fabs fraction 1

Percentage Accurate: 91.9% → 99.7%

Time: 10.8s

Alternatives: 15

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 3.6e-43)
   (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 <= 3.6e-43) {
		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 <= 3.6e-43)
		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, 3.6e-43], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(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 < 3.5999999999999999e-43

   1. Initial program 87.6%

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

   3. Simplified 98.3%

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

   if 3.5999999999999999e-43 < y

   1. Initial program 98.6%

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

      1. associate-*l/ 90.8%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

Alternative 2: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ t_1 := z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_0 - t\_1 \leq 10^{+308}:\\ \;\;\;\;\left|t\_1 - t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{-1 + z}{y\_m} - \frac{4}{y\_m \cdot x}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (* z (/ x y_m))))
   (if (<= (- t_0 t_1) 1e+308)
     (fabs (- t_1 t_0))
     (fabs (* x (- (/ (+ -1.0 z) y_m) (/ 4.0 (* y_m x))))))))

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) <= 1e+308) {
		tmp = fabs((t_1 - t_0));
	} else {
		tmp = fabs((x * (((-1.0 + z) / y_m) - (4.0 / (y_m * x)))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    t_1 = z * (x / y_m)
    if ((t_0 - t_1) <= 1d+308) then
        tmp = abs((t_1 - t_0))
    else
        tmp = abs((x * ((((-1.0d0) + z) / y_m) - (4.0d0 / (y_m * x)))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= 1e+308) {
		tmp = Math.abs((t_1 - t_0));
	} else {
		tmp = Math.abs((x * (((-1.0 + z) / y_m) - (4.0 / (y_m * x)))));
	}
	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) <= 1e+308:
		tmp = math.fabs((t_1 - t_0))
	else:
		tmp = math.fabs((x * (((-1.0 + z) / y_m) - (4.0 / (y_m * x)))))
	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) <= 1e+308)
		tmp = abs(Float64(t_1 - t_0));
	else
		tmp = abs(Float64(x * Float64(Float64(Float64(-1.0 + z) / y_m) - Float64(4.0 / Float64(y_m * x)))));
	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) <= 1e+308)
		tmp = abs((t_1 - t_0));
	else
		tmp = abs((x * (((-1.0 + z) / y_m) - (4.0 / (y_m * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(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], 1e+308], N[Abs[N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(4.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1e308

   1. Initial program 98.4%

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

   if 1e308 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 39.4%

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

      1. fabs-sub 39.4%

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

      2. associate-*l/ 69.7%

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

      3. associate-*r/ 69.7%

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

      4. fma-neg 81.8%

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

      5. distribute-neg-frac 81.8%

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

      6. +-commutative 81.8%

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

      7. distribute-neg-in 81.8%

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

      8. unsub-neg 81.8%

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

      9. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in x around inf 97.0%

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

      1. associate--r+ 97.0%

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

      2. div-sub 97.0%

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

      3. sub-neg 97.0%

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

      4. remove-double-neg 97.0%

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

      5. neg-mul-1 97.0%

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

      6. metadata-eval 97.0%

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

      7. metadata-eval 97.0%

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

      8. distribute-lft-in 97.0%

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

      9. +-commutative 97.0%

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

      10. neg-mul-1 97.0%

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

      11. associate-*r/ 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 10^{+308}:\\ \;\;\;\;\left|z \cdot \frac{x}{y} - \frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{-1 + z}{y} - \frac{4}{y \cdot x}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs((t_1 - t_0));
	} else {
		tmp = x * ((-1.0 + z) / y_m);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	t_1 = z * (x / y_m);
	tmp = 0.0;
	if ((t_0 - t_1) <= Inf)
		tmp = abs((t_1 - t_0));
	else
		tmp = x * ((-1.0 + z) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(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], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.5%

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

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

   1. Initial program 0.0%

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

      1. fabs-sub 0.0%

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

      2. associate-*l/ 50.0%

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

      3. associate-*r/ 50.0%

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

      4. fma-neg 70.0%

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

      5. distribute-neg-frac 70.0%

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

      6. +-commutative 70.0%

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

      7. distribute-neg-in 70.0%

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

      8. unsub-neg 70.0%

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

      9. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

      1. add-sqr-sqrt 45.0%

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

      2. fabs-sqr 45.0%

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

      3. add-sqr-sqrt 45.0%

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

      4. fma-undefine 30.0%

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

      5. associate-*r/ 30.0%

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

      6. associate-*l/ 0.0%

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

      7. div-inv 0.0%

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

      8. sub-neg 0.0%

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

      9. metadata-eval 0.0%

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

      10. distribute-neg-in 0.0%

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

      11. +-commutative 0.0%

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

      12. cancel-sign-sub-inv 0.0%

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

      13. div-inv 0.0%

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

      14. associate-*l/ 30.0%

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

      15. sub-div 55.0%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in x around inf 55.0%

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

      1. associate-/l* 55.0%

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

      2. sub-neg 55.0%

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

      3. metadata-eval 55.0%

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

   9. Simplified 55.0%

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

4. Final simplification 95.1%

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

Alternative 4: 92.6% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := 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. Initial program 90.8%

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

   1. associate-*l/ 92.3%

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

   2. associate-*r/ 93.8%

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

   3. clear-num 93.8%

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

   4. un-div-inv 93.8%

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

4. Applied egg-rr 93.8%

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

Alternative 5: 79.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+116}:\\ \;\;\;\;\frac{x + 4}{y\_m} + \frac{-1}{y\_m} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.6e-12)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 1.75e+116)
     (+ (/ (+ x 4.0) y_m) (* (/ -1.0 y_m) (* x z)))
     (* (/ x y_m) (+ z 1.0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1.75e+116) {
		tmp = ((x + 4.0) / y_m) + ((-1.0 / y_m) * (x * z));
	} else {
		tmp = (x / y_m) * (z + 1.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.6d-12)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 1.75d+116) then
        tmp = ((x + 4.0d0) / y_m) + (((-1.0d0) / y_m) * (x * z))
    else
        tmp = (x / y_m) * (z + 1.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1.75e+116) {
		tmp = ((x + 4.0) / y_m) + ((-1.0 / y_m) * (x * z));
	} else {
		tmp = (x / y_m) * (z + 1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.6e-12:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 1.75e+116:
		tmp = ((x + 4.0) / y_m) + ((-1.0 / y_m) * (x * z))
	else:
		tmp = (x / y_m) * (z + 1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.6e-12)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 1.75e+116)
		tmp = Float64(Float64(Float64(x + 4.0) / y_m) + Float64(Float64(-1.0 / y_m) * Float64(x * z)));
	else
		tmp = Float64(Float64(x / 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 <= -2.6e-12)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 1.75e+116)
		tmp = ((x + 4.0) / y_m) + ((-1.0 / y_m) * (x * z));
	else
		tmp = (x / y_m) * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.6e-12], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75e+116], N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] + N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999983e-12

   1. Initial program 84.3%

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

      1. fabs-sub 84.3%

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

      2. associate-*l/ 87.7%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. add-sqr-sqrt 56.0%

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

      2. fabs-sqr 56.0%

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

      3. add-sqr-sqrt 56.4%

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

      4. fma-undefine 53.3%

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

      5. associate-*r/ 48.9%

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

      6. associate-*l/ 47.1%

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

      7. div-inv 47.0%

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

      8. sub-neg 47.0%

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

      9. metadata-eval 47.0%

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

      10. distribute-neg-in 47.0%

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

      11. +-commutative 47.0%

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

      12. cancel-sign-sub-inv 47.0%

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

      13. div-inv 47.1%

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

      14. associate-*l/ 48.9%

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

      15. sub-div 53.6%

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

   6. Applied egg-rr 53.6%

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

   7. Taylor expanded in x around inf 53.6%

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

      1. associate-/l* 57.9%

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

      2. sub-neg 57.9%

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

      3. metadata-eval 57.9%

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

   9. Simplified 57.9%

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

   if -2.59999999999999983e-12 < x < 1.74999999999999998e116

   1. Initial program 96.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 56.7%

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

      2. fabs-sqr 56.7%

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

      3. add-sqr-sqrt 57.7%

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

      4. fma-undefine 57.7%

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

      5. distribute-rgt-in 57.8%

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

      6. sub-neg 57.8%

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

      7. metadata-eval 57.8%

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

      8. distribute-neg-in 57.8%

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

      9. +-commutative 57.8%

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

      10. frac-2neg 57.8%

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

      11. metadata-eval 57.8%

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

      12. div-inv 57.8%

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

      13. frac-2neg 57.8%

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

   6. Applied egg-rr 57.8%

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

   if 1.74999999999999998e116 < x

   1. Initial program 82.6%

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

      1. fabs-sub 82.6%

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

      2. associate-*l/ 77.6%

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

      3. associate-*r/ 90.4%

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

      4. fma-neg 94.2%

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

      5. distribute-neg-frac 94.2%

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

      6. +-commutative 94.2%

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

      7. distribute-neg-in 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   3. Simplified 94.2%

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

      1. fma-undefine 90.4%

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

      2. associate-*r/ 77.6%

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

      3. associate-*l/ 82.6%

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

      4. div-inv 82.4%

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

      5. sub-neg 82.4%

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

      6. metadata-eval 82.4%

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

      7. distribute-neg-in 82.4%

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

      8. +-commutative 82.4%

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

      9. cancel-sign-sub-inv 82.4%

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

      10. div-inv 82.6%

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

      11. fabs-sub 82.6%

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

      12. add-sqr-sqrt 32.5%

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

      13. fabs-sqr 32.5%

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

      14. add-sqr-sqrt 33.0%

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

      15. sub-neg 33.0%

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

      16. distribute-rgt-neg-in 33.0%

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

   6. Applied egg-rr 33.0%

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

   7. Taylor expanded in x around inf 33.0%

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

      1. *-un-lft-identity 33.0%

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

      2. *-commutative 33.0%

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

      3. distribute-rgt-out 42.6%

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

      4. add-sqr-sqrt 21.3%

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

      5. sqrt-unprod 31.7%

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

      6. sqr-neg 31.7%

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

      7. sqrt-unprod 19.4%

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

      8. add-sqr-sqrt 37.0%

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

      9. +-commutative 37.0%

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

   9. Applied egg-rr 37.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+116}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{-1}{y} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.6e-12)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 1.05e+116)
     (/ (- (+ 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 <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1.05e+116) {
		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 <= (-2.6d-12)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 1.05d+116) 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 <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1.05e+116) {
		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 <= -2.6e-12:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 1.05e+116:
		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 <= -2.6e-12)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 1.05e+116)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	else
		tmp = Float64(Float64(x / 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 <= -2.6e-12)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 1.05e+116)
		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, -2.6e-12], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+116], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999983e-12

   1. Initial program 84.3%

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

      1. fabs-sub 84.3%

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

      2. associate-*l/ 87.7%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. add-sqr-sqrt 56.0%

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

      2. fabs-sqr 56.0%

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

      3. add-sqr-sqrt 56.4%

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

      4. fma-undefine 53.3%

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

      5. associate-*r/ 48.9%

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

      6. associate-*l/ 47.1%

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

      7. div-inv 47.0%

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

      8. sub-neg 47.0%

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

      9. metadata-eval 47.0%

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

      10. distribute-neg-in 47.0%

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

      11. +-commutative 47.0%

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

      12. cancel-sign-sub-inv 47.0%

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

      13. div-inv 47.1%

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

      14. associate-*l/ 48.9%

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

      15. sub-div 53.6%

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

   6. Applied egg-rr 53.6%

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

   7. Taylor expanded in x around inf 53.6%

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

      1. associate-/l* 57.9%

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

      2. sub-neg 57.9%

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

      3. metadata-eval 57.9%

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

   9. Simplified 57.9%

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

   if -2.59999999999999983e-12 < x < 1.0500000000000001e116

   1. Initial program 96.8%

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

      1. fabs-sub 96.8%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.8%

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

      4. fma-neg 95.8%

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

      5. distribute-neg-frac 95.8%

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

      6. +-commutative 95.8%

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

      7. distribute-neg-in 95.8%

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

      8. unsub-neg 95.8%

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

      9. metadata-eval 95.8%

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

   3. Simplified 95.8%

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

      1. fma-undefine 95.8%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 96.8%

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

      4. div-inv 96.7%

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

      5. sub-neg 96.7%

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

      6. metadata-eval 96.7%

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

      7. distribute-neg-in 96.7%

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

      8. +-commutative 96.7%

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

      9. cancel-sign-sub-inv 96.7%

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

      10. div-inv 96.8%

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

      11. fabs-sub 96.8%

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

      12. add-sqr-sqrt 53.9%

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

      13. fabs-sqr 53.9%

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

      14. add-sqr-sqrt 55.0%

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

      15. associate-*l/ 57.8%

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

      16. sub-div 57.8%

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

   6. Applied egg-rr 57.8%

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

   if 1.0500000000000001e116 < x

   1. Initial program 82.6%

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

      1. fabs-sub 82.6%

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

      2. associate-*l/ 77.6%

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

      3. associate-*r/ 90.4%

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

      4. fma-neg 94.2%

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

      5. distribute-neg-frac 94.2%

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

      6. +-commutative 94.2%

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

      7. distribute-neg-in 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   3. Simplified 94.2%

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

      1. fma-undefine 90.4%

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

      2. associate-*r/ 77.6%

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

      3. associate-*l/ 82.6%

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

      4. div-inv 82.4%

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

      5. sub-neg 82.4%

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

      6. metadata-eval 82.4%

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

      7. distribute-neg-in 82.4%

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

      8. +-commutative 82.4%

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

      9. cancel-sign-sub-inv 82.4%

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

      10. div-inv 82.6%

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

      11. fabs-sub 82.6%

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

      12. add-sqr-sqrt 32.5%

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

      13. fabs-sqr 32.5%

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

      14. add-sqr-sqrt 33.0%

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

      15. sub-neg 33.0%

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

      16. distribute-rgt-neg-in 33.0%

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

   6. Applied egg-rr 33.0%

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

   7. Taylor expanded in x around inf 33.0%

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

      1. *-un-lft-identity 33.0%

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

      2. *-commutative 33.0%

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

      3. distribute-rgt-out 42.6%

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

      4. add-sqr-sqrt 21.3%

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

      5. sqrt-unprod 31.7%

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

      6. sqr-neg 31.7%

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

      7. sqrt-unprod 19.4%

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

      8. add-sqr-sqrt 37.0%

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

      9. +-commutative 37.0%

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

   9. Applied egg-rr 37.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.6e-12)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 5.2e-9) (/ (- 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 <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 5.2e-9) {
		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 <= (-2.6d-12)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 5.2d-9) 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 <= -2.6e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 5.2e-9) {
		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 <= -2.6e-12:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 5.2e-9:
		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 <= -2.6e-12)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 5.2e-9)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = Float64(Float64(x / 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 <= -2.6e-12)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 5.2e-9)
		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, -2.6e-12], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-9], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999983e-12

   1. Initial program 84.3%

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

      1. fabs-sub 84.3%

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

      2. associate-*l/ 87.7%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. add-sqr-sqrt 56.0%

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

      2. fabs-sqr 56.0%

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

      3. add-sqr-sqrt 56.4%

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

      4. fma-undefine 53.3%

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

      5. associate-*r/ 48.9%

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

      6. associate-*l/ 47.1%

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

      7. div-inv 47.0%

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

      8. sub-neg 47.0%

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

      9. metadata-eval 47.0%

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

      10. distribute-neg-in 47.0%

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

      11. +-commutative 47.0%

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

      12. cancel-sign-sub-inv 47.0%

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

      13. div-inv 47.1%

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

      14. associate-*l/ 48.9%

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

      15. sub-div 53.6%

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

   6. Applied egg-rr 53.6%

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

   7. Taylor expanded in x around inf 53.6%

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

      1. associate-/l* 57.9%

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

      2. sub-neg 57.9%

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

      3. metadata-eval 57.9%

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

   9. Simplified 57.9%

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

   if -2.59999999999999983e-12 < x < 5.2000000000000002e-9

   1. Initial program 96.9%

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

      1. fabs-sub 96.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.0%

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

      4. fma-neg 95.0%

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

      5. distribute-neg-frac 95.0%

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

      6. +-commutative 95.0%

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

      7. distribute-neg-in 95.0%

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

      8. unsub-neg 95.0%

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

      9. metadata-eval 95.0%

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

   3. Simplified 95.0%

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

      1. fma-undefine 95.0%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 96.9%

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

      4. div-inv 96.9%

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

      5. sub-neg 96.9%

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

      6. metadata-eval 96.9%

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

      7. distribute-neg-in 96.9%

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

      8. +-commutative 96.9%

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

      9. cancel-sign-sub-inv 96.9%

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

      10. div-inv 96.9%

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

      11. fabs-sub 96.9%

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

      12. add-sqr-sqrt 53.5%

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

      13. fabs-sqr 53.5%

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

      14. add-sqr-sqrt 54.7%

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

      15. associate-*l/ 57.2%

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

      16. sub-div 57.2%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in x around 0 57.1%

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

   if 5.2000000000000002e-9 < x

   1. Initial program 86.9%

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

      1. fabs-sub 86.9%

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

      2. associate-*l/ 84.9%

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

      3. associate-*r/ 93.4%

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

      4. fma-neg 96.0%

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

      5. distribute-neg-frac 96.0%

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

      6. +-commutative 96.0%

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

      7. distribute-neg-in 96.0%

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

      8. unsub-neg 96.0%

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

      9. metadata-eval 96.0%

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

   3. Simplified 96.0%

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

      1. fma-undefine 93.4%

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

      2. associate-*r/ 84.9%

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

      3. associate-*l/ 86.9%

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

      4. div-inv 86.8%

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

      5. sub-neg 86.8%

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

      6. metadata-eval 86.8%

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

      7. distribute-neg-in 86.8%

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

      8. +-commutative 86.8%

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

      9. cancel-sign-sub-inv 86.8%

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

      10. div-inv 86.9%

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

      11. fabs-sub 86.9%

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

      12. add-sqr-sqrt 40.0%

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

      13. fabs-sqr 40.0%

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

      14. add-sqr-sqrt 40.6%

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

      15. sub-neg 40.6%

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

      16. distribute-rgt-neg-in 40.6%

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

   6. Applied egg-rr 40.6%

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

   7. Taylor expanded in x around inf 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. *-commutative 37.9%

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

      3. distribute-rgt-out 45.6%

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

      4. add-sqr-sqrt 20.2%

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

      5. sqrt-unprod 30.7%

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

      6. sqr-neg 30.7%

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

      7. sqrt-unprod 17.8%

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

      8. add-sqr-sqrt 32.1%

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

      9. +-commutative 32.1%

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

   9. Applied egg-rr 32.1%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+16}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.32e-14)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 2.95e+16) (/ (+ 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 <= -1.32e-14) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 2.95e+16) {
		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 <= (-1.32d-14)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 2.95d+16) 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 <= -1.32e-14) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 2.95e+16) {
		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 <= -1.32e-14:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 2.95e+16:
		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 <= -1.32e-14)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 2.95e+16)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(Float64(x / 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 <= -1.32e-14)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 2.95e+16)
		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, -1.32e-14], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.95e+16], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e-14

   1. Initial program 84.3%

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

      1. fabs-sub 84.3%

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

      2. associate-*l/ 87.7%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. add-sqr-sqrt 56.0%

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

      2. fabs-sqr 56.0%

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

      3. add-sqr-sqrt 56.4%

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

      4. fma-undefine 53.3%

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

      5. associate-*r/ 48.9%

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

      6. associate-*l/ 47.1%

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

      7. div-inv 47.0%

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

      8. sub-neg 47.0%

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

      9. metadata-eval 47.0%

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

      10. distribute-neg-in 47.0%

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

      11. +-commutative 47.0%

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

      12. cancel-sign-sub-inv 47.0%

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

      13. div-inv 47.1%

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

      14. associate-*l/ 48.9%

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

      15. sub-div 53.6%

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

   6. Applied egg-rr 53.6%

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

   7. Taylor expanded in x around inf 53.6%

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

      1. associate-/l* 57.9%

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

      2. sub-neg 57.9%

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

      3. metadata-eval 57.9%

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

   9. Simplified 57.9%

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

   if -1.32e-14 < x < 2.95e16

   1. Initial program 97.1%

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

      1. fabs-sub 97.1%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.3%

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

      4. fma-neg 95.3%

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

      5. distribute-neg-frac 95.3%

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

      6. +-commutative 95.3%

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

      7. distribute-neg-in 95.3%

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

      8. unsub-neg 95.3%

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

      9. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

      1. fma-undefine 95.3%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 97.1%

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

      4. div-inv 97.1%

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

      5. sub-neg 97.1%

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

      6. metadata-eval 97.1%

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

      7. distribute-neg-in 97.1%

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

      8. +-commutative 97.1%

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

      9. cancel-sign-sub-inv 97.1%

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

      10. div-inv 97.1%

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

      11. fabs-sub 97.1%

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

      12. add-sqr-sqrt 54.1%

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

      13. fabs-sqr 54.1%

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

      14. add-sqr-sqrt 55.3%

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

      15. associate-*l/ 57.6%

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

      16. sub-div 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in z around 0 42.9%

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

   if 2.95e16 < x

   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/ 83.2%

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

      3. associate-*r/ 92.7%

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

      4. fma-neg 95.6%

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

      5. distribute-neg-frac 95.6%

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

      6. +-commutative 95.6%

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

      7. distribute-neg-in 95.6%

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

      8. unsub-neg 95.6%

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

      9. metadata-eval 95.6%

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

   3. Simplified 95.6%

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

      1. fma-undefine 92.7%

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

      2. associate-*r/ 83.2%

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

      3. associate-*l/ 85.4%

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

      4. div-inv 85.3%

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

      5. sub-neg 85.3%

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

      6. metadata-eval 85.3%

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

      7. distribute-neg-in 85.3%

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

      8. +-commutative 85.3%

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

      9. cancel-sign-sub-inv 85.3%

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

      10. div-inv 85.4%

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

      11. fabs-sub 85.4%

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

      12. add-sqr-sqrt 37.4%

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

      13. fabs-sqr 37.4%

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

      14. add-sqr-sqrt 38.0%

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

      15. sub-neg 38.0%

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

      16. distribute-rgt-neg-in 38.0%

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

   6. Applied egg-rr 38.0%

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

   7. Taylor expanded in x around inf 38.0%

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

      1. *-un-lft-identity 38.0%

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

      2. *-commutative 38.0%

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

      3. distribute-rgt-out 46.7%

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

      4. add-sqr-sqrt 22.0%

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

      5. sqrt-unprod 31.4%

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

      6. sqr-neg 31.4%

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

      7. sqrt-unprod 17.6%

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

      8. add-sqr-sqrt 32.5%

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

      9. +-commutative 32.5%

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

   9. Applied egg-rr 32.5%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+16}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 1020000:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2.5e-12)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 1020000.0) (/ (+ x 4.0) y_m) (* x (/ (- 1.0 z) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.5e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1020000.0) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - z) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.5d-12)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 1020000.0d0) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = x * ((1.0d0 - z) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2.5e-12) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 1020000.0) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - z) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2.5e-12:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 1020000.0:
		tmp = (x + 4.0) / y_m
	else:
		tmp = x * ((1.0 - z) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2.5e-12)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 1020000.0)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(x * Float64(Float64(1.0 - z) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2.5e-12)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 1020000.0)
		tmp = (x + 4.0) / y_m;
	else
		tmp = x * ((1.0 - z) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2.5e-12], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1020000.0], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.49999999999999985e-12

   1. Initial program 84.3%

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

      1. fabs-sub 84.3%

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

      2. associate-*l/ 87.7%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. add-sqr-sqrt 56.0%

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

      2. fabs-sqr 56.0%

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

      3. add-sqr-sqrt 56.4%

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

      4. fma-undefine 53.3%

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

      5. associate-*r/ 48.9%

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

      6. associate-*l/ 47.1%

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

      7. div-inv 47.0%

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

      8. sub-neg 47.0%

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

      9. metadata-eval 47.0%

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

      10. distribute-neg-in 47.0%

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

      11. +-commutative 47.0%

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

      12. cancel-sign-sub-inv 47.0%

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

      13. div-inv 47.1%

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

      14. associate-*l/ 48.9%

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

      15. sub-div 53.6%

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

   6. Applied egg-rr 53.6%

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

   7. Taylor expanded in x around inf 53.6%

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

      1. associate-/l* 57.9%

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

      2. sub-neg 57.9%

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

      3. metadata-eval 57.9%

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

   9. Simplified 57.9%

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

   if -2.49999999999999985e-12 < x < 1.02e6

   1. Initial program 97.1%

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

      1. fabs-sub 97.1%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.2%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. fma-undefine 95.2%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 97.1%

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

      4. div-inv 97.1%

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

      5. sub-neg 97.1%

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

      6. metadata-eval 97.1%

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

      7. distribute-neg-in 97.1%

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

      8. +-commutative 97.1%

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

      9. cancel-sign-sub-inv 97.1%

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

      10. div-inv 97.1%

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

      11. fabs-sub 97.1%

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

      12. add-sqr-sqrt 54.2%

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

      13. fabs-sqr 54.2%

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

      14. add-sqr-sqrt 55.3%

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

      15. associate-*l/ 57.7%

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

      16. sub-div 57.7%

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

   6. Applied egg-rr 57.7%

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

   7. Taylor expanded in z around 0 42.8%

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

   if 1.02e6 < x

   1. Initial program 85.8%

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

      1. fabs-sub 85.8%

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

      2. associate-*l/ 83.6%

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

      3. associate-*r/ 92.9%

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

      4. fma-neg 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. unsub-neg 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

      1. fma-undefine 92.9%

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

      2. associate-*r/ 83.6%

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

      3. associate-*l/ 85.8%

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

      4. div-inv 85.7%

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

      5. sub-neg 85.7%

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

      6. metadata-eval 85.7%

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

      7. distribute-neg-in 85.7%

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

      8. +-commutative 85.7%

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

      9. cancel-sign-sub-inv 85.7%

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

      10. div-inv 85.8%

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

      11. fabs-sub 85.8%

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

      12. add-sqr-sqrt 37.7%

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

      13. fabs-sqr 37.7%

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

      14. add-sqr-sqrt 38.4%

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

      15. sub-neg 38.4%

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

      16. distribute-rgt-neg-in 38.4%

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

   6. Applied egg-rr 38.4%

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

   7. Taylor expanded in x around -inf 46.8%

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

      1. mul-1-neg 46.8%

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

      2. div-sub 46.8%

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

      3. associate-/l* 40.1%

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

      4. sub-neg 40.1%

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

      5. metadata-eval 40.1%

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

      6. distribute-lft-in 40.1%

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

      7. *-commutative 40.1%

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

      8. neg-mul-1 40.1%

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

      9. remove-double-neg 40.1%

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

      10. distribute-rgt-neg-in 40.1%

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

      11. mul-1-neg 40.1%

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

      12. distribute-lft-neg-in 40.1%

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

      13. *-rgt-identity 40.1%

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

      14. distribute-lft-in 40.1%

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

      15. neg-mul-1 40.1%

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

      16. +-commutative 40.1%

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

      17. associate-*r* 40.1%

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

      18. associate-*r/ 40.1%

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

      19. mul-1-neg 40.1%

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

      20. distribute-neg-frac2 40.1%

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

   9. Simplified 46.8%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 1020000:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 8.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.52) (/ x (- y_m)) (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.52) {
		tmp = x / -y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.52d0)) then
        tmp = x / -y_m
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = x / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.52) {
		tmp = x / -y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.52:
		tmp = x / -y_m
	elif x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(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 <= -1.52)
		tmp = x / -y_m;
	elseif (x <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.52], N[(x / (-y$95$m)), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   1. Initial program 84.0%

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

      1. fabs-sub 84.0%

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

      2. associate-*l/ 87.5%

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

      3. associate-*r/ 91.9%

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

      4. fma-neg 95.1%

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

      5. distribute-neg-frac 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. unsub-neg 95.1%

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

      9. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

      1. add-sqr-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. add-sqr-sqrt 57.3%

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

      4. fma-undefine 54.1%

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

      5. associate-*r/ 49.7%

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

      6. associate-*l/ 47.8%

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

      7. div-inv 47.8%

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

      8. sub-neg 47.8%

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

      9. metadata-eval 47.8%

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

      10. distribute-neg-in 47.8%

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

      11. +-commutative 47.8%

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

      12. cancel-sign-sub-inv 47.8%

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

      13. div-inv 47.8%

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

      14. associate-*l/ 49.7%

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

      15. sub-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in z around 0 36.5%

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

      1. associate-*r/ 36.5%

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

      2. distribute-lft-in 36.5%

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

      3. metadata-eval 36.5%

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

      4. neg-mul-1 36.5%

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

      5. sub-neg 36.5%

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

   9. Simplified 36.5%

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

   10. Taylor expanded in x around inf 36.5%

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

       1. neg-mul-1 36.5%

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

   12. Simplified 36.5%

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

   if -1.52 < x < 4

   1. Initial program 97.0%

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

      1. fabs-sub 97.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 95.1%

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

      4. fma-neg 95.1%

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

      5. distribute-neg-frac 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. unsub-neg 95.1%

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

      9. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

      1. fma-undefine 95.1%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 97.0%

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

      4. div-inv 97.0%

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

      5. sub-neg 97.0%

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

      6. metadata-eval 97.0%

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

      7. distribute-neg-in 97.0%

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

      8. +-commutative 97.0%

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

      9. cancel-sign-sub-inv 97.0%

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

      10. div-inv 97.0%

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

      11. fabs-sub 97.0%

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

      12. add-sqr-sqrt 54.2%

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

      13. fabs-sqr 54.2%

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

      14. add-sqr-sqrt 55.4%

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

      15. associate-*l/ 57.9%

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

      16. sub-div 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in x around 0 41.1%

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

   if 4 < x

   1. Initial program 86.4%

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

      1. fabs-sub 86.4%

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

      2. associate-*l/ 84.3%

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

      3. associate-*r/ 93.2%

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

      4. fma-neg 95.9%

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

      5. distribute-neg-frac 95.9%

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

      6. +-commutative 95.9%

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

      7. distribute-neg-in 95.9%

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

      8. unsub-neg 95.9%

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

      9. metadata-eval 95.9%

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

   3. Simplified 95.9%

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

      1. fma-undefine 93.2%

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

      2. associate-*r/ 84.3%

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

      3. associate-*l/ 86.4%

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

      4. div-inv 86.3%

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

      5. sub-neg 86.3%

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

      6. metadata-eval 86.3%

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

      7. distribute-neg-in 86.3%

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

      8. +-commutative 86.3%

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

      9. cancel-sign-sub-inv 86.3%

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

      10. div-inv 86.4%

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

      11. fabs-sub 86.4%

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

      12. add-sqr-sqrt 38.9%

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

      13. fabs-sqr 38.9%

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

      14. add-sqr-sqrt 39.5%

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

      15. sub-neg 39.5%

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

      16. distribute-rgt-neg-in 39.5%

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

   6. Applied egg-rr 39.5%

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

   7. Taylor expanded in x around -inf 46.0%

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

      1. mul-1-neg 46.0%

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

      2. div-sub 46.0%

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

      3. associate-/l* 39.6%

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

      4. sub-neg 39.6%

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

      5. metadata-eval 39.6%

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

      6. distribute-lft-in 39.6%

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

      7. *-commutative 39.6%

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

      8. neg-mul-1 39.6%

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

      9. remove-double-neg 39.6%

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

      10. distribute-rgt-neg-in 39.6%

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

      11. mul-1-neg 39.6%

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

      12. distribute-lft-neg-in 39.6%

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

      13. *-rgt-identity 39.6%

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

      14. distribute-lft-in 39.6%

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

      15. neg-mul-1 39.6%

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

      16. +-commutative 39.6%

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

      17. associate-*r* 39.6%

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

      18. associate-*r/ 39.6%

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

      19. mul-1-neg 39.6%

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

      20. distribute-neg-frac2 39.6%

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

   9. Simplified 46.0%

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

   10. Taylor expanded in z around 0 22.8%

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

4. Final simplification 34.7%

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

Alternative 11: 71.0% accurate, 9.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.2e-14) (* x (/ (+ -1.0 z) y_m)) (/ (+ x 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.2e-14) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.2d-14)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else
        tmp = (x + 4.0d0) / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.2e-14) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -5.2e-14:
		tmp = x * ((-1.0 + z) / y_m)
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5.2e-14)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	else
		tmp = Float64(Float64(x + 4.0) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -5.2e-14)
		tmp = x * ((-1.0 + z) / y_m);
	else
		tmp = (x + 4.0) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -5.2e-14], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999993e-14

   1. Initial program 84.3%

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

      1. fabs-sub 84.3%

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

      2. associate-*l/ 87.7%

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

      3. associate-*r/ 92.0%

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

      4. fma-neg 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. add-sqr-sqrt 56.0%

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

      2. fabs-sqr 56.0%

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

      3. add-sqr-sqrt 56.4%

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

      4. fma-undefine 53.3%

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

      5. associate-*r/ 48.9%

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

      6. associate-*l/ 47.1%

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

      7. div-inv 47.0%

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

      8. sub-neg 47.0%

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

      9. metadata-eval 47.0%

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

      10. distribute-neg-in 47.0%

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

      11. +-commutative 47.0%

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

      12. cancel-sign-sub-inv 47.0%

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

      13. div-inv 47.1%

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

      14. associate-*l/ 48.9%

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

      15. sub-div 53.6%

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

   6. Applied egg-rr 53.6%

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

   7. Taylor expanded in x around inf 53.6%

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

      1. associate-/l* 57.9%

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

      2. sub-neg 57.9%

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

      3. metadata-eval 57.9%

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

   9. Simplified 57.9%

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

   if -5.19999999999999993e-14 < x

   1. Initial program 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate-*l/ 93.9%

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

      3. associate-*r/ 94.4%

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

      4. fma-neg 95.4%

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

      5. distribute-neg-frac 95.4%

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

      6. +-commutative 95.4%

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

      7. distribute-neg-in 95.4%

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

      8. unsub-neg 95.4%

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

      9. metadata-eval 95.4%

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

   3. Simplified 95.4%

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

      1. fma-undefine 94.4%

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

      2. associate-*r/ 93.9%

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

      3. associate-*l/ 92.9%

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

      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 92.9%

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

      11. fabs-sub 92.9%

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

      12. add-sqr-sqrt 48.1%

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

      13. fabs-sqr 48.1%

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

      14. add-sqr-sqrt 49.1%

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

      15. associate-*l/ 49.7%

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

      16. sub-div 51.2%

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

   6. Applied egg-rr 51.2%

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

   7. Taylor expanded in z around 0 35.4%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.7% accurate, 11.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(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 84.0%

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

      1. fabs-sub 84.0%

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

      2. associate-*l/ 87.5%

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

      3. associate-*r/ 91.9%

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

      4. fma-neg 95.1%

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

      5. distribute-neg-frac 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. unsub-neg 95.1%

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

      9. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

      1. add-sqr-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. add-sqr-sqrt 57.3%

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

      4. fma-undefine 54.1%

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

      5. associate-*r/ 49.7%

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

      6. associate-*l/ 47.8%

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

      7. div-inv 47.8%

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

      8. sub-neg 47.8%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 47.8%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 47.8%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 47.8%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 47.8%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 47.8%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 49.7%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 54.4%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in z around 0 36.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 36.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 36.5%

         \[\leadsto \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 36.5%

         \[\leadsto \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 36.5%

         \[\leadsto \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 36.5%

         \[\leadsto \frac{\color{blue}{-4 - x}}{y} \]

   9. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   10. Taylor expanded in x around inf 36.5%

       \[\leadsto \frac{\color{blue}{-1 \cdot x}}{y} \]
   11. Step-by-step derivation

       1. neg-mul-1 36.5%

          \[\leadsto \frac{\color{blue}{-x}}{y} \]

   12. Simplified 36.5%

       \[\leadsto \frac{\color{blue}{-x}}{y} \]

   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/ 93.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 94.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 95.4%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 95.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 95.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 95.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 95.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 95.4%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 95.4%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 94.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 93.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      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 48.3%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 48.3%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 49.3%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 49.9%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 51.5%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 51.5%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 35.2%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.7% 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 92.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 92.5%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 95.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 94.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 95.1%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 95.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 95.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 95.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 95.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 95.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 94.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 95.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 92.5%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 92.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 92.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 92.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 92.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 92.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 92.5%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 92.5%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 92.5%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 48.1%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 48.1%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 49.0%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 51.2%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 52.3%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 86.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 86.4%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 84.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 93.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg 95.9%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 95.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 95.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 95.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 95.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 95.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 93.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 84.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 86.4%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 86.3%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 86.3%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 86.3%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 86.3%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 86.3%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 86.3%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 86.4%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 86.4%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 38.9%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 38.9%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 39.5%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. sub-neg 39.5%

          \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      16. distribute-rgt-neg-in 39.5%

          \[\leadsto \frac{x + 4}{y} + \color{blue}{\frac{x}{y} \cdot \left(-z\right)} \]

   6. Applied egg-rr 39.5%

      \[\leadsto \color{blue}{\frac{x + 4}{y} + \frac{x}{y} \cdot \left(-z\right)} \]

   7. Taylor expanded in x around -inf 46.0%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 46.0%

         \[\leadsto \color{blue}{-x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)} \]

      2. div-sub 46.0%

         \[\leadsto -x \cdot \color{blue}{\frac{z - 1}{y}} \]

      3. associate-/l* 39.6%

         \[\leadsto -\color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]

      4. sub-neg 39.6%

         \[\leadsto -\frac{x \cdot \color{blue}{\left(z + \left(-1\right)\right)}}{y} \]

      5. metadata-eval 39.6%

         \[\leadsto -\frac{x \cdot \left(z + \color{blue}{-1}\right)}{y} \]

      6. distribute-lft-in 39.6%

         \[\leadsto -\frac{\color{blue}{x \cdot z + x \cdot -1}}{y} \]

      7. *-commutative 39.6%

         \[\leadsto -\frac{x \cdot z + \color{blue}{-1 \cdot x}}{y} \]

      8. neg-mul-1 39.6%

         \[\leadsto -\frac{x \cdot z + \color{blue}{\left(-x\right)}}{y} \]

      9. remove-double-neg 39.6%

         \[\leadsto -\frac{\color{blue}{\left(-\left(-x \cdot z\right)\right)} + \left(-x\right)}{y} \]

      10. distribute-rgt-neg-in 39.6%

          \[\leadsto -\frac{\left(-\color{blue}{x \cdot \left(-z\right)}\right) + \left(-x\right)}{y} \]

      11. mul-1-neg 39.6%

          \[\leadsto -\frac{\left(-x \cdot \color{blue}{\left(-1 \cdot z\right)}\right) + \left(-x\right)}{y} \]

      12. distribute-lft-neg-in 39.6%

          \[\leadsto -\frac{\color{blue}{\left(-x\right) \cdot \left(-1 \cdot z\right)} + \left(-x\right)}{y} \]

      13. *-rgt-identity 39.6%

          \[\leadsto -\frac{\left(-x\right) \cdot \left(-1 \cdot z\right) + \color{blue}{\left(-x\right) \cdot 1}}{y} \]

      14. distribute-lft-in 39.6%

          \[\leadsto -\frac{\color{blue}{\left(-x\right) \cdot \left(-1 \cdot z + 1\right)}}{y} \]

      15. neg-mul-1 39.6%

          \[\leadsto -\frac{\color{blue}{\left(-1 \cdot x\right)} \cdot \left(-1 \cdot z + 1\right)}{y} \]

      16. +-commutative 39.6%

          \[\leadsto -\frac{\left(-1 \cdot x\right) \cdot \color{blue}{\left(1 + -1 \cdot z\right)}}{y} \]

      17. associate-*r* 39.6%

          \[\leadsto -\frac{\color{blue}{-1 \cdot \left(x \cdot \left(1 + -1 \cdot z\right)\right)}}{y} \]

      18. associate-*r/ 39.6%

          \[\leadsto -\color{blue}{-1 \cdot \frac{x \cdot \left(1 + -1 \cdot z\right)}{y}} \]

      19. mul-1-neg 39.6%

          \[\leadsto -\color{blue}{\left(-\frac{x \cdot \left(1 + -1 \cdot z\right)}{y}\right)} \]

      20. distribute-neg-frac2 39.6%

          \[\leadsto -\color{blue}{\frac{x \cdot \left(1 + -1 \cdot z\right)}{-y}} \]

   9. Simplified 46.0%

      \[\leadsto \color{blue}{x \cdot \frac{1 - z}{y}} \]

   10. Taylor expanded in z around 0 22.8%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.3% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ 4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return 4.0 / y_m;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = 4.0d0 / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return 4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return 4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = 4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 90.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 90.8%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 92.3%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 93.8%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fma-neg 95.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 95.3%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 93.8%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

   2. associate-*r/ 92.3%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

   3. associate-*l/ 90.8%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

   4. div-inv 90.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

   5. sub-neg 90.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

   6. metadata-eval 90.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

   7. distribute-neg-in 90.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

   8. +-commutative 90.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

   9. cancel-sign-sub-inv 90.7%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

   10. div-inv 90.8%

       \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

   11. fabs-sub 90.8%

       \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

   12. add-sqr-sqrt 45.4%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   13. fabs-sqr 45.4%

       \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   14. add-sqr-sqrt 46.3%

       \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   15. associate-*l/ 47.1%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   16. sub-div 49.1%

       \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

6. Applied egg-rr 49.1%

   \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

7. Taylor expanded in x around 0 20.8%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
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 90.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 90.8%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 92.3%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 93.8%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fma-neg 95.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 95.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 95.3%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 48.1%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

   2. fabs-sqr 48.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

   3. add-sqr-sqrt 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   4. fma-undefine 47.9%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

   5. associate-*r/ 46.5%

      \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

   6. associate-*l/ 45.8%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

   7. div-inv 45.8%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

   8. sub-neg 45.8%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

   9. metadata-eval 45.8%

      \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

   10. distribute-neg-in 45.8%

       \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

   11. +-commutative 45.8%

       \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

   12. cancel-sign-sub-inv 45.8%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

   13. div-inv 45.8%

       \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

   14. associate-*l/ 46.5%

       \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

   15. sub-div 48.4%

       \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

6. Applied egg-rr 48.4%

   \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

7. Taylor expanded in x around 0 17.6%

   \[\leadsto \color{blue}{\frac{-4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))