
		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 <= -3.7)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 0.00045)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(x / Float64(y_m / 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 <= -3.7)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 0.00045)
		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, -3.7], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00045], 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 < -3.7000000000000002

   1. Initial program 85.4%

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

      1. fabs-sub 85.4%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 89.0%

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

      4. fmm-def 92.6%

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

      5. distribute-neg-frac 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. unsub-neg 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

      1. add-sqr-sqrt 45.2%

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

      2. fabs-sqr 45.2%

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

      3. add-sqr-sqrt 45.8%

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

      4. fma-undefine 45.8%

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

      5. associate-*r/ 42.4%

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

      6. associate-*l/ 44.0%

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

      7. div-inv 43.9%

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

      8. sub-neg 43.9%

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

      9. metadata-eval 43.9%

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

      10. distribute-neg-in 43.9%

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

      11. +-commutative 43.9%

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

      12. cancel-sign-sub-inv 43.9%

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

      13. div-inv 44.0%

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

      14. associate-*l/ 42.4%

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

      15. sub-div 46.1%

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

   6. Applied egg-rr 46.1%

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

   7. Taylor expanded in x around inf 45.4%

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

      1. associate-/l* 48.7%

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

      2. sub-neg 48.7%

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

      3. metadata-eval 48.7%

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

   9. Simplified 48.7%

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

   if -3.7000000000000002 < x < 4.4999999999999999e-4

   1. Initial program 97.8%

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

      1. fabs-sub 97.8%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 91.0%

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

      4. fmm-def 91.0%

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

      5. distribute-neg-frac 91.0%

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

      6. +-commutative 91.0%

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

      7. distribute-neg-in 91.0%

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

      8. unsub-neg 91.0%

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

      9. metadata-eval 91.0%

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

   3. Simplified 91.0%

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

      1. fma-undefine 91.0%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 97.8%

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

      4. div-inv 97.8%

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

      5. sub-neg 97.8%

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

      6. metadata-eval 97.8%

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

      7. distribute-neg-in 97.8%

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

      8. +-commutative 97.8%

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

      9. cancel-sign-sub-inv 97.8%

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

      10. div-inv 97.8%

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

      11. fabs-sub 97.8%

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

      12. add-sqr-sqrt 39.1%

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

      13. fabs-sqr 39.1%

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

      14. add-sqr-sqrt 40.4%

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

      15. associate-*l/ 42.5%

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

      16. sub-div 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in z around 0 33.6%

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

   if 4.4999999999999999e-4 < x

   1. Initial program 83.3%

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

      1. fabs-sub 83.3%

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

      2. associate-*l/ 77.7%

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

      3. associate-*r/ 86.2%

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

      4. fmm-def 92.3%

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

      5. distribute-neg-frac 92.3%

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

      6. +-commutative 92.3%

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

      7. distribute-neg-in 92.3%

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

      8. unsub-neg 92.3%

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

      9. metadata-eval 92.3%

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

   3. Simplified 92.3%

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

      1. fma-undefine 86.2%

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

      2. associate-*r/ 77.7%

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

      3. associate-*l/ 83.3%

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

      4. div-inv 83.0%

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

      5. sub-neg 83.0%

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

      6. metadata-eval 83.0%

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

      7. distribute-neg-in 83.0%

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

      8. +-commutative 83.0%

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

      9. cancel-sign-sub-inv 83.0%

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

      10. div-inv 83.3%

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

      11. fabs-sub 83.3%

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

      12. add-sqr-sqrt 34.6%

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

      13. fabs-sqr 34.6%

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

      14. add-sqr-sqrt 35.3%

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

      15. sub-neg 35.3%

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

      16. distribute-rgt-neg-in 35.3%

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

   6. Applied egg-rr 35.3%

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

   7. Taylor expanded in x around -inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. div-sub 45.4%

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

      3. associate-/l* 44.0%

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

      4. sub-neg 44.0%

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

      5. metadata-eval 44.0%

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

      6. distribute-rgt-in 44.0%

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

      7. neg-mul-1 44.0%

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

      8. sub-neg 44.0%

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

      9. *-lft-identity 44.0%

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

      10. sub-neg 44.0%

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

      11. distribute-rgt-in 44.0%

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

      12. *-rgt-identity 44.0%

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

      13. remove-double-neg 44.0%

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

      14. *-commutative 44.0%

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

      15. mul-1-neg 44.0%

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

      16. associate-*r* 44.0%

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

      17. neg-mul-1 44.0%

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

      18. distribute-rgt-neg-in 44.0%

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

      19. mul-1-neg 44.0%

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

      20. distribute-lft-in 44.0%

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

      21. neg-mul-1 44.0%

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

      22. +-commutative 44.0%

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

      23. associate-*r* 44.0%

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

   9. Simplified 45.4%

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

       1. clear-num 45.4%

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

       2. un-div-inv 45.5%

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

       3. sub-neg 45.5%

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

       4. add-sqr-sqrt 15.0%

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

       5. sqrt-unprod 31.9%

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

       6. sqr-neg 31.9%

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

       7. sqrt-unprod 18.4%

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

       8. add-sqr-sqrt 40.8%

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

       9. +-commutative 40.8%

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

   11. Applied egg-rr 40.8%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 0.00042:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z + 1}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.85)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 0.00042) (/ (+ x 4.0) y_m) (* x (/ (+ z 1.0) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.85) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 0.00042) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((z + 1.0) / 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 <= (-2.85d0)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 0.00042d0) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = x * ((z + 1.0d0) / 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 <= -2.85) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 0.00042) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((z + 1.0) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.85:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 0.00042:
		tmp = (x + 4.0) / y_m
	else:
		tmp = x * ((z + 1.0) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.85)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 0.00042)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(x * Float64(Float64(z + 1.0) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.85)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 0.00042)
		tmp = (x + 4.0) / y_m;
	else
		tmp = x * ((z + 1.0) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.85], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00042], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(N[(z + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.85000000000000009

   1. Initial program 85.4%

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

      1. fabs-sub 85.4%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 89.0%

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

      4. fmm-def 92.6%

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

      5. distribute-neg-frac 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. unsub-neg 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

      1. add-sqr-sqrt 45.2%

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

      2. fabs-sqr 45.2%

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

      3. add-sqr-sqrt 45.8%

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

      4. fma-undefine 45.8%

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

      5. associate-*r/ 42.4%

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

      6. associate-*l/ 44.0%

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

      7. div-inv 43.9%

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

      8. sub-neg 43.9%

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

      9. metadata-eval 43.9%

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

      10. distribute-neg-in 43.9%

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

      11. +-commutative 43.9%

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

      12. cancel-sign-sub-inv 43.9%

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

      13. div-inv 44.0%

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

      14. associate-*l/ 42.4%

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

      15. sub-div 46.1%

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

   6. Applied egg-rr 46.1%

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

   7. Taylor expanded in x around inf 45.4%

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

      1. associate-/l* 48.7%

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

      2. sub-neg 48.7%

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

      3. metadata-eval 48.7%

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

   9. Simplified 48.7%

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

   if -2.85000000000000009 < x < 4.2000000000000002e-4

   1. Initial program 97.8%

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

      1. fabs-sub 97.8%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 91.0%

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

      4. fmm-def 91.0%

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

      5. distribute-neg-frac 91.0%

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

      6. +-commutative 91.0%

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

      7. distribute-neg-in 91.0%

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

      8. unsub-neg 91.0%

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

      9. metadata-eval 91.0%

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

   3. Simplified 91.0%

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

      1. fma-undefine 91.0%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 97.8%

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

      4. div-inv 97.8%

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

      5. sub-neg 97.8%

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

      6. metadata-eval 97.8%

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

      7. distribute-neg-in 97.8%

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

      8. +-commutative 97.8%

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

      9. cancel-sign-sub-inv 97.8%

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

      10. div-inv 97.8%

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

      11. fabs-sub 97.8%

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

      12. add-sqr-sqrt 39.1%

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

      13. fabs-sqr 39.1%

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

      14. add-sqr-sqrt 40.4%

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

      15. associate-*l/ 42.5%

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

      16. sub-div 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in z around 0 33.6%

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

   if 4.2000000000000002e-4 < x

   1. Initial program 83.3%

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

      1. fabs-sub 83.3%

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

      2. associate-*l/ 77.7%

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

      3. associate-*r/ 86.2%

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

      4. fmm-def 92.3%

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

      5. distribute-neg-frac 92.3%

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

      6. +-commutative 92.3%

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

      7. distribute-neg-in 92.3%

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

      8. unsub-neg 92.3%

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

      9. metadata-eval 92.3%

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

   3. Simplified 92.3%

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

      1. fma-undefine 86.2%

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

      2. associate-*r/ 77.7%

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

      3. associate-*l/ 83.3%

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

      4. div-inv 83.0%

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

      5. sub-neg 83.0%

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

      6. metadata-eval 83.0%

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

      7. distribute-neg-in 83.0%

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

      8. +-commutative 83.0%

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

      9. cancel-sign-sub-inv 83.0%

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

      10. div-inv 83.3%

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

      11. fabs-sub 83.3%

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

      12. add-sqr-sqrt 34.6%

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

      13. fabs-sqr 34.6%

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

      14. add-sqr-sqrt 35.3%

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

      15. sub-neg 35.3%

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

      16. distribute-rgt-neg-in 35.3%

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

   6. Applied egg-rr 35.3%

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

   7. Taylor expanded in x around -inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. div-sub 45.4%

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

      3. associate-/l* 44.0%

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

      4. sub-neg 44.0%

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

      5. metadata-eval 44.0%

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

      6. distribute-rgt-in 44.0%

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

      7. neg-mul-1 44.0%

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

      8. sub-neg 44.0%

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

      9. *-lft-identity 44.0%

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

      10. sub-neg 44.0%

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

      11. distribute-rgt-in 44.0%

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

      12. *-rgt-identity 44.0%

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

      13. remove-double-neg 44.0%

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

      14. *-commutative 44.0%

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

      15. mul-1-neg 44.0%

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

      16. associate-*r* 44.0%

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

      17. neg-mul-1 44.0%

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

      18. distribute-rgt-neg-in 44.0%

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

      19. mul-1-neg 44.0%

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

      20. distribute-lft-in 44.0%

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

      21. neg-mul-1 44.0%

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

      22. +-commutative 44.0%

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

      23. associate-*r* 44.0%

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

   9. Simplified 45.4%

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

       1. sub-neg 45.4%

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

       2. add-sqr-sqrt 14.9%

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

       3. sqrt-unprod 31.8%

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

       4. sqr-neg 31.8%

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

       5. sqrt-unprod 18.3%

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

       6. add-sqr-sqrt 40.7%

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

       7. +-commutative 40.7%

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

   11. Applied egg-rr 40.7%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 0.00042:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{x}{-y\_m}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.55) (/ 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 <= -1.55) {
		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 <= (-1.55d0)) 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 <= -1.55) {
		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 <= -1.55:
		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 <= -1.55)
		tmp = Float64(x / Float64(-y_m));
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = Float64(x / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1.55)
		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, -1.55], 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 < -1.55000000000000004

   1. Initial program 85.4%

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

      1. fabs-sub 85.4%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 89.0%

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

      4. fmm-def 92.6%

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

      5. distribute-neg-frac 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. unsub-neg 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

      1. add-sqr-sqrt 45.2%

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

      2. fabs-sqr 45.2%

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

      3. add-sqr-sqrt 45.8%

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

      4. fma-undefine 45.8%

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

      5. associate-*r/ 42.4%

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

      6. associate-*l/ 44.0%

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

      7. div-inv 43.9%

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

      8. sub-neg 43.9%

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

      9. metadata-eval 43.9%

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

      10. distribute-neg-in 43.9%

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

      11. +-commutative 43.9%

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

      12. cancel-sign-sub-inv 43.9%

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

      13. div-inv 44.0%

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

      14. associate-*l/ 42.4%

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

      15. sub-div 46.1%

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

   6. Applied egg-rr 46.1%

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

   7. Taylor expanded in z around 0 39.6%

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

      1. associate-*r/ 39.6%

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

      2. distribute-lft-in 39.6%

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

      3. metadata-eval 39.6%

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

      4. neg-mul-1 39.6%

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

      5. sub-neg 39.6%

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

   9. Simplified 39.6%

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

   10. Taylor expanded in x around inf 38.9%

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

       1. neg-mul-1 38.9%

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

   12. Simplified 38.9%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 97.8%

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

      1. fabs-sub 97.8%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 91.1%

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

      4. fmm-def 91.1%

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

      5. distribute-neg-frac 91.1%

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

      6. +-commutative 91.1%

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

      7. distribute-neg-in 91.1%

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

      8. unsub-neg 91.1%

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

      9. metadata-eval 91.1%

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

   3. Simplified 91.1%

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

      1. fma-undefine 91.1%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 97.8%

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

      4. div-inv 97.8%

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

      5. sub-neg 97.8%

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

      6. metadata-eval 97.8%

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

      7. distribute-neg-in 97.8%

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

      8. +-commutative 97.8%

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

      9. cancel-sign-sub-inv 97.8%

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

      10. div-inv 97.8%

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

      11. fabs-sub 97.8%

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

      12. add-sqr-sqrt 38.8%

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

      13. fabs-sqr 38.8%

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

      14. add-sqr-sqrt 40.1%

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

      15. associate-*l/ 42.2%

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

      16. sub-div 42.2%

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

   6. Applied egg-rr 42.2%

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

   7. Taylor expanded in x around 0 33.0%

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

   if 4 < x

   1. Initial program 83.0%

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

      1. fabs-sub 83.0%

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

      2. associate-*l/ 77.3%

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

      3. associate-*r/ 86.0%

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

      4. fmm-def 92.2%

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

      5. distribute-neg-frac 92.2%

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

      6. +-commutative 92.2%

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

      7. distribute-neg-in 92.2%

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

      8. unsub-neg 92.2%

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

      9. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

      1. fma-undefine 86.0%

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

      2. associate-*r/ 77.3%

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

      3. associate-*l/ 83.0%

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

      4. div-inv 82.8%

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

      5. sub-neg 82.8%

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

      6. metadata-eval 82.8%

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

      7. distribute-neg-in 82.8%

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

      8. +-commutative 82.8%

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

      9. cancel-sign-sub-inv 82.8%

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

      10. div-inv 83.0%

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

      11. fabs-sub 83.0%

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

      12. add-sqr-sqrt 35.1%

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

      13. fabs-sqr 35.1%

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

      14. add-sqr-sqrt 35.8%

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

      15. sub-neg 35.8%

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

      16. distribute-rgt-neg-in 35.8%

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

   6. Applied egg-rr 35.8%

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

   7. Taylor expanded in x around -inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. div-sub 46.1%

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

      3. associate-/l* 44.7%

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

      4. sub-neg 44.7%

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

      5. metadata-eval 44.7%

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

      6. distribute-rgt-in 44.7%

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

      7. neg-mul-1 44.7%

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

      8. sub-neg 44.7%

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

      9. *-lft-identity 44.7%

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

      10. sub-neg 44.7%

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

      11. distribute-rgt-in 44.7%

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

      12. *-rgt-identity 44.7%

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

      13. remove-double-neg 44.7%

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

      14. *-commutative 44.7%

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

      15. mul-1-neg 44.7%

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

      16. associate-*r* 44.7%

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

      17. neg-mul-1 44.7%

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

      18. distribute-rgt-neg-in 44.7%

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

      19. mul-1-neg 44.7%

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

      20. distribute-lft-in 44.7%

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

      21. neg-mul-1 44.7%

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

      22. +-commutative 44.7%

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

      23. associate-*r* 44.7%

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

   9. Simplified 46.1%

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

   10. Taylor expanded in z around 0 22.3%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.8% accurate, 9.2× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.7) (* x (/ (+ -1.0 z) y_m)) (/ (+ x 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.7) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.7d0)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else
        tmp = (x + 4.0d0) / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.7) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.7:
		tmp = x * ((-1.0 + z) / y_m)
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.7)
		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 <= -1.7)
		tmp = x * ((-1.0 + z) / y_m);
	else
		tmp = (x + 4.0) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.7], 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 < -1.69999999999999996

   1. Initial program 85.4%

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

      1. fabs-sub 85.4%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 89.0%

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

      4. fmm-def 92.6%

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

      5. distribute-neg-frac 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. unsub-neg 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

      1. add-sqr-sqrt 45.2%

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

      2. fabs-sqr 45.2%

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

      3. add-sqr-sqrt 45.8%

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

      4. fma-undefine 45.8%

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

      5. associate-*r/ 42.4%

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

      6. associate-*l/ 44.0%

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

      7. div-inv 43.9%

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

      8. sub-neg 43.9%

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

      9. metadata-eval 43.9%

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

      10. distribute-neg-in 43.9%

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

      11. +-commutative 43.9%

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

      12. cancel-sign-sub-inv 43.9%

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

      13. div-inv 44.0%

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

      14. associate-*l/ 42.4%

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

      15. sub-div 46.1%

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

   6. Applied egg-rr 46.1%

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

   7. Taylor expanded in x around inf 45.4%

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

      1. associate-/l* 48.7%

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

      2. sub-neg 48.7%

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

      3. metadata-eval 48.7%

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

   9. Simplified 48.7%

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

   if -1.69999999999999996 < 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/ 92.6%

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

      3. associate-*r/ 89.4%

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

      4. fmm-def 91.4%

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

      5. distribute-neg-frac 91.4%

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

      6. +-commutative 91.4%

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

      7. distribute-neg-in 91.4%

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

      8. unsub-neg 91.4%

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

      9. metadata-eval 91.4%

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

   3. Simplified 91.4%

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

      1. fma-undefine 89.4%

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

      2. associate-*r/ 92.6%

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

      3. associate-*l/ 93.0%

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

      4. div-inv 92.9%

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

      5. sub-neg 92.9%

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

      6. metadata-eval 92.9%

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

      7. distribute-neg-in 92.9%

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

      8. +-commutative 92.9%

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

      9. cancel-sign-sub-inv 92.9%

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

      10. div-inv 93.0%

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

      11. fabs-sub 93.0%

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

      12. add-sqr-sqrt 37.6%

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

      13. fabs-sqr 37.6%

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

      14. add-sqr-sqrt 38.7%

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

      15. associate-*l/ 39.7%

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

      16. sub-div 43.1%

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

   6. Applied egg-rr 43.1%

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

   7. Taylor expanded in z around 0 29.9%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 11.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 85.4%

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

      1. fabs-sub 85.4%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 89.0%

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

      4. fmm-def 92.6%

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

      5. distribute-neg-frac 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. unsub-neg 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

      1. add-sqr-sqrt 45.2%

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

      2. fabs-sqr 45.2%

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

      3. add-sqr-sqrt 45.8%

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

      4. fma-undefine 45.8%

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

      5. associate-*r/ 42.4%

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

      6. associate-*l/ 44.0%

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

      7. div-inv 43.9%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 43.9%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 43.9%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 43.9%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 43.9%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 43.9%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 44.0%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 42.4%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 46.1%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in z around 0 39.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 39.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 39.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 39.6%

         \[\leadsto \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 39.6%

         \[\leadsto \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 39.6%

         \[\leadsto \frac{\color{blue}{-4 - x}}{y} \]

   9. Simplified 39.6%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   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/ 92.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 89.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 91.4%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 89.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 92.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 93.0%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 92.9%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 93.0%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 93.0%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 37.6%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 37.6%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 38.7%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 39.7%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 43.1%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 29.9%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.9% 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(x / Float64(-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 85.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 85.4%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 85.7%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 89.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.6%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 45.2%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 45.2%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 45.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 45.8%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 42.4%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 44.0%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 43.9%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 43.9%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 43.9%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 43.9%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 43.9%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 43.9%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 44.0%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 42.4%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 46.1%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in z around 0 39.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 39.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 39.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 39.6%

         \[\leadsto \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 39.6%

         \[\leadsto \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 39.6%

         \[\leadsto \frac{\color{blue}{-4 - x}}{y} \]

   9. Simplified 39.6%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   10. Taylor expanded in x around inf 38.9%

       \[\leadsto \frac{\color{blue}{-1 \cdot x}}{y} \]
   11. Step-by-step derivation

       1. neg-mul-1 38.9%

          \[\leadsto \frac{\color{blue}{-x}}{y} \]

   12. Simplified 38.9%

       \[\leadsto \frac{\color{blue}{-x}}{y} \]

   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/ 92.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 89.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 91.4%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 91.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 89.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 92.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 93.0%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 92.9%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 93.0%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 93.0%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 37.6%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 37.6%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 38.7%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 39.7%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 43.1%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 29.9%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = x / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = Float64(x / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 94.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 94.2%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 95.8%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 90.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 91.5%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 90.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 95.8%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 94.2%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 94.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 94.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 94.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 94.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 94.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 94.2%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 94.2%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 94.2%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 39.6%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 39.6%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 40.7%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 42.7%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 44.8%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 44.8%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in x around 0 24.5%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 83.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 83.0%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 77.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 86.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.2%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 86.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 77.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 83.0%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 82.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 82.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 82.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 82.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 82.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 82.8%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 83.0%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 83.0%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 35.1%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 35.1%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 35.8%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. sub-neg 35.8%

          \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      16. distribute-rgt-neg-in 35.8%

          \[\leadsto \frac{x + 4}{y} + \color{blue}{\frac{x}{y} \cdot \left(-z\right)} \]

   6. Applied egg-rr 35.8%

      \[\leadsto \color{blue}{\frac{x + 4}{y} + \frac{x}{y} \cdot \left(-z\right)} \]

   7. Taylor expanded in x around -inf 46.1%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 46.1%

         \[\leadsto \color{blue}{-x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)} \]

      2. div-sub 46.1%

         \[\leadsto -x \cdot \color{blue}{\frac{z - 1}{y}} \]

      3. associate-/l* 44.7%

         \[\leadsto -\color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]

      4. sub-neg 44.7%

         \[\leadsto -\frac{x \cdot \color{blue}{\left(z + \left(-1\right)\right)}}{y} \]

      5. metadata-eval 44.7%

         \[\leadsto -\frac{x \cdot \left(z + \color{blue}{-1}\right)}{y} \]

      6. distribute-rgt-in 44.7%

         \[\leadsto -\frac{\color{blue}{z \cdot x + -1 \cdot x}}{y} \]

      7. neg-mul-1 44.7%

         \[\leadsto -\frac{z \cdot x + \color{blue}{\left(-x\right)}}{y} \]

      8. sub-neg 44.7%

         \[\leadsto -\frac{\color{blue}{z \cdot x - x}}{y} \]

      9. *-lft-identity 44.7%

         \[\leadsto -\frac{\color{blue}{1 \cdot \left(z \cdot x - x\right)}}{y} \]

      10. sub-neg 44.7%

          \[\leadsto -\frac{1 \cdot \color{blue}{\left(z \cdot x + \left(-x\right)\right)}}{y} \]

      11. distribute-rgt-in 44.7%

          \[\leadsto -\frac{\color{blue}{\left(z \cdot x\right) \cdot 1 + \left(-x\right) \cdot 1}}{y} \]

      12. *-rgt-identity 44.7%

          \[\leadsto -\frac{\color{blue}{z \cdot x} + \left(-x\right) \cdot 1}{y} \]

      13. remove-double-neg 44.7%

          \[\leadsto -\frac{\color{blue}{\left(-\left(-z \cdot x\right)\right)} + \left(-x\right) \cdot 1}{y} \]

      14. *-commutative 44.7%

          \[\leadsto -\frac{\left(-\left(-\color{blue}{x \cdot z}\right)\right) + \left(-x\right) \cdot 1}{y} \]

      15. mul-1-neg 44.7%

          \[\leadsto -\frac{\left(-\color{blue}{-1 \cdot \left(x \cdot z\right)}\right) + \left(-x\right) \cdot 1}{y} \]

      16. associate-*r* 44.7%

          \[\leadsto -\frac{\left(-\color{blue}{\left(-1 \cdot x\right) \cdot z}\right) + \left(-x\right) \cdot 1}{y} \]

      17. neg-mul-1 44.7%

          \[\leadsto -\frac{\left(-\color{blue}{\left(-x\right)} \cdot z\right) + \left(-x\right) \cdot 1}{y} \]

      18. distribute-rgt-neg-in 44.7%

          \[\leadsto -\frac{\color{blue}{\left(-x\right) \cdot \left(-z\right)} + \left(-x\right) \cdot 1}{y} \]

      19. mul-1-neg 44.7%

          \[\leadsto -\frac{\left(-x\right) \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(-x\right) \cdot 1}{y} \]

      20. distribute-lft-in 44.7%

          \[\leadsto -\frac{\color{blue}{\left(-x\right) \cdot \left(-1 \cdot z + 1\right)}}{y} \]

      21. neg-mul-1 44.7%

          \[\leadsto -\frac{\color{blue}{\left(-1 \cdot x\right)} \cdot \left(-1 \cdot z + 1\right)}{y} \]

      22. +-commutative 44.7%

          \[\leadsto -\frac{\left(-1 \cdot x\right) \cdot \color{blue}{\left(1 + -1 \cdot z\right)}}{y} \]

      23. associate-*r* 44.7%

          \[\leadsto -\frac{\color{blue}{-1 \cdot \left(x \cdot \left(1 + -1 \cdot z\right)\right)}}{y} \]

   9. Simplified 46.1%

      \[\leadsto \color{blue}{x \cdot \frac{1 - z}{y}} \]

   10. Taylor expanded in z around 0 22.3%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.6% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ 4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return 4.0 / y_m;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = 4.0d0 / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return 4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return 4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = 4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 91.4%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 91.4%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 91.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 89.3%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 91.7%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 91.7%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 89.3%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

   2. associate-*r/ 91.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

   3. associate-*l/ 91.4%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

   4. div-inv 91.3%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

   5. sub-neg 91.3%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

   6. metadata-eval 91.3%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

   7. distribute-neg-in 91.3%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

   8. +-commutative 91.3%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

   9. cancel-sign-sub-inv 91.3%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

   10. div-inv 91.4%

       \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

   11. fabs-sub 91.4%

       \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

   12. add-sqr-sqrt 38.5%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   13. fabs-sqr 38.5%

       \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   14. add-sqr-sqrt 39.5%

       \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   15. associate-*l/ 40.6%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   16. sub-div 44.9%

       \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

6. Applied egg-rr 44.9%

   \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

7. Taylor expanded in x around 0 19.0%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Alternative 15: 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 91.4%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 91.4%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 91.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 89.3%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 91.7%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 91.7%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 91.7%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 52.2%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

   2. fabs-sqr 52.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

   3. add-sqr-sqrt 53.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   4. fma-undefine 52.0%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

   5. associate-*r/ 52.0%

      \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

   6. associate-*l/ 53.4%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

   7. div-inv 53.3%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

   8. sub-neg 53.3%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

   9. metadata-eval 53.3%

      \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

   10. distribute-neg-in 53.3%

       \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

   11. +-commutative 53.3%

       \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

   12. cancel-sign-sub-inv 53.3%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

   13. div-inv 53.4%

       \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

   14. associate-*l/ 52.0%

       \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

   15. sub-div 53.5%

       \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

6. Applied egg-rr 53.5%

   \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

7. Taylor expanded in x around 0 25.8%

   \[\leadsto \color{blue}{\frac{-4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024169 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))